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QUANTUM AND BRAIDED LIE ALGEBRAS

arXiv:hep-th/9303148v1 26 Mar 1993DAMTP/93-4QUANTUM AND BRAIDED LIE ALGEBRASS. Majid1Department of Applied Mathematics& Theoretical PhysicsUniversity of CambridgeCambridge CB3 9EW, U.K.20 March 1993ABSTRACT We introduce the notion of a braided Lie algebra consisting of afinite-dimensional vector space L equipped with a bracket [ , ] : L ⊗L →L and aYang-Baxter operator Ψ : L ⊗L →L ⊗L obeying some axioms.

We show that suchan object has an enveloping braided-bialgebra U(L). We show that every genericR-matrix leads to such a braided Lie algebra with [ , ] given by structure constantscIJ K determined from R. In this case U(L) = B(R) the braided matrices introducedpreviously.

We also introduce the basic theory of these braided Lie algebras, in-cluding the natural right-regular action of a braided-Lie algebra L by braided vectorfields, the braided-Killing form and the quadratic Casimir associated to L. Theseconstructions recover the relevant notions for usual, colour and super-Lie algebrasas special cases. In addition, the standard quantum deformations Uq(g) are under-stood as the enveloping algebras of such underlying braided Lie algebras with [ , ]on L ⊂Uq(g) given by the quantum adjoint action.Contents1Introduction22Quantum Lie Algebras43Properties of the Braided-Adjoint Action124Braided-Lie Algebras and their Enveloping Algebras165Matrix Braided Lie algebras266Braided-Vector Fields347Braided Killing Form and the Quadratic Casimir421SERC Fellow and Fellow of Pembroke College, Cambridge1

1IntroductionMany authors have sought a description of Uq(g) and other quantum groups as generated, insome sense, by a finite-dimensional ‘quantum Lie algebra’ via some kind of enveloping algebraconstruction (just as U(g) is the universal enveloping algebra of the Lie algebra g). Such a notionwould be useful since one would only have to work with the finite-dimensional Lie algebra insteadof the whole quantum group.

It is also important for geometrical applications where we mightbe interested in quantum vector fields generated by the action not of general elements of thequantum group but by the action of the ‘Lie algebra’ elements.We are in a situation here where a new mathematical concept is needed: quantum groupsUq(g) have various interesting choices of generators but which ones should we look at, and whataxioms should they obey? One idea would be to attempt to build this on g itself but with somekind of deformed bracket obeying some kind of new axioms.

In [10, Sec. 2] we have initiated adifferent approach based on the subspace {l+Sl−} ⊂Uq(g) (where l± are the FRT generatorsof Uq(g)[3]).

This subspace is already well-known to be useful for certain kinds of computationsand we introduced on it some kind of ‘quantum Lie bracket’ [ , ] based on the quantum adjointaction and defined by structure constants cIJ K. This is recalled briefly in Section 2. Our goalin the present paper is to develop this further onto an axiomatic framework for this bracket.The natural bracket here does not obey of course the Jacobi identities, but rather we find thatit obeys naturally some kind of ‘braided-Jacobi identity’.

In this notion, which we introduce,the Yang-Baxter operator associated to the action of Uq(g) in the adjoint representation plays acentral role. Armed with suitable identities we show quite generally that brackets obeying themallow one to generate an entire enveloping algebra.The problem of defining some kind of braided-Lie algebra has been an open one for sometime.

The reason is that in a braided setting the Yang-Baxter operator or braided-transpositionΨ does not have square 1. As a result there is no action of the symmetric group and no notionof the Jacobi identity as 0 = [ξ, [η, ζ]]+cyclic.If we do suppose that Ψ2 = id then we arein the symmetric or unbraided situation as studied in [4][21] and elsewhere.

In this situationeverything goes through just as in the case of usual or super-Lie algebras. Unfortunately, thiscase is extremely similar to the usual or super case (because Ψ basically has eigenvalues ±1)2

so no really new phenomena are obtained. Moreover, it is too restrictive to deal with quantumgroups of interest, such as Uq(sl2).Our approach to the problem is the following.

In a series of papers we have introduced thenotion of braided group, see [16][17][12] and others. These are a generalization of quantum groupsin which the elements are allowed to have braid statistics.

This means that they live in a braidedtensor category where the tensor product ⊗is commutative only up to a braided-transpositionΨ. Most importantly for us now, we introduced the notion of a braided-cocommutative object ofthis type.

Only such braided-cocommutative objects could be truly expected to be some kind ofenveloping algebra. Thus we know the object which we wish to emerge as the enveloping algebraof some kind of braided Lie algebra.

By studying the properties of the braided-adjoint actionof such objects, we can then deduce the right properties of the braided-Lie algebra itself. Thesebraided groups and the necessary Jacobi-like properties of the braided-adjoint action form thetopic of Section 3.In Section 4 we take the properties of the braided-adjoint action formally as a set of axiomsfor a braided-Lie algebra.

Here we no longer assume that we are given a braided group butrather our main theorem is to show that such braided-Lie algebras indeed generate a braidedgroup or semigroup. By the latter we mean a bialgebra in a braided category without necessarilyan antipode.

Our main example is developed in Section 5 where we see that the quantum-Liealgebras of Section 2 fit naturally into this axiomatic framework. The construction works for ageneral R-matrix and in this case U(L) recovers the braided matrices B(R) introduced in [17].There they were introduced as a braided version of a quantum function algebra (like functionson Mn) but the same braided matrices arise as a braided enveloping bialgebra.

It is interestingthat only after further quotienting by determinant-type (and other relations) does one recoverprecisely Uq(g) in this way: the braided matrices seem to be a natural covering algebra ofthese objects and yet have properties like an enveloping algebra. We have already identifiedthe braided matrices covering Uq(sl2) as a form of Sklyanin algebra at degenerate parametervalue[10], which we understand now as U(L) where L is a braided-deformation of gl2.In a different direction we note that the action of such braided-Lie algebras should naturallybe some kind of braided-vector field.

We demonstrate this in Section 6 where we compute theright-regular action of the generators on Uq(g). This was announced in [8] and our goal here is3

to give the full details. These braided-vector fields are characterised by a matrix-Leibniz rule(ab)←−∂ij = a · Ψ(b ⊗←−∂ik)←−∂kjand are the left-invariant (and bicovariant) ‘vector fields’ generated by right-translations ofthe braided-Lie algebra generators on the braided group.

This can be contrasted with otherconstructions for differential operators on quantum groups.The main difference is that weabandon in our notion of braided-Lie algebras and braided-vector fields a commitment to theusual linear form of the Leibniz rule. This is tied to the linear coproduct ∆ξ = ξ ⊗1 + 1 ⊗ξ.In general for a quantum group there are few such primitive elements.

Instead, we work moregenerally and consider the notions as subordinate to a choice of (braided) coproduct ∆. Asidefrom the standard linear one, the matrix coproduct then suggests this matrix-notion of Liealgebras and vector fields.

For expressions that reduce in the classical limit to usual infinitesimalsone need only work with ∂ij −δij.Finally, in Section 7 we give another application where the notion of a natural finite-dimensional Lie algebra object is useful, namely to the definition of braided-Killing form. Thisis provided by the quantum or braided-trace in the adjoint representation of U(L) on L. Onethe generators uij −δij one recovers in the classical limit and for standard R-matrices the usualKilling form.

As an unusual phenomenon we find that the Killing form is made non-degenerateon gl2 = sl2 ⊕u(1) by the process of braided q-deformation. Moreover, our constructions workfor any bi-invertible R-matrix and we give formulae for the braided-Killing form gIJ in termsof it.

In the invertible case it can be used to raise and lower indices (i.e. to identify L andL∗) and is Ad-invariant and braided-symmetric in a suitable sense.

As an application of thebraided-Killing form we compute the corresponding quadratic CasimirC = uIuJgIJin this invertible case.2Quantum Lie AlgebrasThis section provides some motivation for the constructions of the paper from the point of viewof quantum groups. It is perfectly possible to proceed directly to the braided version in the4

next section and return only for some details needed for the examples in subsequent sections.Throughout the present section some familiarity with quantum groups is assumed. We workover a field k or (with care) a commutative ring (the reader can keep in mind C or C[[ℏ]]) anduse the usual notations and methods for a quasitriangular Hopf algebra (H, ∆, ǫ, S, R).

HereH is a unital algebra, ∆: H →H ⊗H is the coproduct, ǫ : H →k the counit, S : H →Hthe antipode and (for a strict quantum group) R ∈H ⊗H is the quasitriangular structure orso-called ‘universal R-matrix’. It obeys the axioms of Drinfeld[2],(∆⊗id)(R) = R13R23,(id ⊗∆)(R) = R13R12,Xh(2) ⊗h(1) = R(∆h)R−1.

(1)For an introduction one can see [15]. Here and below we use the Sweedler notation[24] ∆h =P h(1) ⊗h(2) for the coproduct.The problem which we consider is the following.

It is well-known that the standard quantum-groups Oq(G) of function algebra type can be obtained by an R-matrix method as quotientsof the quantum matrices A(R) by determinant and other relations[3]. On the other hand, theknown treatments of the quantum enveloping algebras to which these are dual, are quite abit different.

There is the approach of Drinfeld and Jimbo in terms of the roots[2][5] and anapproach in [3] with twice as many generators l± which have to be cut down somehow (usuallyby means of some imaginative ansatz). These l± are roughly speaking the matrix elements ofthe fundamental and conjugate-fundamental representation of A(R).

Here we recall a somewhatdifferent approach based on the quantum Killing form and a single braided-matrix of generatorsu = (uij) and developed in [10].Just as Lie algebras like sln can be defined both via root systems and via matrices, so we givein this way a matrix approach to the standard quantum enveloping algebras. At the same timea remarkable correspondence principle or self-duality emerges between Oq(G) as a quotient ofquantum matrices A(R) and Uq(g) as a corresponding quotient of the braided matrices B(R).

Inthe present section we shall try to give a self-contained picture of this using well-known quantumgroup formulae without too much direct dependence on the theory of braided groups.Let R in Mn ⊗Mn be an invertible matrix solution of the quantum Yang-Baxter equations(QYBE) R12R13R23 = R23R13R12.We recall that A(R) denotes the matrix bialgebra withgenerators t = (tij) and relations Rt1t2 = t2t1R in the usual compact notation (where the5

numerical suffices refer to the position in a matrix tensor product). We recall also that B(R)denotes the quadratic algebra with n2 generators {uij} and 1, and relationsRkaibubcRcjadudl = ukaRabicucdRdjbli.e.R21u1R12u2 = u2R21u1R12.

(2)These relations have been known for some time to be convenient for describing Uq(g) but theyhave been studied formally as a quadratic algebra for the first time in [17]. We will come to thebraided aspect[17] in Section 5.

For now we just work with B(R) as a quadratic algebra.Proposition 2.1 The algebra B(R) is dual to A(R) in the following sense. Let (H, R) be aquasitriangular bialgebra which is dually paired by < , > with A(R) such that < t1 ⊗t2, R >= R.Letl = (t ⊗id)(Q),Q = R21R12.Here lij are elements of H. Then there is an algebra map B(R) →H such that l is the imageof u, i.e.

H is a realization of B(R).ProofThis is motivated by ideas for Uq(g) implicit in the literature, see [20][19]. The newpart in our approach however, is to formulate the result at the level of bialgebras.

Both A(R)and B(R) are quadratic algebras and no antipode is needed. This approach arises out of thetransmutation theory of braided groups that related A(R) to B(R) in [13][12], where we showthe useful identityl1R12l2 = R12(t1t2 ⊗id)(Q).

(3)To be self-contained we can also give a direct proof of this easily enough asl1R12l2=XR21 < t1, Q(1) >< t1 ⊗t2, R >< t2, Q′(1) > Q(2)Q′(2)=X< t1, R(2)R′(1)R′′′′(1) >< t2, R′′′′(2)R′′(2)R′′′(1) > R(1)R′(2)R′′(1)R′′′(2)=XR(1)R′′(1)R′(2)R′′′(2) < t1, R(2)R′′′′(1)R′(1) >< t2, R′′(2)R′′′′(2)R′′′(1) >=X< t1, R(2)(2)R′′′′(1)R′(1)(1) >< t2, R(2)(1)R′′′′(2)R′(2) > R(1)R′(2)=X< t1, R′′′′(1)R(2)(1)R′(1)(1) >< t2, R′′′′(2)R(2)(2)R′(2) > R(1)R′(2)=XR12 < t1t2, R(2) >< t1t2, R′(1) > R(1)R′(2) = R12 < t1t2 ⊗id, Q(1) > Q(2)where R′, R′′ etc denote further copies of R = P R(1) ⊗R(2).For the second equality werecognised the matrix form of the coproduct of the t as paired to multiplication in H. For the6

third equality we used the QYBE for R. For the fourth and fifth we used the axioms (1) directly.We then wrote the expressions as products in A(R) for the sixth equality and recognized theresult.By permuting the matrix position labels we have equally well l2R21l1 = R21(t2t1 ⊗id)(Q).HenceR21l1R12l2 = R21R12(t1t2 ⊗id)(Q) = (t1t2 ⊗id)(Q)R21R12 = l2R21l1R12using the relations Rt1t2 = t2t1R repeatedly. ⊔⊓In this sense then, B(R) is some kind of universal dual algebra to A(R).

Just as A(R) hasto be cut down by determinant and other relations to obtain an honest Hopf algebra, likewiseif H is a Hopf algebra then B(R) is generally a little too big to coincide with H: it too hasto be cut down by additional relations. Note that in this case where H is a Hopf algebra theelementary identity R−1 = (S ⊗id)(R) means that l = l+Sl−relating this description of H tothe FRT approach in [3].

For the next proposition we concentrate on those standard quantumgroups Uq(g) which can be put in this FRT form (this includes the deformations of at least thenon-exceptional semisimple Lie algebras).Proposition 2.2 [10] Let H = Uq(g) be of FRT form[3] with associated R-matrix R and duallypaired with A(R). Then the map B(R) →Uq(g) has kernel given by ‘braided versions’ of thedeterminant and other relations associated to the Lie group G. Hence Uq(g) can be identified asB(R) modulo such relations.ProofThe argument in [10] is as a non-trivial corollary of the process of transmutation[13].This turns the matrix bialgebra A(R) into the braided matrix B(R) and also turns the quotientquantum groups Oq(G) into their braided versions Bq(G).

This is done in a categorical way (byshifting categories) and transmutes at the same time all constructions such as quantum planesetc on which these objects act. Hence (by these rather general arguments) Bq(G) is obtained ina braided version of the way that Oq(G) is obtained.

On the other hand, there is also a braidedversion BUq(g) of Uq(g) coinciding as an algebra. Unlike the untransmuted theory the quantumKilling form Q : Bq(G) →BUq(g) is not just a linear map but a map of braided Hopf algebras.For the standard deformations of semisimple Lie algebras it is even an isomorphism.

This is the7

general reason for the correspondence between Oq(G) and Uq(g) as quotients of matrices. Fora truly self-contained picture one can of course verify the proposition directly by computing indetail the required quotients of B(R).

For example, for Uq(sl2) on must divide BSLq(2) by thebraided determinant ab −q2cb = 1[17]. ⊔⊓Note that the additional relations needed to obtain a Hopf algebra like Uq(g) in this way fromB(R) are such that there exists a braided antipode S with uSu = 1 = (Su)u.

This exhibits theremarkable similarity with what is done to obtain a quantum group from A(R), but with onecatch: the matrix coproduct ∆u = u ⊗u does not give a bialgebra in the usual sense (it is not analgebra homomorphism to the usual tensor product). This explains why for a full appreciationof this approach one must understand B(R) correctly as a bialgebra with braid-statistics[17].Even without such a full picture, Proposition 2.2 does however, provide a quick way ofcomputing Uq(g) as well as the corresponding quantum enveloping algebra in a general non-standard but factorizable case.

Namely, compute the quadratic algebra B(R) and then imposefurther determinant-type and other relations. Factorizable means here by definition that themap from the relevant dual of H to H given by evaluation against the first factor of Q = R21R12is a surjection[20].

This ensures in Proposition 2.1 that for such H the l are generators. Note alsothat the existence of a quasitriangular Hopf algebra dually paired to A(R) is not possible for allR.

A necessary condition is thet R has a second inverse eR =< t1 ⊗t2, (id ⊗S)R >= ((Rt2)−1)t2(where t2 is transposition in the second matrix factor).For the moment we can just note then that Uq(g) and in general any quasitriangular Hopfalgebra dual to A(R) has a subspaceL = span < lij >⊂H(4)Proposition 2.3 [10]i) In the factorizable case[20], the subspace L = span < lij > and 1 generate all of H. ii)The subspace L is stable under the quantum adjoint action of H on itself.iii) The quantum adjoint action as a map [ , ] : L ⊗L →L looks explicitly like[lI, lJ] = cIJ KlK,cIJ K = eRai1j0bR−1bk0i0cRk1ncmRmanj1where lI = li0i1 and I = (i0, i1) is a multi-index notation (running from (1, 1), · · · , (n, n)).8

ProofAgain, the proof in [10] is based on the theory of transmutation in [13][12]. The linearspace of B(R) can be identified with that of A(R) with the generators u = t identified (but nottheir products as we have seen above).

Then (ii) is automatic because t transforms to a linearcombination under the quantum coadjoint action, hence so does u under the quantum adjointaction. To be self-contained we can also give a direct proof using more familiar methods asfollows.

(i) In the present setting this is (as we have mentioned) more or less by the definition offactorizable. In our usage this notion is subordinate to the choice of a bialgebra or Hopf algebradually paired with H in the sense of [15].

Here the choice is A(R) or its quotients such as Oq(G). (ii) We use the form l = l+Sl−valid in the Hopf algebra case and let ⊲= Ad denote thequantum adjoint action of H on itself given by h⊲b = P h(1)bSh(2).

We show that[14]l+2 ⊲l1 = R−1l1R,l−1 ⊲l2 = Rl2R−1(5)using the definition of Ad, elementary properties of the antipode and the fact that l± obey therelations R−1l±1 l±2 = l±2 l±1 R−1 and R−1l−1 l+2 = l+2 l−1 R−1 as in [3] (These relations are not tied tothe standard Uq(g) as in [3] if one uses the general formulation as in [15][14]). Thusl+2 ⊲(l+1 Sl−1 )= l+2 l+1 Sl−1 Sl+2 = l+2 l+1 S(l+2 l−1 ) = l+2 l+1 R−1S(Rl+2 l−1 )= R−1l+1 l+2 S(l−1 l+2 )R = R−1l+1 (l+2 Sl+2 )Sl−1 R = R−1l+1 Sl−1 Rl−1 ⊲(l+2 Sl−2 )= l−1 l+2 Sl−2 Sl−1 = l−1 l+2 S(l−1 l−2 ) = l−1 l+2 RS(R−1l−1 l−2 )= Rl+2 l−1 S(l−2 l−1 )R−1 = Rl+2 (l−1 Sl−1 )Sl−2 R−1 = Rl+2 Sl−2 R−1.

(iii) We can also deduce from this the action of Sl± from l+Sl+ = id = (Sl+)l+ etc (theidentity matrix times the action of the identity). In particular,(Sl−ij)⊲lkl = eRajkmlmnRianl.

(6)where eR obeys eRiablRajkb = δijδkl = Riabl eRajkb. Combining this with (5) we can computeli0i1⊲lj0j1 = l+i0a⊲((Sl−ai1)⊲lj0j1 to find the result stated.

⊔⊓Thus L is some kind of ‘quantum Lie algebra’ for H because it is a finite-dimensional subspacethat generates H and at the same time is closed under the quantum adjoint action, which9

provides a kind of ‘quantum Lie bracket’ [ξ, η] = ξ⊲η. We have introduced this point of viewin [10] and pointed out that this bracket obeys a number of Lie-algebra-like identities inheritedfrom the standard properties of the quantum adjoint action, such as(L0)[ξ, η] ∈Lforξ, η ∈L(L1)[ξ, [η, ζ]] = P[[ξ(1), η], [ξ(2), ζ]](L1′)[[ξ, η], ζ] = P[ξ(1), [η, [Sξ(2), ζ]]]The second of these is just the statement that Ad is a covariant action of the Hopf algebraon itself, while the third follows from the definition of Ad.

We see here two problems with thisapproach. Firstly, these properties (L1) and (L1′) cannot be taken as abstract properties ofsome kind of Lie algebra structure on L because they involve the coproduct and this does notin general act on L in a simple way (its just gives some subspace of H ⊗H).

Secondly, theyhold for any Hopf algebra and so do not express the fact that quantum groups such as Uq(g) areclose to being cocommutative. The usual Lie bracket has properties inherited from the fact thatU(g) is cocommutative, and our quantum Lie algebra, to be convincing, should deform some ofthese.

For later reference,Proposition 2.4 If H is a cocommutative Hopf algebra then the usual Hopf algebra adjointaction [ , ] = Ad obeys in addition to the identities above, the identities(L2)P ξ(2) ⊗[ξ(1), η] = P ξ(1) ⊗[ξ(2), η](L3)P[ξ, η](1) ⊗[ξ, η](2) = P[ξ(1), η(1)] ⊗[ξ(2), η(2)]for all ξ, η in H.Proof(L2) needs no comment except to say that we have written the cocommutativity inthis way because later we shall adopt something like this without assuming that the Hopfalgebra is completely cocommutative.This weak notion of cocommutativity (as relative tosomething on which the Hopf algebra acts) is useful in other contexts also. For (L3) we have∆[ξ, η] = P ξ(1)(1)η(1)Sξ(2)(2) ⊗ξ(2)(1)η(2)Sξ(2)(1) using that S is an anticoalgebra map.

In thecocommutative case the numbering of the suffices does not matter so we have at once the righthand side of (L3). For the record we give here also the proof of (L1).

This holds for any Hopfalgebra and readsX[[ξ(1), η], [ξ(2), ζ]]=X(ξ(1)(1)ηSξ(1)(2))(1)(ξ(2)(1)ζSξ(2)(2))S(ξ(1)(1)ηSξ(1)(2))(2)10

=Xξ(1)η(1)(Sξ(4))ξ(5)ζ(Sξ(6))(S2ξ(3))(Sη(2))(Sξ(2))=Xξ(1)η(1)ζ(Sξ(4))(S2ξ(3))(Sη(2))(Sξ(2)) = ξ(1)η(1)ζ(Sη(2))(Sξ(2)) = [ξ, [η, ζ]]expanding out the definitions, the properties of the antipode and the Sweedler notation [24]to renumber the suffices to base 10 (keeping the order). The third and fourth equalities thensuccessively collapse using the axioms of an antipode.

⊔⊓Another aspect of our matrix approach, which is not a problem but a convention is thatour chosen finite-dimensional subspace L is a mixture of ‘group-like’ elements with coproduct∆ξ ∼ξ ⊗ξ and more usual Lie-algebra-like elements where ∆ξ ∼ξ ⊗1 + 1 ⊗ξ (with a suitabledeformation). The latter are how off-diagonal elements of lij tend to behave, while the formerare how diagonal elements tend to behave.

Another good convention is to take as ‘quantum Liealgebra’ the subspaceX = span < χij >⊂H,χij = lij −δij(7)This is a matter of taste and is entirely equivalent. The subspace is also closed under [ , ] = Adwhich now has structure constants[χI, χJ] = [lI, lJ] + δIδJ −δIδJ −δIlJ = (cIJ K −δIδJ K)χK(8)using the elementary properties of the quantum adjoint action (notably lI⊲1 = ǫ(lI) = δI).

HereδI = δi0i1 and δJ K are Kronecker delta-functions. The last equality uses thatcI J KδK = δIδJ(9)which follows at once from the expression for c in the proposition.

These χI equally generate Halong with 1 and have a better-behaved semi-classical limit in the standard cases. This aspectof our approach has been stressed in [22].

It is also quite natural from the point of view ofbicovariant differential calculus as explained in [23]. We note also that some combinations ofthe basis elements of L or X can have trivial quantum Lie bracket and have to be decoupled ifwe want to have the minimum number of generators.Finally, to complete the picture of B(R) as a some kind of dual of A(R) we have an elementarylemma which we will need later in Section 6.11

Lemma 2.5 The generators of A(R) define matrix elements of the representation ρ of B(R)given byρ2(u1) =< u1, t2 >= Q12,Q = R21R12ProofWe have to show that this extends consistently to all of B(R) as an algebra represen-tation,ρ3(R21u1R12u2)=R21ρ3(u1)R12ρ3(u2)=R21Q13R12Q23 = Q23R21Q13R12=ρ3(u2)R21ρ3(u1)R12 = ρ3(u2R21u1R12).The middle equality follows from repeated use of the QYBE. Thus the extension is consistentwith the algebra relations of B(R).

⊔⊓One can also see over C that if R obeys a certain reality condition then B(R) is a ∗-algebrawith uij∗= uji. We call this the hermitian real form of the braided matrices B(R).

At the levelof the standard Uq(g) with real q the corresponding lij∗= ljj recovers the standard compactreal form of the these Hopf algebras. These remarks confirm that the braided-matrix approachto Uq(g) is quite natural.3Properties of the Braided-Adjoint ActionIn this section we recall some basic facts about Hopf algebras in braided categories (braidedHopf-algebras) and their braided adjoint action.

It is these categorical constructions that leadto the notion of braided-Lie algebras in the next section. The idea is that we know what is abraided group[9] (or in physical terms a group-like object with braid statistics[16]) and we justhave to infinitesimalize this notion.One of the novel aspects of braided groups is that results fully analogous to those familiarin algebra or group theory are proven now using braid and knot diagrams.

This is because wework in a braided or quasitensor category.This means (C, ⊗, 1, Φ, Ψ) where C is a category(a collection of objects and allowed morphisms or maps between them), ⊗is a tensor productbetween two objects, with 1 a unit object for the tensor product. The isomorphisms ΦV,W,Z :V ⊗(W ⊗Z) →(V ⊗W) ⊗Z express associativity and say that we can forgot about brackets12

=BBS.∆B B B BB B B B..∆∆=.∆=BBS.∆BBεηBBε=B=BBBε∆∆=BB=BBηη..=ε.B B B BεεFigure 1: Axioms of a Braided Hopf Algebra(all tensor products can be put into a canonical form in a consistent way). Finally, there is abraided-transposition or quasisymmetry ΨV,W : V ⊗W →W ⊗V saying that the tensor productis commutative up to this isomorphism.

The difference between this setting and the standardone for symmetric monoidal categories[7] is that we do not suppose that ΨW,V ◦ΨV,W = id. Putanother way, we distinguish carefully between ΨV,W and (ΨW,V )−1 which are both morphismsV ⊗W →W ⊗V for any two objects.

To avoid confusion a good notation here is to write themorphisms not in the usual way as single arrows, but downward as braid crossings,V WW VΨW,VΨ-1==V WW VV,WThese braided-transpositions do however obey other obvious properties of usual transpositionsuch as ΨV ⊗W,Z = ΨV,Z◦ΨW,Z and similarly for ΨV,W ⊗Z. These ensure that different sequencesof braided transpositions that connect two composite objects coincide if the corresponding braidsin the notation above coincide.

Also, these isomorphisms are functorial in that they are compat-ible with any morphisms between objects. If we write any morphisms also pointing downwardsas notes with input lines and output lines, then the functoriality says that we can pull suchnodes through braid crossings much as beads on a string.This describes the diagrammatic notation that we shall use.For a formal treatment ofbraided categories see [6] and for on introduction to our methods see [8, Sec.3].In thisnotation, the axioms of a Hopf algebra in a braided category are recalled in Figure 1.

They arelike a usual Hopf algebra B except that the product, coproduct ∆, antipode S, unit η and counitǫ are all morphisms in the braided category. In the diagrammatic notation we write the unitobject as the empty set.

The first axiom shown (the bialgebra axiom) is the most important:it says that the braided coproduct B →B⊗B is an algebra homomorphism where B⊗B is the13

======SSSSSSSSSSSSBB B BS∆B B BB B BBB.SSSSB B BB B BBBS2BB B BB B BBS∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆∆.......................Figure 2: Proof that the braided adjoint action obeys the braided Jacobi identitybraided tensor product algebra structure on B ⊗B. This is like a super-tensor product andinvolves transposition by Ψ.

The two factors B in B⊗B do not commute but instead enjoybraid statistics given by Ψ.The reader can keep in mind the trivially braided group (the ordinary group Hopf algebra)B = CG with ∆g = g ⊗g and Sg = g−1. The antipode axiom says that if we split g into g, g,apply S to one factor and then multiply up, we get something trivial.

The diagrams on the rightin Figure 1 just say this abstractly as morphisms.It is remarkable that such objects defined in this way really behave like usual groups orquantum groups. For example, the usual adjoint action of a group on itself consists in takingg, a, splitting g to give g, g, a, applying S to give g, g−1, a, transposing g−1 past the a, and thenmultiplying up.

When written as diagrams or morphisms in our braided category, this is thebraided adjoint action. It is shown in the box in Figure 2.Figure 2 itself is the diagrammatic proof of the main result of this section.

It shows thatapplying the braided adjoint action twice as on the right in Figure 2, is the same as the lefthand expression. This consists in applying the tensor product braided adjoint action of B on14

ηB B= >ηηBBε=(a)(b)B B B=B B BAdAdAdAd∆BBAd(c)B B=B BB BB BB B=B BB BAdAdAdAdAd∆∆∆∆∆(d)B B BBB B BB=.AdAdAd=BBBBB B BB=AdAdAd.∆BB B B.BBAdAdFigure 3: Summary of Properties of the Braided Adjoint Action (a) an action (b) a mod-ule-algebra under the action (c) the braided Jacobi identity (d) compatibility with the coproductimplied by the assumption of braided-cocommutativity with respect to AdB ⊗B and then applying the adjoint action again to the result (all together three applicationsof the braided adjoint action on the left in Figure 2). We call this the braided-Jacobi identity.The proof reads as follows.

Starting on the left, use the bialgebra axiom that ∆is an algebrahomomorphism to expand the expression on the left. For the second equality we use the fact thatS is a braided anti-coalgebra map ∆◦S = (S ⊗S)Ψ◦∆[18].

We then identify (the dotted line) aclosed loop which will after reorganization or the branches using associativity and coassociativitycancel according to the antipode axioms in Figure 1. We make this cancellation for the thirdequality.

We then use that S is a braided anti-algebra map for the fourth equality and identifyanother loop. This cancels in a similar way to the antipode loop giving the fifth equality.

Thefinal equality is the easier fact already proven in [12] that the braided adjoint action is indeedan action. We summarise this along with some other known properties.Proposition 3.1 Let B be a Hopf algebra in a braided or quasitensor category and let Ad =·2 ◦(id ⊗ΨB,B)(id ⊗S ⊗id)(∆⊗id) denote the braided adjoint action as above.It is (a) anaction of B on itself and (b) respects its own product (a braided module algebra) as in Figure 3.Further (c) it obeys the braided Jacobi identity.

Finally, if B is cocommutative with respect toAd in the sense of [16] (as shown) then (d) holds.ProofWe have spelled out the proofs of (a),(b) and (d) in [12] in a dual form with comodules15

and coactions (for the braided adjoint coaction). We ask the reader to turn the diagrammaticproofs for these in [12] up-side down (a 180 rotation) and read them again.

They read exactlyas the required proof for the Ad action. This is part of the self-duality of the axioms of a Hopfalgebra.

For the new part (c) we have given the proof above. ⊔⊓Note that for a usual non-cocommutative Hopf algebra the quantum adjoint action does notrespect the coproduct in the sense of (c) above.

One needs a cocommutativity condition. Theidea in [16] was not to try to define this intrinsically (the naive notion does not work well) butin a week form as cocommutative with respect to a module.

This is the form that we have used:we suppose that B is cocommutative in this weak sense. This corresponds to directly assuming(L2) in Proposition 2.4.

This is then enough to derive (d) which corresponds to (L3) in thatproposition.To this extent then, the kind of Hopf algebras in braided categories that we consider are trulylike groups or enveloping algebras in the sense that they are supposed braided-cocommutative atleast with respect to their own braided adjoint action. This completes our review of the braidedadjoint action and the derivation of the identities that we will need in the next section.

We willtake them as the defining properties of a braided-Lie algebra.4Braided-Lie Algebras and their Enveloping AlgebrasWe have seen that if we do have a braided group as in the last section then the braided-Adjointaction obeys some Lie-algebra like identities as in the second line in Figure 3. If the braidedgroup has some generating subobject which is closed under Ad then these identities hold forit also.

Motivated by this, we are going to adopt these as abstract axioms for a braided Liealgebra and prove a theorem in the converse direction. Thus every such braided Lie algebra willhave (at least in a category with direct sums) an enveloping braided-bialgebra returning us tosomething like the the kind of braided group we might have began with.

One surprise will bethat the enveloping algebra here seems more naturally to be a bialgebra (in a braided sense)rather than a Hopf algebra with antipode. Of course one can add further conditions to forcea braided-antipode but they do not appear to be very natural from the point of view of theunderlying braided Lie algebra.16

εεε=L L LL L L∆[ , ][ , ][ , ][ , ]L[ , ]LL LL L[ , ]=L LL L[ , ]∆[ , ]L LL L=∆∆L LL L[ , ]∆∆L L=L L[ , ][ , ](L1)(L2)(L3)Figure 4: Axioms of a Braided Lie Algebra (a) Braided-Jacobi identity axiom (b) Cocommuta-tivity axiom (c) Coalgebra compatibility axiomDefinition 4.1 A braided Lie algebra is (L, ∆, ǫ, [ , ]) where L is an object in a braided orquasitensor category, ∆: L →L ⊗L and ǫ : L →1 are morphisms forming a coalgebra in thecategory, and [ , ] : L ⊗L →L is a morphism obeying the conditions (L1),(L2),(L3) in Figure 4.The idea of introducing a coalgebra here is one of the novel aspects of the approach. In theusual definition of a Lie algebra a coalgebra structure ∆ξ = ξ ⊗1 + 1 ⊗ξ and ǫξ = 0 is implicit.We do not want to be tied to a specific form such as this and hence bring the implicit ∆to theforeground as part of the axiomatic structure.

The only requirements of a coalgebra are(∆⊗id) ◦∆= (id ⊗∆) ◦∆,(ǫ ⊗id) ◦∆= id = (id ⊗ǫ) ◦∆(10)as usual.There is no bialgebra axiom here because after all L is not being required to have an asso-ciative product. It is typically some finite-dimensional vector space.

Instead axiom (L1) saysthat L is being equipped with some kind of Lie bracket [ , ]. This braided-Jacobi identity is aform of associativity.

If one imagines momentarily the usual linear form for ∆then the left handside of (L1) has two terms then we have something like the usual Jacobi identity as discussedin Section 2. We of course do not suppose this (we do not even suppose that L has an elementthat can be called 1).

We do however suppose that ∆is braided-cocommutative with respect tothis Lie bracket [ , ] in the sense of (L2) and that ∆respects it in the sense of (L3). This (L3)is a Lie form of the bialgebra condition in Figure 1.Proposition 4.2 Let (L, ∆, ǫ, [ , ]) be a braided Lie algebra in an Abelian braided tensor category(we suppose that we have direct sums with the usual properties).

Then there is a braided bialgebra17

L LL L..∆[ , ]=LLUUFigure 5: Defining relations of the braided enveloping algebra U(L)U(L) in the sense of Section 3, generated by 1 and L with relations as shown in Figure 5. Wecall it the universal enveloping algebra of the braided Lie algebra.ProofFormally U(L) is the free tensor algebra generated by L modulo these relations withcoproduct given by ∆extended to products as a braided bialgebra.

We have to show that thisextension is compatible with the relations of U(L). This is shown in Figure 6.

The first equalityis the definition of how ∆extends to products. The second assumes the relations in U(L).The third is coassociativity and functoriality.

The fourth uses the cocommutativity axiom (L2)applied in reverse. The fifth uses functoriality and coassociativity again to reorganise.

The sixthequality is (L3). The result the coincides with the extension of ∆to products when the relationsof U(L) are used first.

The proof to higher order proceeds similarly by induction. The proofthat ǫ also extends to a counit on U(L) is equally straightforward.

⊔⊓The motivation here is as follows. In any Hopf algebra one has the identity P[ξ(1), η]ξ(2) =P ξ(1)(1)η(Sξ(1)(2))ξ(2) = ξη.

For example for the usual U(g) with linear coproduct this is [ξ, η]+ηξ = ξη as expected. We have a similar definition but without any specific form of coalgebra,and of course in the braided setting.

We conclude with some general properties of these braidedenveloping algebras U(L). Following the usual ideas about Lie algebras representations we haveDefinition 4.3 A representation of a braided Lie algebra (L, ∆, ǫ, [ , ]) is an object V andmorphism α : L ⊗V →V such that the polarised form of the braided-Jacobi identity (L1) holds.This is shown in Figure 7 (a).

We say that L is cocommutative with respect to V if the polarisedform of the cocommutativity axiom (L2) holds. This is shown in part (b).One can tensor product representations of a braided Lie algebra (using the coproduct ∆)just as for braided Hopf algebras.

The class O(L) of representations with respect to which L is18

LULULULULULULULULULULULULULULULUL L.∆=======L LL LL LL LL LL L[ , ][ , ]∆∆∆∆..L L..∆∆[ , ][ , ].∆∆∆[ , ][ , ]∆∆∆.....∆∆∆∆∆∆[ , ][ , ]∆∆[ , ][ , ]...∆Figure 6: Proof that ∆extends to U(L) as a braided bialgebra=∆[ , ]=∆∆(a)(b)L LL LLLLLVVVVVVVVααααααFigure 7: Definition (a) of representation of a braided Lie algebra and (b) cocommutativity withrespect to it19

∆[ , ]==[ , ]=∆. (a)(b)∆.=======..∆∆.∆∆∆∆∆∆∆∆∆.......∆∆.L LL LL LL LL LL LL LL LL LL LL LL LLLLLLLLLααααααααααααααααααVVVVVVVVVVVVVVVVVVVVVVVVFigure 8: Proof that (a) a representation on V extends from L to U(L) and (b) cocommutativityalso extendscocommutative is also closed under tensor product and braided with braiding given by Ψ. Thefacts are just as for the representation theory of braided Hopf algebra or bialgebra[18].

Thediagrammatic proofs are similar. Alternatively, these facts follow from the following propositionthat connects representations of L to those of U(L) for which the bialgebra theory alreadydeveloped applies.Proposition 4.4 Every representation (α, V ) of a braided Lie algebra L extends a representa-tion of U(L) on V .

If L is cocommutative with respect to V in the sense of (L2) then U(L) iscocommutative with respect to V in a similar sense (as in [16]).ProofThis is shown in Figure 8. Part (a) verifies that the relations of U(L) are representedcorrectly.

We define the action of U(L) by the repeated application of the Lie algebra action asshown. The representation axiom in Definition 4.3 ensures that this coincides with the action ofU(L) if its relations are used first.

Part (b) verifies that the resulting action is cocommutativeif the representation is cocommutative. We show it on elements of U(L) with are products of L.20

L L∆[ , ][ , ][ , ][ , ]LL L=====(a)===(b)L LLL L.∆[ , ]L LLL L∆[ , ][ , ]L LLL L∆∆[ , ][ , ][ , ]L LLL L∆∆[ , ][ , ][ , ][ , ]∆L LLL L..∆∆∆[ , ][ , ]L LLL L.∆[ , ][ , ]∆[ , ][ , ].L LLL L[ , ][ , ][ , ]L LLL LL L∆∆[ , ][ , ][ , ][ , ][ , ]LL L∆∆[ , ][ , ][ , ]..L LLL L[ , ][ , ][ , ]∆.L LLL L==Figure 9: Proof that (a) property (L1) and (b) property (L3) extend to representation [ , ] ofU(L) on LThe proof proceeds similarly by induction to all orders. The first equality uses Proposition 4.2that U(L) is a bialgebra.

The second equality is functoriality to pull one of the products intothe position shown, and that α is a representation for the other product. The third equalityis functoriality again to pull one of the α’s up to the right.

We then use the cocommutativityassumption for the fourth equality, and then again for the fifth. We then use that α is an actionand the bialgebra property of U(L) in reverse.

⊔⊓An important example is of course provided by [ , ] itself. It was the model for the definitionsand is clearly a representation and L is cocommutative with respect to it.

We call it the adjointrepresentation of L on itself. By the last proposition then, it extends to a representation (alsodenoted [ , ]) of U(L) on L with respect to which U(L) is cocommutative.Lemma 4.5 The adjoint representation [ , ] of U(L) on L defined via Proposition 4.4 obeys anextended form of the braided-Jacobi identity (L1) and the coalgebra compatibility property (L3)in which the left-most input L in Figure 4 is extended to U(L).ProofThis is shown in Figure 9.

Part (a) verifies the extended braided-Jacobi identity onelements of U(L) which are products of L.The first equality uses that U(L) is a braided21

bialgebra from Proposition 4.2. The second that [ , ] is a representation of U(L) as obtainedfrom Proposition 4.4.

We then successively use the braided Jacobi identity axiom (L1) twice.The final equality uses again that [ , ] is an action. Exactly the same proof holds with theelements in U(L) is a higher order composite element, provided only that the result has beenproved already at lower orders so that we can use it for the third and fourth equalities.

Hencethe result is proven to all orders by induction. Part (b) is proved in a similar way.

We verify(L3) extended to products in its first input. The first equality is that [ , ] is a representation.The second and third successively use (L3).

The fourth then uses that [ , ] is an action and thefifth that U(L) is a bialgebra. The proof extends to all orders by induction.

⊔⊓Proposition 4.6 The adjoint representation [ , ] of U(L) on L defined via Proposition 4.4extends to a representation on U(L) itself as a braided module algebra. We call it the adjointaction of U(L) itself.

U(L) remains braided-cocommutative with respect to this action.ProofThe proof is indicated in Figure 10.We show in part (a) that the representationconstructed in the previous proposition extends consistently as a braided-module algebra. Thefirst equality is the definition of the extension in this way.

The second uses the relations inU(L), the third that U(L) acts cocommutatively on L from part (b) of the last proposition. Thefourth is axiom (L3).

The fifth equality is a reorganization using coassociativity and functorialityand the sixth is the cocommutativity again. The seventh requires the preceding lemma that theextended [ , ] continues to obey a braided-Jacobi identity as in (L1) but with the first L replacedby U(L).

Assuming this we see that the result is the same as first using the relations in U(L)and then extending [ , ] as a braided module algebra. This proves the result when acting onproducts or two L. The proof on higher products proceeds by induction.

Note that in doingthis we have to prove Lemma 4.4 again with the second input of (L1) now also extended toproducts. The proof of this is similar to the strategy here (namely consider composites) andneeds the module algebra property of [ , ] as just proven in Figure 10.

Thus the induction hereproceeds hand in hand with this extension of Lemma 4.5.Part (b) contains the proof that the resulting action of U(L) remains cocommutative onproducts. The first equality is functoriality while the second is the module-algebra property justproven.

The third and then the fourth each use the cocommutativity of the U(L) action from22

LUL LLULULULULULULULULULULUL LLULULULULULUL LLUL LLULU∆. [ , ]LUL LLULU[ , ][ , ]∆.LUL LLULULUL LLULULUL LLULULUL LLULU=(a)===L L[ , ][ , ]∆..[ , ]∆∆∆[ , ][ , ][ , ][ , ].=====L L[ , ][ , ]∆.L L.[ , ][ , ]∆∆[ , ]L L.[ , ][ , ][ , ]∆∆[ , ][ , ].∆∆[ , ]L L∆[ , ][ , ][ , ].∆∆L L[ , ]∆∆∆[ , ][ , ].

[ , ]======(b)UL LLULLU[ , ].∆∆. [ , ]∆[ , ][ , ].∆∆[ , ][ , ].

[ , ]∆∆[ , ]. [ , ]Figure 10: Proof that [ , ] extends to a cocommutative action of U(L) on itself23

the preceding proposition. Coassociativity is expressed by combining branches into multiplenodes (keeping the order).

The fifth equality uses cocommutativity one more time. Finally weuse the module algebra property again to obtain the result.

Again the proof on higher productsproceeds in the same way by induction, this time hand in hand with the extension of the property(L3) in Lemma 4.5 to U(L) in its second input. This is proven by the same strategy and usesbraided-commutativity of the action of U(L) on products of a lower order.

⊔⊓In the course of the last proof (and using similar techniques) we see that the braided Jacobiidentity and the coalgebra compatibility property also extend from L to U(L). In short, allthe properties of Ad summarized in Figure 3 hold for this extended [ , ].

We remark that if∆on U(L) happens to have an antipode making U(L) into a braided Hopf algebra then theaction [ , ] indeed coincides with the braided-adjoint action Ad. This follows easily from thedefinitions.

On the other hand, for a general coproduct such as the matrix example in the nextsection, there is no reason for U(L) to be a braided Hopf algebra. It is remarkable that [ , ]nevertheless plays the role of the adjoint action even in this case.

Further properties of thesebraided enveloping algebras can be developed using similar techniques to those above.Finally, we note that that L1 = 1 ⊕L ⊂U(L) is also a coalgebra and closed under thebracket [ , ] extended as in Proposition 4.6. Of course the enveloping algebra for this unitalcoalgebra L1 should be defined without adding another copy of 1.

Otherwise the construction isjust the same as above. Moreover, it may be that another choice of decomposition of this unitalcoalgebra L1 is possible.

For example L1 = 1 ⊕X where X is a subobject of the form∆χ = χ ⊗1 + 1 ⊗χ + ∆1χ,ǫχ = 0,∆1 : X →X ⊗X(11)for χ ∈X in concrete cases, and like L is closed under [ , ]. This is expressed in our categoryby diagrams as in Figure 11 part (a).In the other direction if ∆1 is a morphism which iscoassociative (we do not require it to have a counit) then (11) defines a coalgebra structure with∆1 = 1 ⊗1 and ǫ1 = 1 in the concrete case.

Some L1 of interest below will be of this form andin this case U(L) can be regarded as generated just as well by X as U(X ). From this pointof view a braided Lie algebra of this type is determined by (X , ∆1, [ , ]) in a braided categoryobeying axioms obtained by putting (11) into Figure 4.

We use that [ , ] extends U(L) as abraided-module algebra. The resulting form of (L1) and (L2) is shown in Figure 11 and (L3) is24

X XXU∆1. [ , ]X XXU[ , ]X XXU.X XXU.=-+=X X XX∆1[ , ][ , ][ , ][ , ][ , ]X X XX(L1)[ , ]X XX X∆1∆1X XX X [ , ]X XX XX XX XX X XX[ , ][ , ]X X XX[ , ][ , ]ε1 X1 X1X XX∆1-=-(L2)(b)(a)++ηη+== 0∆XXXXX1 X+Figure 11: For coalgebras of the form (a) on X ⊂1 ⊕L the axioms (L1) and (L2) (and also asimilar (L3)) of a braided-Lie algebra in terms of (X, ∆1, [ , ]) look more familiar.

The braidedenveloping algebra in terms of X has relations (b)obtained in just the same way. In each case nothing is gained by working in this form (there arejust two extra terms) and this is why we have developed the theory with (L, ∆, [ , ]).

On theother hand the extra terms bring out the sense in which these generators precisely generalise theusual notion of Lie algebra, with a ‘braided-correction’ ∆1. Apart from this we see that (L1)becomes the obvious Jacobi identity in a familiar form.

The enveloping algebra as generatednow by the (X , ∆1, [ , ]) is also of the obvious Ψ-commutator form with this ∆1 correction.Note that from (L2) in Figure 11 we see that ∆1 ̸= 0 if we are to obey this braided-cocommutativity axiom, unless it happens that Ψ2 = id.Thus, our notion of braided-Liealgebra in terms of (X , ∆1, [ , ]) reduces to precisely the usual notion of Lie-algebra with threeterms in the Jacobi identity etc, only if the category is symmetric and not truly braided. In thetruly braided case there is no advantage to considering the X and we may as well work with the‘group-like’ generators L.25

5Matrix Braided Lie algebrasThe constructions in the last two sections have been rather abstract (and can be phrased evenmore formally). In this section we want to show how they look in a concrete case where thecategory is generated by a matrix solution of the QYBE and ∆has a matrix form.Firstly, let us recall that our notion of braided Lie algebra is subordinate to a choice ofcoalgebra structure on L. Whatever form we fix determines how the axioms look in concreteterms for braided Lie algebras of that type.

It need not be the usual implicit linear form. Thussuppose that L is a vector space with basis {uI} say and fix a coalgebra structure ∆, ǫ on it.These are determined in the basis by tensors∆uI = ∆IJKuJ ⊗uK,ǫuI = δI,∆IAL∆AJK = ∆IJA∆AKL,∆IAJδA = δI J = ∆IJAδI(12)where δI J is the Kronecker delta function.

The underlines on ∆and ǫ are to remind is that theseare not an ordinary Hopf algebra coproduct and counit. Repeated indices are to be summed asusual.

These are obviously the coassociativity and counity axioms in tensor form.With this chosen coalgebra in the background, the content of Definition 4.1 in this basis isas follows.Proposition 5.1 Let L be a vector space with a basis {uI} and coalgebra ∆IJK, δI. Then abraided-Lie algebra on L is determined by tensors R = RIJ KL and cIJ K such that R is aninvertible solution of the QYBE and the following three sets of identities hold(L0a)δARJ AIB = δIBδJ and δBRJ AIB = δIδJ A(L0b)∆IMN RKAN BRALM J = RKLIA ∆AJB and ∆KMN RM AIB RN LBJ = RKBIJ ∆BAL(L0c)RKM J B RM LIA cABN = cIJ A RKLAN and RIAKM RJBM L cABN = cIJ A RAN KL(L1)∆KP Q RIAQB cP AM cBJ N cMN L = cIJ A cKAL(L2)∆IP Q RJ AQB cP AM RBLM K = ∆ILB cBJ K(L3)cIJ A ∆AKL = ∆IMN ∆J P Q RP AN BcBQL cMAK and cIJ KδK = δIδJ.In this case the corresponding braided-Lie algebra structure isΨ(uI ⊗uK) = RKLIJuL ⊗uJ,[uI, uJ] = cIJ KuK.26

εεεεε=εεIJKIJKIJKJIJIJIJIJIJIJIε(L2)(L3)(L1)========IJIJIJ=IKMNJLIKJLIKJJIKIJKKJIIJKABLLNAMABNLMANLLLAMNQPKKKKKLLLLNPQMAMBPQNNLL(L0)M==IδIδI=IJJIJ= ∆IJKIJ K= cIJKIKJRLKLKJ IABBABAABBAABBAABABABFigure 12: Tensor version of braided Lie algebra axioms is obtained by assigning indices to arcsand tensors as shownThe enveloping bialgebra of L is generated by the relationsuIuK = ∆IAM RKBM L cABJ uJuL.ProofWe are simply writing the axioms of a braided-Lie algebra as in Definition 4.1 in ourbasis. To do this is is convenient to write all operations as tensors, as we have done already for∆.

To read offthe tensor equations simply assign labels to all arcs of the diagram, assign tensorsas shown in Figure 12 and sum over repeated indices. These can be called braided-Feynmandiagrams or braided-Penrose diagrams according to popular terminology.

It is nothing otherthan our diagrammatic notation in a basis. The group (L0) are the morphism properties arisingfrom the fact that ∆, ǫ, [ , ] are morphisms in the category and the braiding is functorial withrespect to them, and have been used freely in preceding sections.

In the converse direction,given such matrices, one has to check that they define a braided Lie algebra. The category in27

which this lives is the braided category of left A-comodules where (in the present conventions)A is a quotient of the dual-quasitriangular bialgebra A(R). It is in a certain sense the categorygenerated by R and the braiding is R on the vector space L and extended as a braiding toproducts.

The morphism properties ensure that the relevant maps are morphisms (intertwinersfor the coaction). The other properties needed are (L1)-(L3) which clearly hold in our basis ifthe tensor equations hold.

Likewise we read-offthe relations for the enveloping bialgebra fromFigure 5. ⊔⊓To give some concrete examples we now take ∆and ǫ to be of matrix form.

Thus we workwith vector spaces of dimension n2 and let {ui0i1} denote our basis. Here I = (i0, i1) is regardedas a multiindex.

We fix∆ui0i1 = ui0a ⊗uai1,ǫui0i1 = δi0i1,i.e.,∆IJK = δi0j0δj1k0δk1i1, δI = δi0i1. (13)Braided Lie algebras defined with respect to this implicit coalgebra can naturally be calledmatrix braided Lie algebras.Proposition 5.2 Let R ∈Mn ⊗Mn be a bi-invertible solution of the QYBE (so both R−1 andeR = ((Rt2)−1)t2 exist).

ThenRKLIJ = Ri0adl0R−1aj0l1bRj1cbk1 ˜Rci1k0d,cIJ K = eRai1j0bR−1bk0i0cRk1ncmRmanj1obey the conditions in the preceding proposition and hence define a matrix braided Lie algebra(L, Ψ, [ , ]). Its braided enveloping bialgebra is the braided-matrices bialgebra introduced in [17],U(L) = B(R)with matrix coalgebra ∆u = u ⊗u, ǫu = id.ProofIn fact, most of the work for this was done in [17] where we proved that B(R) wasa braided bialgebra.

Apart for an abstract proof (by transmutation from A(R)) we also gavea direct proof in which we verified directly the relevant identities. This includes most of theabove, and the rest as similar.

The matrix R with components RKLIJ was denoted ΨKLIJ in[17] to avoid confusion with the initial Rijkl, while the matrix Q in [17] is basically our cIJ K.The relations of the enveloping algebra areuIuK = c(i0,a)BJRKB(a,i1)L uJuL = R−1dj0i0aRj1bal0Rl1cbk1 eRci1k0d uJuL28

by multiplying out and canceling some inverses. This is the matrix Ψ′ in [17] and defines therelations of B(R).

One can move two of the R′s to the left hand side for the more compact formin Section 2. ⊔⊓Thus the quantum-Lie algebras in Section 2 are successfully axiomatized but only as braided-Lie algebras.

This is therefore the structure that generates quantum enveloping algebras suchas Uq(g). For such standard R-matrices which are deformations of the identity matrix, a moreappropriate choice of generators of U(L) is χI = uI −δI.

It is standard in the theory of non-commutative differential calculus to take for the ‘infinitesimals’ elements such that ǫ = 0, andthis is what the shift to these generators achieves. This works fairly generally as follows.Proposition 5.3 Let L be a braided-Lie algebra in tensor form as in Proposition 5.1 and UL)its braided enveloping algebra with bracket extended to U(L) as in Proposition 4.6.

Then thesubspace X = span{χI} ⊂U(L) where χI = uI −δI, is closed under the braiding and bracketwith structure constantsΨ(χI ⊗χK) = RKLIJχL ⊗χJ,[χI, χJ] = (cIJ K −δIδJ K)χKand has coalgebra∆χI = χI ⊗1 + 1 ⊗χI + ∆IJKχJ ⊗χK,ǫχI = 0.ProofFor the braiding we use the morphism properties (L0a) for the counit, to computeΨ(χI ⊗χK) noting that in its extension to U(L) as a braiding, the braiding of 1 with anythingis trivial (the usual permutation). For the coproduct ∆we use the counity property in (12) andthat ∆1 = 1 ⊗1 in U(L).

For the bracket we note that the extension in Proposition 4.6 is as abraided-module algebra. In particular, [ , 1] = ǫ and [1, ] = id so that we can compute it on theχI.

⊔⊓This subspace X equally well generates U(L) along with 1, but in general it is not any moreconvenient to work than L because the coproduct just has two extra terms and the same terminvolving ∆IJK. For example in our matrix setting (13) we have∆χ = χ ⊗1 + 1 ⊗χ + χ ⊗χ29

where the χI = χi0i1 are regarded as a matrix. This not better to work with than our matrixform on u.

It is however, useful in the following case.Corollary 5.4 Let (L, Ψ, [ , ]) be the braided-Lie algebra in Proposition 5.2 corresponding to amatrix solution R ∈Mn ⊗Mn of the QYBE, taken in the form generated by X in Proposition 5.3with its inherited bracket and braiding. If R is triangular in the sense R21R12 = 1 then Ψ is asymmetry and the braided-Lie bracket vanishes,Ψ2 = id,[χI, χJ] = 0.Moreover, the enveloping algebra U(L) in this case is Ψ-commutative in the sense · ◦Ψ = ·.Suppose now that R is not triangular but a deformation R = R0 + O(ℏ) of a triangularsolution R0.

If f IJK is the semiclassical part of the bracket according to[χI, χJ] = ℏf IJKχK + O(ℏ2)say on these generators and if we rescale to ¯χI = ℏ−1χI then[¯χI, ¯χJ] = f IJK ¯χK + O(ℏ),∆¯χI = ¯χI ⊗1 + 1 ⊗¯χI + O(ℏ)and f IJK obeys the usual axioms of a Ψ-Lie algebra where Ψ = Ψ(R0) is the symmetry (thisincludes usual, super and colour Lie-algebras etc).ProofFor the first part we have already pointed out in [17] that in the construction of B(R)the braiding is symmetric if R is triangular and cIJ K is trivial in the sense cIJ K = δIδJ K. In anycase these facts follow easily from the explicit forms of Ψ, c given in Proposition 5.2. Note that in[17] this was interpreted as B(R) being like the Ψ-commutative bialgebra of functions on a ‘space’(like a super-space), while in the present case we put these observations into Proposition 5.3 withthe interpretation of B(R) as the enveloping algebra of a Ψ-commutative Ψ-Lie algebra.

For thesecond part it is clear from the description of the braided-Jacobi identity and other axioms inSection 4 for the form of the coproduct in Proposition 5.3 that the semiclassical term f IJK obeysprecisely the obvious notion of an R0-Lie algebra (where R0 is triangular, as studied for examplein [4][21]). If R0 = id we have the usual braiding Ψ to lowest order and hence an ordinary Liealgebra.

Another triangular solution is (RS)ijkl = δijδkl(−1)p(i)p(k) where p(i) = i −1 and its30

deformations in the above framework have super-Lie algebras as their semiclassical structure.⊔⊓Our formalism is not at all limited to deformations of triangular solutions of the QYBE,so the matrix braided-Lie algebras in Proposition 5.2 may not resemble usual Lie algebras orsuper-Lie algebras or their usual generalizations. But in the case when R is a deformation ofa triangular solution then they will be deformations of such usual ideas for generalising Liealgebras when one looks at the generators X.We conclude with two of the simplest matrix examples, namely for the initial Rijkl given byRgl2 =q00001q −q−100010000q,Rgl1|1 =q00001q −q−100010000−q−1.Here the rows label (i, k) and the columns (j, l).

We denote the matrix generators asu = abcdand compute from Proposition 5.2. We assume q2 ̸= 1, 0.

The corresponding braidings Ψ andbraided enveloping algebras B(R) have already been computed in [17] to which we refer fordetails of these.Example 5.5 cf[10]. Let R = Rgl2 be the standard GLq(2) R-matrix associated to the Jonesknot polynomial.

A convenient basis for the corresponding braided-Lie algebra L is γ = q−2a +d, ξ = d −a, b, c and the non-zero braided Lie-brackets are[ξ, ξ] = (q2 + 1)(q2 −1)2ξ,[γ, γ] = (q−2 + 1)γ,[b, c] = (q2 −1)q−2ξ = −[c, b][ξ, b] = (q−2 + 1)(q2 −1)b = −q2[b, ξ],[ξ, c] = −(q−2 + 1)(q2 −1)q−2c = −q−2[c, ξ],[γ, ξ] = (q6 + 1)q−4ξ,[γ, b] = (q6 + 1)q−4b,[γ, c] = (q6 + 1)q−4c.A convenient basis of X is ξ, b, c and γ −ǫ(γ) which we rescale by a uniform factor (q2 −1)−1to a basis ¯ξ,¯b, ¯c, ¯γ. Then the braided-Lie algebra takes the form[¯ξ,¯b] = (q−2 + 1)¯b = −q2[¯b, ¯ξ],[¯ξ, ¯c] = −(q−2 + 1)q−2¯c = −q−2[¯c, ¯ξ],[¯b, ¯c] = q−2¯ξ = −[¯c,¯b][¯ξ, ¯ξ] = (q4 −1)¯ξ,[¯γ, ¯ξ] = (1 −q−4)ξ,[¯γ,¯b] = (1 −q−4)¯b,[¯γ, ¯c] = (1 −q−4)¯c.31

with zero for the remaining six brackets. As q →1 the braiding Ψ becomes the usual transpositionand the space X with its bracket becomes the Lie algebra sl2 ⊕u(1).

The bosonic generator ¯γ ofthe U(1) decouples completely in this limit.ProofThis is from the definition on Proposition 5.2. It is similar to the computation of theaction of Uq(sl2) on for the degenerate Sklyanin algebra in [10].

We computed B(R) in [13][17]and already noted the importance of the element d −a = ξ, and that the element q−2a + d = γas bosonic and central in B(R). It is remarkable that its braided Lie bracket is not entirely zeroeven though the action of Uq(sl2) on it is trivial.

The shift to the barred variables follows thegeneral theory explained above since R here is a deformation of a triangular solution (namelythe identity). To compute the brackets [¯γ, ] we note that ǫ(γ) = (q−2 + 1) and that the bracketobeys [1, ] = id and [ , 1] = ǫ.

Hence[γ −ǫ(γ), b] = [γ, b] −(q−2 + 1)[1, b] = ((q6 + 1)q−4 −(q−2 + 1))b = (q2 −1)(1 −q−4)b[γ −ǫ(γ), γ −ǫ(γ)] = [γ, γ] −(q−2 + 1)[1, γ] = 0etc. The other computations are similar.

The braiding Ψ and the structure of the envelopingalgebra are in [17] ⊔⊓Note that braided enveloping bialgebra U(L) in terms of these rescaled generators must inthe limit q →1 tend to U(gl2). It can be called BUq(gl2) because it is a braided object.

Wehave identified it in [10] as the degenerate Sklyanin algebra. On the other hand this same B(R)in terms of the original generators u tends to the commutative algebra algebra generated bythe co-ordinate functions on the space of matrices M2 which was our original point of view in[13][17].

Thus for generic q we can think of the braided bialgebra U(L) = B(R) from either ofthese points of view. The same applies in the next example where we took the view in [17] thatB(R) tends as q →1 to the super-bialgebra of super-matrices M1|1.

This time, after rescalingit becomes in the limit the super-enveloping algebra U(gl1|1).Example 5.6 Let R = Rgl1|1 be non-standard R-matrix associated to the Alexander-Conwayknot polynomial. A convenient basis for the corresponding braided-Lie algebra L is a, ξ = d −a, b, c and the non-zero braided-Lie brackets are[b, c] = −(q2 −1)ξ = q2[c, b],[b, a] = (q2 −1)q−2b,[c, a] = −(q2 −1)c32

[ξ, a] = −(q2 −1)2q−2ξ,[a, ξ] = ξ,[a, a] = a,[a, b] = q−2b,[a, c] = q2cA convenient basis for X is a−1, b, c, ξ which we rescale by a uniform factor (q2 −1)−1 to obtaina basis ¯a,¯b, ¯c, ¯ξ. Then the braided-Lie algebra takes the form[¯b, ¯c] = −¯ξ = q2[¯c,¯b],[¯ξ, ¯a] = (q−2 −1)¯ξ,[¯a,¯b] = −q−2¯b = −[¯b, ¯a],[¯a, ¯c] = ¯c = −[¯c, ¯a]with zero for the remaining nine brackets.

As q →1 the braiding Ψ is such that X becomes asuper-vector space with ¯a, ¯ξ even degree (bosonic) and ¯b, ¯c odd degree (fermionic), and its bracketbecomes that for the super-Lie algebra gl1|1.ProofThis is by direct computation from Proposition 5.2. The enveloping algebra B(R) wasstudied in [17] where we identified the element ξ = d −a as bosonic and central.

The passageto the barred variables follows the same steps as the previous example. The braiding Ψ and thestructure of the enveloping algebra are in [17].

⊔⊓This example tends as q →1 to a super-Lie algebra, as it must from the general theorydescribed above. This is because R tends to the matrix RS which is the critical limit point forsuper-Lie algebras.

The corresponding braiding Ψ for this is the usual super-transpositions. Itis a triangular solution of the QYBE and all its deformations lead by the above to super-Liealgebras.In this way we see that our general R-matrix construction for braided algebras unifies thenotions of Lie algebras and super-Lie algebras, colour-Lie algebras etc., into a single framework.These usual notions are the semiclassical part of the structure as we approach a certain subset(the triangular solutions) in the moduli space of all solutions of the quantum Yang-Baxterequations.

On the other hand we are not at all tied in principle to such usual deformations.For example if we consider our braided-Lie algebras at other points R in the moduli space itis natural to call the corresponding semi-classical structures R-Lie algebras. They control thedeformations of B(R) (the enveloping algebra at R).

One possible application may be thatby solving some kind of R-classical Yang-Baxter equation for general R (based on an R-Liealgebra) one should be able to exponentiate to paths in the moduli space. Moreover, the usualquantum groups are precisely quotients of such enveloping algebras so we have the possibility ofconnecting them by paths in the moduli space.

This is a problem for further work.33

6Braided-Vector FieldsIn this section we show that the braided enveloping algebras U(L) act quite naturally as braided-vector fields on braided-function algebras. We have already seen one example namely the bracket[ , ] consisting of one copy of U(L) acting on another.

In the construction of Proposition 5.2the braided enveloping algebra can also be thought of as the braided-matrix function algebraand we do so for the copy of U(L) which is acted upon. The vector-fields in this case are (in abraided-group quotient) those induced by the adjoint action.

In this section we give by contrastvector fields corresponding to the right action on functions induced by left-multiplication in thegroup (the right regular representation).In the case of usual matrix groups recall that these vector fields are literally given by matrixmultiplication of the Lie algebra elements realised as matrices on the group elements. Thus, ifuij are the matrix co-ordinate functions on the matrix group in the defining representation ρ, ga group element and ξ a Lie algebra element, we have(uij⊳ξ)(g) = uij(ξg) = ρ(ξ)ikukj(g).Our constructions in this section give in the matrix case of Proposition 5.2 precisely a q-deformation of this situation.

We realise our matrix braided-Lie algebras concretely as matricesacting by a deformation of matrix multiplication. This is in marked contrast to usual quantumgroups, but mirrors well the situation for super groups and super-matrices and their super-Liealgebras.Our strategy to obtain this result is to go back to the abstract situation where we have abraided Hopf algebra B in a braided category, formulate the construction there and afterwardscompute its matrix form.

Because the relevant braided matrices and braided groups that concernus are related (in the nice cases) to quantum groups by a process of transmutation, we obtainon the way vector-fields on quantum groups also.The general construction of the regular representation proceeds in our categorical setting inSection 3 along the same lines as the braided-adjoint action. Namely, one writes the usual groupor Hopf algebra construction in diagrammatic form.

Note that the coproduct of B encodes thegroup multiplication law if B is like the algebra of functions on a group. The evaluation of thisagainst an element of the dual B∗is then like the action of the enveloping or group algebra in34

SS SSSSSS====(a)(b)∆S.BB B B*∆∆∆∆∆∆.....∆.====∆∆B B* B*BB B* B*B∆B B* B*∆S∆SB∆S.B B* B*B∆S∆BB B* B*∆S∆SB B B*BBB B B*.BB B B*BB B B*Figure 13: The braided right action of B∗on B is shown in the box. It is (a) a right action and(b) a right braided module algebrathe regular representation.

This gives the following construction.Proposition 6.1 Let B be a Hopf algebra in a braided category as in Section 3 and supposethat it has a dual B∗. Then B∗acts on B from the right as depicted in the box in Figure 13.Moreover, the action respects the product on B in the sense that B-becomes a B∗-module algebra.We call this the right-regular action.ProofHere B∗assumes that our category is equipped with dual objects (in this case left duals)and the cup and cap denote evaluation ev : B∗⊗B →1 and coevaluation coev : 1 →B ⊗B∗respectively.

They obey a natural compatibility(ev ⊗id)(id ⊗coev) = id,(id ⊗ev)(coev ⊗id) = id(14)which in diagrammatic form says that certain horizontal double-bends can be pulled straight.The unusual ingredient in the right action is the braided antipode S which converts a left actionto a right action and is needed in the strictly braided case for the module algebra property towork out without getting tangled. The proof that this is an action is in part (a).

The firstequality is the definition of the product in B∗in terms of the coproduct in B. In terms of mapsthis is equivalent to the characterizationev ◦(id ⊗ev ⊗id) ◦(id ⊗∆B) = ev ◦(·B∗⊗id).

(15)35

The second equality is the double-bend cancellation property of left duals. We then use that thefact that the braided-antipode is a braided anti-coalgebra map and functoriality to recognize theresult.

That this makes B a braided right module algebra is shown in part (b). The first equalityis the bialgebra axiom, the second is the fact that the braided-antipode is a braided-antialgebramap, the third functoriality and the fourth the definition of the coproduct in B∗in terms of theproduct in B.

This is determined in a similar way to (15) via pairing by ev. An introduction tothe methods is in [8].

⊔⊓Now let H be an ordinary quasitriangular Hopf algebra dually paired as as in Section 2 withsuitable dual A. There are associated braided groups B(H, H) and B(A, A) corresponding tothese by transmutation[16][13].

They can both be viewed in the braided category of H-modulesand as such B(H, H) = B(A, A)∗at least in the finite dimensional case. We can therefore applythe above diagrammatic construction and compute the action of B(H, H) on B(A, A).Theresulting formulae can also be used with care even in the infinite dimensional case.Proposition 6.2 The canonical right-action of B(H, H) on B(A, A) (the braided group of func-tion algebra type) comes out asa⊳b =X< R(2)⊲b, a(1) > a(2) < R(1), a(3) >,a ∈B(A, A), b ∈B(H, H).This makes B(A, A) into a right braided B(H, H)-module algebra in the category of left H-modules.ProofWe compute from the formulae for B(H, H) in [16] using standard Hopf algebra tech-niques.

Its product is that same as that of H and it lives in the stated category by the quantumadjoint action ⊲. We need the explicit formulae∆b =Xb(1)(SR(2)) ⊗R(1)⊲b(2),Sb = u−1(SR(2))(Sb)R(1)for the braided coproduct and braided-comultiplication, where u = P(SR(2))R(1) implementsthe square of the antipode.

Finally, B(A, A) has the same coproduct as A, transforms underthe quantum coadjoint action and is dually paired by the map B(H, H) →B(A, A)∗given byb 7→< Sb, >. Armed with these explicit formulae we compute the box in Figure 13 asa⊳b=X< S(R(2)⊲Sb), R(1)(1)⊲a(1) > R(1)(2)⊲a(2)36

=X< (SR(1)(1))(S(R(2)⊲Sb))R(1)(2), a(1) > a(3) < SR(1)(3), a(2) >< R(1)(4), a(4) >=X< (SR(1))S(R(2)R′(2)⊲Sb), a(1) > a(2) < R′(1), a(3) >=X< (SR(1))S(R(2)⊲S(R′(2)⊲b)), a(1) > a(2) < R′(1), a(3) >=X< R′(2)⊲b, a(1) > a(2) < R′(1), a(3) > .Here the first equality follows from the form of ∆and of the braiding Ψ in the category ofH-modules (it is given by the action of R followed by usual permutation). The second equalityputs the coadjoint action as an adjoint action on the other side of the pairing in one case, andcomputes it in the other case.

The third equality writes the coproduct in A as a product in Hand cancels using the antipode axioms. We also used the axioms of a quasitriangular structure(1).

The fourth uses that S is a morphism in the category (an intertwiner). Finally we use forthe last equality the definition of S in the reverse formX(R(2)⊲Sb)R(1) = u−1(Sb)u = S−1b,∀b ∈B(H, H)easily obtained from the formula above.

We apply this to the element R′(2)⊲b.From the general categorical construction above, we know that this right action has all theproperties of a braided-module algebra. One can (in principle) verify some of these directly.

Forexample, that ⊳as stated is a morphism in the category meansh⊲(a⊳b) =X(h(1)⊲a)⊳(h(2)⊲b),∀h ∈H(16)which can be verified directly using the standard properties of quasitriangular Hopf algebras ascan that ⊳is indeed an action. The module algebra property is more difficult to see directly.

⊔⊓In the infinite-dimensional case we take here the category of A-comodules and write R as adual-quasitriangular structure A ⊗A →C. For H we can then take for example Uq(g) in FRTform.

The braided-version B(H, H) has isomorphic algebra and coincides in this factorizablecase to a quotient of U(L) for the corresponding braided Lie algebra L. For A we can takethe quantum function algebra Oq(G) and as seen in [13][17] its corresponding braided versionB(A, A) is a quotient of B(R). In this case we can compute the action in Proposition 6.2 asuij⊳lkl=< (SR(2))⊲lkl, tia > uab < SR(1), tbj >37

=< Sl−bj⊲lkl, tia > uab =< eRajkmlmnRbanl, tia > uab= uab eRmjknQnpiaRbmplusing the notations in Section 2. We used (6) and the definition of l−in terms of the quasitrian-gular structure R. Moreover, we know that the construction the covariant under a backgroundcopy of Uq(g) in the sense of (16) with action as in (5 on u.

Clearly the same constructions applyfor any R which is sufficiently nice that we have a factorizable quantum group in the picture.On the other hand, we are now ready to verify directly that this whole construction lifts to thebialgebra level. It is quite natural at the level of braided-Lie algebras.Proposition 6.3 Let R be a bi-invertible solution of the QYBE as in Proposition 5.2 and Lthe braided Lie algebra introduced there.

Let B(R) be the braided-matrix bialgebra. Then Lacts from the right on the algebra of B(R) by braided-automorphisms (B(R) is a right-braidedmodule algebra for the action of (L, ∆)).

We write ⊳ui0i1 = ←−∂i0i1 = ←−∂I for the correspondingoperators. Thenui0i1←−∂j0j1 = uk0k1 eRci1j0bQbai0k0Rk1caj1and the extension is according to the braided-Leibniz rule(ab)←−∂i0i1 = a · Ψ(b ⊗←−∂i0k)←−∂ki1,∀a, b ∈B(R).ProofWe no longer need a quantum group, but if there is one it remains a backgroundcovariance of the system as above.

For our direct verification it is convenient to write the actioncompactly asu1R12←−∂2 = Q21u1R12 = ρ1(u2)u1R12(17)where ρ is the fundamental representation of U(L) defined in Lemma 2.5. From this it is clearthat the operators ←−∂are truly a representation of U(L) as required, and hence also of L in thesense of Definition 4.3.

Next we need to check that the extension of this action to products asa right-braided module algebra,(u1R−123 u2R23)←−∂3 = (u1←−∂3)(R−123 u2R23←−∂3)(18)etc, respects the relations of B(R). In proving this it is convenient to insert some R-matricesand prove compatibility with the relations in an equivalent form.

Thus,(R21R−113 u1R13R12R−123 u2R23)←−∂3 = (R−123 R−113 R21u1R12u2R13R23)←−∂338

= (R−123 R−113 u2R21u1R12R13R23)←−∂3 = (R−123 u2R23R21R−113 u1R13)←−∂3R12= (R−123 u2R23R21←−∂3)(R−113 u1R13←−∂3)R12 = R32R31u2R21u1R23R13R12.Here the first equality is a few applications of the QYBE, the second the relations in B(R) andthe third the QYBE again (this combination is the relations of B(R) transformed under l+⊲asin Section 2). The fourth equality is our supposed extension according to (18).

We compute thederivatives from (17) and use the QYBE for the fifth. On the other hand if we begin from thesame starting point and use (17) we have(R21R−113 u1R13R12←−∂3)(R−123 u2R23←−∂3) = R32R31R21u1R12u2R13R23which gives the same result as above using the relations in B(R).

From this it follows thatthese relations are compatible with the action of L. The direct computation with tensor indices(rather than the compact notation) is also possible. ⊔⊓This is the natural right action of B(R) regarded as a braided enveloping algebra U(L)on itself regarded as a braided function algebra.

Just as in Corollary 5.4, it is trivial if R istriangular. It is natural in this case to define the action of the infinitesimal generators χI.

Thisis ⊳χI =←−∂I −←−δI =←−DI say, and from (17) it is clear that it vanishes if R is triangular.Corollary 6.4 If R is a solution of the QYBE such that R = R0+O(ℏ) where R0 is a triangularsolution, then←−DI = O(ℏ) and the action of the rescaled generators ⊳¯χI = ℏ−1←−DI =←−¯DI is a usualΨ-derivation. Here Ψ is from Proposition 5.2 with R = R0 and is a symmetry.ProofAs in Figure 11, we compute the form of the right-module algebra property in Figure 13for the form of ∆on the ¯χ.

Explicitly,(ab)←−¯DI = a(b←−¯DI) + aΨ(b ⊗←−¯DI) + ℏaΨ(b ⊗←−¯D (i0,k))←−¯D (k,i1). (19)The last ∆1 term enters at order ℏas does the deformation of the braiding.

Hence to lowestorder the ℏ−1←−DI obey the usual axioms of a right-vector field in a symmetric monoidal category.⊔⊓Recall that it is these rescaled generators that behave like usual Lie algebras or super-Liealgebras etc to lowest order as we approach the critical variety of triangular solutions of the39

QYBE. We see that in this case it is exactly these that act on the braided matrices B(R) inthis corollary.

Here B(R) itself becomes in the triangular limit the Ψ-commutative algebra offunctions on some kind of matrix space. Moreover, these constructions work at the braided-group level so the underlying space here can be regarded as some kind of group-manifold in thesense of a supergroup or ordinary group etc.Example 6.5 For Rgl2 as in Example 5.5 the matrix-braided vector fields are←−∂11=q200(q −q−1)20q20000100001,←−∂12=001 −q−20000q2 −100000000←−∂21=0−(1 −q−2)2000000q2 −100(q −q−1)201 −q−200,←−∂22=q2 + q−2 −100−(1 −q−2)20q2 + q−2 −10000q20000q2.From this we obtain the action of the rescaled generators ¯χ as abcd⊳¯ξ = −q−2a−q−2bcd + (q−4 −1)a, abcd⊳¯b =00q−2ab abcd⊳¯c = cq−2d −(1 −q−2)q−2a0(1 −q−2)c, abcd⊳¯γ = abcd.As q →1 this becomes the usual right action of the lie algebra gl2 on the co-ordinate functionsof M2.ProofThis is by direct computation from Proposition 6.3.

The ←−∂act on the row vector(a, b, c, d) by the matrices shown. From this by subtracting the identity matrix from ←−∂11 and←−∂22 we obtain the action of the χij variables.

This then gives the action of the rescaled basis¯ξ,¯b, ¯c, ¯γ, where the rescaling is by (q2 −1)−1 as before. These also act by 4 × 4 matrices on thegenerators of B(R), which we write now in a more explicit form as shown.

From this explicitform we see that as q →1 the actions become abcd⊳¯ξ = −1001 abcd, abcd⊳¯b = 0010 abcd, abcd⊳¯c = 0100 abcdwhich is the usual action of the sl2 generators by left-invariant vector fields on the functionsalgebra of SL2 or M2 as here.40

Note that at the level of U(L) and its action on B(R), the choice of normalization of thisinitial R is not important. It does not change the algebras and simply scales the ←−∂in Propo-sition 6.3.

On the other hand since the action of 1 is not scaled, the action of the χ generatorscan change more significantly. For the present example the so-called quantum-group normaliza-tion for the present R-matrix requires an additional factor q−12 in Rgl2.

This means a uniformfactor q−1 in the ←−∂as well as for the ⊳¯b, ⊳¯c, ⊳¯ξ, while ⊳¯γ now acts by a different multiple ofthe identity. This normalization is the one needed for the representation of BUq(gl2) to descendto the quantum group Uq(sl2), for which γ becomes proportional to its quadratic Casimir.

Onthe other hand, we are not tied to this consideration and have retained the normalization thatseems more suitable for the braided enveloping bialgebra. ⊔⊓We see that when q →1 the action of the braided-vector fields becomes the usual action byleft-multiplication of the Lie algebra on the co-ordinate functions, as it must by the constructionsabove.

On the other hand for general q or other non-standard R-matrices it is not possible towrite the actions of our braided-vector fields as a matrix product of the Lie algebra matrix onthe group matrix. This problem is well-known even in the case of super-Lie algebras acting bysuper-vector-fields.Example 6.6 For Rgl1|1 as in Example 5.6 the matrix-braided vector fields are←−∂11=q200(q −q−1)20q20000100001,←−∂12=001 −q−20000q−2 −100000000←−∂21=0(q −q−1)2000000q2 −100(q −q−1)201 −q200,←−∂22=q2 + q−2 −100(q −q−1)20q2 + q−2 −10000q−20000q−2.From this we obtain the action of the rescaled generators ¯χ as abcd⊳¯a = ab0(1 −q−2)a, abcd⊳¯b =00q−2a−q−2b abcd⊳¯c = c−d + (1 −q−2)a0(1 −q−2)c, abcd⊳¯ξ = −q−2 abcd.As q →1 this becomes the right action of the super-lie algebra gl1|1 on the super-algebra M1|1.41

ProofThe steps are similar to those in the preceding example. This time as q →1 one hasthe even elements ¯a (and ¯ξ) acting by matrix multiplication while abcd⊳¯b = 0010 abcd 100−1, abcd⊳¯c = 0100 abcd 100−1.Note that this is a feature of super-Lie algebras, in the general braided case (as when q ̸= 1) eventhe possibility of a further matrix on the right hand side will not suffice for a representation as amatrix product.

One can verify directly that these actions represent gl1|1 as super-derivations.⊔⊓Thus we recover a complete geometric picture of braided-Lie algebras acting on braided-commutative algebras of functions (i.e. a classical picture but braided).

The picture unifies thefamiliar theory of left-invariant vector fields on groups, super-groups and its obvious generaliza-tions such as to colour-derivations etc into a single framework based on an R-matrix, which allappear as the semiclassical part of a general braided theory.7Braided Killing Form and the Quadratic CasimirIn this section we give a final application of our notion of braided-Lie algebras, namely to thenotion of braided-Killing form and associated quadratic Casimir. It will be Ad-invariant andbraided-symmetric in a certain sense.

Like the last section, our a result depends on the factthat we have an actual finite-dimensional Lie-algebra like subspace L or X and not merely somekind of Hopf algebra.As before, we do the construction first in a categorical setting with diagrams, and thenafterwards deduce and compute the matrix form. The idea behind the braided Killing form inthe categorical setting is quite straightforward.

In any braided category with duals there is anatural notion of braided-trace of an endomorphism. Assuming that L has a dual L∗(a kindof finite-dimensionality condition) we define the braided-Killing form via the braided-trace inthe adjoint representation of U(L) on L constructed in Proposition 4.4.

We begin with thebraided-trace itself.Proposition 7.1 For an object V in a braided category with dual V ∗, and any morphism φ :W ⊗V →V we define the braided trace as the map Tr (φ) : W →1 obtained as shown in42

======WαVφSαVSαVφφBWVφSαVαVBWφααVεαφφSαVαV(a)WV(b)SBWφTr( )B∆∆∆∆∆BWWBWFigure 14: Definition (a) of braided-trace Tr of a morphism W ⊗Vφ→V and (b) proof of itscyclicity property of invariance under a cocommutative action α of any braided-Hopf algebra B.The extra input W is optionalFigure 14 (a). If B is a braided-Hopf algebra and acts cocommutatively by α on V then Tr (φ)is B-invariant in the manner shown in (b).ProofBy definition Tr (φ) is a morphism W →1 as shown in (a).

Here ∪and ∩denoteevaluation V ∗⊗V →1 and coevaluation 1 →V ⊗V ∗respectively. In part (b) we suppose thata braided-Hopf algebra B acts on V cocommutatively.

The first equality uses functoriality andthe double-bend property of duals (compatibility between evaluation and coevaluation, as usedabove in Proposition 6.1) to pull αV down. The second equality cancels the new double-bendand also pushes φ up.

The third equality is the braided-cocommutativity of B with respectto V .We then use functoriality to reorganise, and that α is an action to cancel using thebraided-antipode axioms. ⊔⊓The invariance here is our braided-analog of the usual ‘cyclicity’ property of the trace.

Notealso that W can be anything, for example W = 1 and φ : V →V an endomorphism. We haveretained the extra input W for greater generality.

In particular, if W = B and φ = α then theinvariance means precisely that Tr (α) is Ad-invariant, where α is the braided-adjoint action of43

Section 3.Proposition 7.2 Let L be a braided-Lie algebra in the setting of Section 4.We define itsbraided-Killing form g : L ⊗L →1 to be the braided-trace of the map [ , ] ◦(id ⊗[ , ]). Inconcrete terms this isg(ξ, η) = Tr ([ξ, [η, ]])for ξ, η ∈L.

If U(L) has an antipode then g is invariant under [ , ] as shown in Figure 15 (c).It is braided-symmetric as shown in Figure 15 (d). The braided-Killing form is defined on all ofU(L) ⊗U(L) and has descendants T and dim(L) as also shown.ProofThe braided-metric is defined as the braided-trace of the iterated braided-adjoint action.This is well-defined as a morphism L ⊗L →1 but can also be viewed as shown in (a) as therestriction of a morphism U(L) ⊗U(L) →1.

In this case, because [ , ] is an action, we canunderstand it as multiplication in U(L) followed by the braided-trace in the braided-adjointrepresentation. In this case its Ad-invariance follows at once in (c) from the Ad-invariance ofT proven in part (b).

This in turn follows from the cyclicity of the braided-trace proven inProposition 7.1.This assumes in the second equality that U(L) has a braided-antipode, inwhich case [ , ] can be identified with the braided-adjoint action as explained in Section 4. Part(d) is the braided-symmetry property.

The first equality is the definition of g, the second is theextended-form of the braided-Jacobi identity in Section 4. For the braided-symmetry only on Lwe need only the braided-Jacobi identity axiom (L1).

Finally, part (e) justifies our terminologyby showing how the property looks on the subspace X ⊂U(L) where the coproduct is as inFigure 11. ⊔⊓Clearly the braided-Killing form is the same as first multiplying in U(L) and then applyingthe braided-trace to [ , ] considered as an action of U(L) from Proposition 4.4.

Also, if L is ofthe form L1 = 1 ⊕X as discussed at the end of Section 4, we can equally well definegχ : X ⊗X →1in just the same way as Tr ([ , ](id ⊗[ , ]) restricted to χ ⊗χ. Both are useful in examples.

Themetric on L is some kind of ‘multiplicative’ Killing form while gχ is more like the classical one.Its diagrammatic properties are in Figure 15(e).44

LULULULUgLULUηηLUgηTTTLULULULULULULUSLULUSggg[ , ]X X X1∆ggg1∆LULUεTLULUεLULULUεgLULULUεT[ , ]LULULUT[ , ].LULULUT.∆[ , ][ , ]LULULUg∆[ , ][ , ]LULULULUgLULU[ , ][ , ]g∆[ , ]LULU(a)L L*[ , ][ , ]===g[ , ]dim( )L===L L*[ , ]=L L*[ , ][ , ][ , ].∆∆[ , ][ , ][ , ](e)X X X[ , ][ , ]++[ , ]=X X XX XX X+=X X[ , ]0====(d)====(c)(b)[ , ]===∆[ , ][ , ][ , ]Figure 15: Definition (a) of braided-Killing form and its descendants (b)-(c) proof of their[ , ]-invariance and (d) braided-symmetry. In the form (e) on X these look more familiar.45

The proof above assumes that U(L) has a braided-antipode. On the other hand the for-mulation of the proposition does not require this if we work with [ , ] instead of an actualbraided-adjoint action.

This was the strategy in Section 4 and we take the same view here. Forexample, in the tensor setting of Proposition 5.1 we can assume that the tensors defining thebraided-Lie algebra are sufficiently nice for L to have a dual object and for the braided-Killingform to be [ , ]-invariant.

We say in this case that the braided-Lie algebra is regular. Also, wedefine tensors for g and the braided trace, as well as the normalization dim(L) byg(uI ⊗uJ) = Tr ([uI, [uJ, ]]) = gIJ,Tr ([uI, ]) = T I,dim(L) = Tr (id)(20)Their properties in tensor form are read of from the braid-diagrams just as for Proposition 5.1.In particular, the invariance and braided-symmetry conditions take the form∆IAB RJM BN cAM P cNKQgP Q = δI gJK,cIJ KT K = δIT J(21)∆IAB RJM BN cAM P gP N = gIJ(22)and likewise for gχ and Tχ on the generators χI = uI −δI.

These are related to gIJ and T I bygIJχ = g(χI ⊗χJ) = gIJ −δIT J −δJT I + dim(L)δIδJ,T Iχ = T I −dim(L)δI. (23)Here g and gχ differ only by the braided-trace of the action of 1 in one or other or both of theinputs.

The fact that these maps are all morphisms in the category means that they obey thecorresponding morphism conditions along the lines of (L0) in Proposition 5.1. Thus, T J obeysthe same equations as for δI in (L0) while g (and gχ) obeyRKM J B RM LIA gAB = gIJδKL,RIAKM RJ BM L gAB = gIJδKL.

(24)We have mentioned in the proof of Proposition 5.1 that the nicest setting is the one in whichthe constructions can be viewed as taking place in the category of left A(R)-comodules, or moreprecisely in the category of A-comodules where A is a dual-quasitriangular quotient of A(R).In the present context one could demand also that A is a Hopf algebra. In this case its categoryof comodules has duals, so this is sufficient to have a quantum trace.

We do not want to limitourselves to this case, but it is convenient for generating the necessary formulae which can thenbe verified directly on the assumption of suitable properties for the structure constants. To46

see that this supposition implies restrictions on A we note that in these terms, the morphismproperties of ∆, ǫ, c, g aretIJ∆J KL = ∆IABtAKtBL,tIJδJ = δI,cIJ KtKL = tIAtJ BcABL(25)gIJ = tIAtJ BgAB,tIJT J = T I(26)where tIJ is the matrix generator of A(R).Proposition 7.3 Let L be a braided-Lie algebra of the general tensor type in Proposition 5.1and suppose that it lives in the category of A-comodules as explained. ThengIJ = cIAB cJLA eRKLBK,T I = cIJ A eRKJ AK,dim(L) = eRKJ J Kwhere e denotes the second-inverse as above but applied now to the multi-index R.ProofWe assume here that the category in which we work is the braided tensor category of leftA-comodules where A is a dual-quasitriangular Hopf algebra given as a quotient of A(R).

It hasat least the additional relations (25) and (26) as explained. The finite-dimensional comodulessuch as L and X here then have duals in the category using the antipode.From this onecomputes the braiding between a basis {uI} of L and a dual basis {fI} say of L∗in a standardway as explained in [12].

The {uI} transform as a vector under the matrix generator of A(R)and {fI} as a covector with right-multiplication by the inverse matrix generator. This givesΨ(uI ⊗fJ) = fK ⊗uL eRKJ IL.Using this for the braid-crossing in the diagrammatic definition of the braided-trace and braided-Killing form and proceeding as in Proposition 5.1 for the other tensors, immediately gives theresults stated.

Note that the Tr that we use here is defined for any endomorphism φIJ byTr (φ) = φBA eRKBAK just as for the usual quantum or braided trace associated to an R-matrix.We are simply using this now applied to the endomorphisms built from the structure constantscIJ K of the braided-Lie algebra. ⊔⊓In our matrix examples of Proposition 5.2, all the data are based on an initial R-matrixRijkl.

In this context we have already introduced the notion for quantum groups that R is47

regular if A(R) has a quotient Hopf algebra A which remains dual-quasitriangular. In this caseB(R) has a quotient which is indeed a braided-Hopf algebra with braided-antipode.

Related tothis, U(L) = B(R) for this class of matrix-braided-Lie algebras is indeed regular in the senseabove. On the other hand, we do not want to limit ourselves to this case.

In fact, it is sufficientto suppose that R obeys certain matrix identities to arrive at the same conclusion.Proposition 7.4 In our matrix examples of Proposition 5.2 we suppose that this is regular inthe sense that the initial R ∈Mn ⊗Mn comes from a quantum group obtained from A(R). Thenthe braided-Killing g is given bygIJ = cIAB cJLA Rb0canRndcbϑbl0 ˜Rdb1l1ain terms of the initial R and its second-inverse ˜R.

Here ϑij = eRikkj. Similarly for T I anddim(L).

If R = R0 + O(ℏ) then gχ = O(ℏ2). On the rescaled generators ¯χI = ℏ−1χI we havegIJ¯χ = g(¯χI, ¯χJ) = KIJ + O(ℏ),T I¯χ = T(¯χI) = O(ℏ)where KIJ defines the Killing form of the R0-Lie algebra in Proposition 5.3 and has its usualAd-invariance and Ψ-symmetry properties (e.g.

for usual, super or colour Lie algebras etc).Here Ψ = Ψ(R0) is symmetric. Meanwhile, the braided trace T on the rescaled generators tendsto zero.ProofOne can either compute eRKJ AK for the particular matrix in Proposition 5.2, or com-pute the braiding Ψ(uI ⊗fJ) between a basis element of uI and a dual-basis element directly inthe same way that the braiding in Proposition 5.2 was obtained in [13][17].

For the latter coursethe category in which we work is that of right A-comodules where A is now a dual-quasitriangularHopf algebra obtained as a quotient of A(R) and Rijkl here is the initial R-matrix in in Propo-sition 5.1. It is related to the general setting above via the bialgebra map A(R) →Aop givenby tIJ 7→ti0j0 ·Aop SAoptj1i1.

This along with the antipode of A converts the left-comodulealgebras in the general setting into right A-comodule algebras. In the latter category the ele-ments u transform under the right adjoint coaction u →t−1ut using a compact notation wheret is the matrix generator of A(R).

This induces on the dual basis {fij} the transformationfij →fmn ⊗(Stjn)S2tmi where S is the antipode. From this one hasΨ(ui0i1 ⊗fj0j1)= fk0k1 ⊗ul0l1R((Sti0l0)tl1i1 ⊗(Stj1k1)S2tk0j0)48

= fk0k1 ⊗ul0l1R(Sti0l0 ⊗(Stak1)S2tk0b)R(tl1i1 ⊗(Stj1a)S2tbj0)= fk0k1 ⊗ul0l1R(ti0c ⊗tak1)R(tcl0 ⊗Stk0b)R(tl1d ⊗S2tbj0)R(tdi1 ⊗Stj1a)= fk0k1 ⊗ul0l1Ri0cak1 ˜Rcl0k0b( ˜R)−1l1dbj0 ˜Rdi1j1a = fK ⊗uL eRKJ ILwhere in the last line we evaluated the dual-quasitriangular structure R on the matrix generators.This gives the matrix eRKJ IL in this example (compare with the braiding in the proof of theprevious proposition). Composing this with evaluation we have< , > ◦Ψ(uI ⊗fJ)= eRKJ IK = Ri0can ˜Rcmmb( ˜R)−1ndbj0 ˜Rdi1j1a= Ri0canϑcb( ˜R)−1ndbj0 ˜Rdi1j1a = Ri0canRndcbϑbj0 ˜Rdi1j1awhere ϑcb = ˜Rcmmb is the matrix used for the quantum or braided trace associated to the initialR-matrix.

It obeys ϑ2( ˜R)−112 ϑ−12= R12 as proven in [12] and we use this now. Putting this intothe preceding proposition gives the results stated.

Note that more generally, one can supposeonly that R is bi-invertible and obeys suitable matrix identities such as[12].ϑ2 eR = R−1ϑ2,ϑ1R−1 = eRϑ1(27)to conclude the proposition directly.From this one sees the limit as R approaches a triangular solution R0. From Figure 15 (e)we see that the semiclassical part K of the braided-Killing has the familiar properties.

Likewisefor the braided-trace. ⊔⊓Thus the braided-Killing form reduces near the triangular solutions to the more usual notionof Killing form which is Ad-invariant and Ψ-symmetric in the more naive sense.

This includes ofcourse the usual Killing form but holds also for super-Lie algebras and colour-Lie algebras. Inthe latter cases we have not found this notion in the literature, perhaps because it need not benon-degenerate as we shall see in an example.

In the former standard case we will recover theusual Killing form which will be non-degenerate on the semisimple part of the classical limit. Wefind here an unusual phenomenon: the process of q-deformation can make a degenerate Killingform non-degenerate.49

Example 7.5 In Example 5.5 where R = Rgl2 the braided-Killing form and trace etc on L isg = [4]qq10q4 + q−2 −100q4 −q2 + 100(1 −q−2)200(q −q−1)200q4 −q2 + 100q4 −q2 + 1 + (1 −q−2)2T I = (1 + [3]q2)δI,dim(L) = [4]q;[n]q = 1 −q−2n1 −q−2 .Here g is non-degenerate for generic q. The braided-Killing form on the rescaled ¯χ with basis¯ξ,¯b, ¯c, ¯γ is also non-degenerate for generic q and given byg¯χ = q−4[4]q[2]q00000q−4[4]q00q−2[4]q00000[3]q[2]q(1 −q−4)As q →1 it becomes the usual Killing form on sl2 and 0 on the ¯γ generator.ProofThis is a direct computation from Proposition 7.2 or 7.3 (the result it the same).

Notethat as q →1 the braided-Killing forms become symmetric and the braided-traces of the ¯χIbecome zero as we should expect from Proposition 7.4. ⊔⊓It is remarkable here that the braided-Killing form on our braided-version of gl2 is non-degenerate for generic q.This reflects the fact that for generic q the U(1) generator ¯γ inExample 5.5 did not fully decouple from the braided-Lie bracket.

This is in spite of the factthat it is central in the braided-enveloping algebra.Example 7.6 In Example 5.6 where R = Rgl1|1 the braided-Killing form on L and on the ¯χ¯a,¯b, ¯c, ¯ξ basis areg = −(q2 −q−2)21001000000001001,g¯χ = −(1 + q−4)1000000000000000T I = −(q −q−1)2δI,dim(L) = 050

ProofThis is likewise a direct computation from Proposition 7.3 or 7.4. ⊔⊓The braided-dimension becomes as q →1 the super-dimension for the R-matrix in thisexample.

Hence its vanishing corresponds in the limit to the equal number of bose and fermimodes in the algebra. This is typical of vanishing theorems in super-symmetry and suggests thatsimilar results can sometimes extend to the braided case.

A similar degeneracy of the braided-Killing form, and vanishing of the braided dimension holds for other non-standard R-matrices(such as the 8-vertex model solution). On the other hand non-degeneracy as in Example 7.5 istypical of the standard R-matrices associated to deformations of semisimple Lie algebras.Armed with non-degeneracy in at least some cases it is natural to define for invertible gIJ, gIJχthe corresponding quadratic Casimirs in U(L),C = uIuJgIJ,Cχ = χIχJgχIJ(28)where the matrixes with lower indices are the matrix inverses.

This can also be said diagram-matically.Corollary 7.7 In the setting of Proposition 7.2 we suppose that the braided-Killing form has aninverse g : 1 →L ⊗L. Then this is [ , ]-invariant and the Casimir · ◦g : 1 →U(L) is invariantand central in U(L).

Moreover, the braided-Killing form and its inverse allow us to identify Land L∗in the category.ProofThe categorical inverse (also denoted g) is defined via Figure 16 (a), along with somerelated maps. The corollary then follows at once from the invariance and braided-symmetryproperties of the braided-Killing form in Figure 15, as shown.

For simplicity we have assumedfor the proof that U(L) has a braided-antipode, but as in Proposition 7.2 we do not limitourselves to this case. In the matrix example of Proposition 5.2 with suitable R one can provethe invariance of the inverse-Killing form directly.

Also, using these maps we can identify Lwith L∗, with the braided-Killing form as evaluation and its inverse as co-evaluation. In thiscase it is natural to consider the twist σ and we have included in part (d) one of its interestingproperties.

⊔⊓Similar properties (with similar proofs) apply to the inverse gχ : 1 →X ⊗X when this exists.51

ggggCLUggLLgLUεLUCLUCLULUCLULUσσLULUC[ , ] .∆gCLULU.LUgLUgεL LgLUSLL[ , ][ , ]∆LUSgggLL[ , ][ , ]∆gggLUSLL[ , ][ , ][ , ][ , ]∆∆g(a)(b)L=LL=LLL(c)=.==[ , ].=LLLL∆[ , ]LL[ , ]∆=(d)==LL=[ , ][ , ]∆LL===∆[ , ]Figure 16: Definition (a) inverse braided Killing form, braided-quadratic Casimir and asso-ciated twist morphism. [ , ]-invariance (b) of g implies invariance and centrality (c) of thebraided-Casimir.

(d) is a property of the twist σ related to braided-antisymmetry of the bracket.The identification of L with L∗(or X with X ∗) when the inverses exist means in tensorial termsthat we can use the braided-metric and its inverse to raise and lower indices in a familiar way.Example 7.8 For the braided-Lie algebra in Example 5.5 the quadratic Casimirs defined fromthe inverse of the braided-Killing form are central and take the formC =[2]q[4]q(1 −q−2)2 (q6 −q2 + 1)q4(1 + q2)(q8 + q4 −q2 + 1)(q−2a + d)2 −(ad −q2cb)!Cχ = q4[4]q(¯ξ2[2]q+ ¯b¯c + q2¯c¯b) +q2[3]q(1 −q−4)2 ¯γ2As q →1 the sl2-part of Cχ tends to the usual quadratic Casimir and the U(1) part tends to ∞.ProofThis is by direct computation from the generators using REDUCE. To put the resultsinto the form shown we made extensive use of the relations in U(L) = BMq(2) from [17].

Weknow that the rescaled generators tend in the classical limit to gl2. We see that the naturalbraided Casimir tends to the usual quadratic Casimir for the sl2 part while the U(1) part blowsup in terms of the rescaled generator ¯γ.

Note that the two terms in Cχ are separately centralfor all q so one an subtract offthis divergent part if desired. ⊔⊓52

Moreover, for q ̸= 1 and the standard R-matrices we can put here the form u = l+Sl−and recover from C the (square of the) quantum quadratic Casimir known previously by othermethods. On the other hand our construction is not tied to such standard cases.This completes our development of the basic theory of braided-Lie algebras and some typicalexamples.

Further applications and examples will be developed elsewhere. The phenomenonseen here for the braided version of gl2 can be expected quite generally and is part of one set ofpotential applications of the theory, namely to a process that can be called ‘q-regularization’ ofsingularities.

The singularity of the inverse-Killing form for gl2 is resolved by q-deformation inour braided context, as a pole at q = 1. The regularization of infinities in physics is one of themotivations for q-deformed physics and q-deformed geometry (another is interesting phenomenaat roots of unity).For one possible physical application of these constructions we note that we have introduceda general quantum-group gauge theory in [1], which should adapt (by transmutation) to ourbraided-setting.

The gauge fields of such a theory should take values in a braided-Lie algebraL or X and the Yang-Mills Lagrangian should involve the braided-Killing form as above. Insuch a theory the SU(2) × U(1) of the standard model could be unified for q ̸= 1 with theU(1) mode not decoupling from the SU(2) mode in the bare Lagrangian.

After renormalizationthe zero in the braided-Killing form above for the U(1) mode may still leave a residue as theq-regularization is removed.For another direction we note that the quadratic Casimirs become represented as differentialoperators on the braided-group function algebras as we have seen in Section 6. Thus←−⊔⊓=←−∂I←−∂JgIJ,←−⊔⊓χ =←−DI←−DJgχIJ(29)should play the role of Laplacian in some kind of braided-geometry of which the braided-groupsare the simplest examples.

They could perhaps be used as propagators in some form of braidedor q-deformed physics. Again, one would have in mind some interaction with the process ofrenormalization, where q is regarded as a regularization parameter and set to 1 at the end.

Itmay also be that q ̸= 1 could be used as a model of feedback to the geometry due to quantumeffects in the context of Planck scale physics.Related to these considerations we note that there are further examples of braided Hopf53

algebras associated to the quantum plane and to the braided-Heisenberg algebras[11], as well aspossibly to the infinite-dimensional exchange algebras in conformal field theory – one would liketo know if they have a braided-Lie algebra underlying them. The deformation of braided-Liealgebras via braided-Poisson brackets is a further question related to these.Apart from these physical directions, there are of a variety of natural mathematical questionsalso to be addressed.

The long term goal is to develop the differential geometry of braidedgroups with braided-Lie algebras and other braided-geometrical constructions in analogy withthe classical theory. In this paper we have taken some of the first steps in such a programme.We introduced brackets, vector-field or matrix realizations and Killing forms for them in somegenerality.

We recover usual notions from any regular R-matrix, which need not be a standarddeformation of the identity. The theory interpolates and unifies with super and other Lie-algebraconstructions also.

Moreover, even in the standard case we have found some unusual phenomenaconcerning the removal of degeneracy.REFERENCES[1] T. Brzezi´nski and S. Majid. Quantum group gauge theory on quantum spaces.

Preprint,DAMTP/92-27, May 1992. [2] V.G.

Drinfeld. Quantum groups.

In A. Gleason, editor, Proceedings of the ICM, pages798–820, Rhode Island, 1987. AMS.

[3] L.D. Faddeev, N.Yu.

Reshetikhin, and L.A. Takhtajan. Quantization of Lie groups and Liealgebras.

Leningrad Math J., 1:193–225, 1990. [4] D.I.

Gurevich. The Yang-Baxter equations and a generalization of formal Lie theory.

Sov.Math. Dokl., 33:758–762, 1986.

[5] M. Jimbo. A q-difference analog of U(g) and the Yang-Baxter equation.

Lett. Math.

Phys.,10:63–69, 1985. [6] A. Joyal and R. Street.

Braided monoidal categories. Mathematics Reports 86008, Mac-quarie University, 1986.

[7] S. Mac Lane. Categories for the Working Mathematician.

Springer, 1974. GTM vol.

5.54

[8] S. Majid.Beyond supersymmetry and quantum symmetry (an introduction to braidedgroups and braided matrices). In M-L. Ge, editor, Proc.

Nankai Workshop, 1992. WorldSci.

To appear. [9] S. Majid.

Braided groups, 1990. J.

Pure Applied Algebra. To appear.

[10] S. Majid. Braided matrix structure of the Sklyanin algebra and of the quantum lorentzgroup, january 1992 (damtp/92-10).

Comm. Math.

Phys. To appear.

[11] S. Majid.Free braided differential calculus, braided binomial theorem and the braidedexponential map. Preprint, DAMTP/93-3.

[12] S. Majid. Quantum and braided linear algebra, february 1992 (damtp/92-12).

J. Math.Phys. To appear.

[13] S. Majid. Rank of quantum groups and braided groups in dual form.

In Proc. of the EulerInstitute, St. Petersberg (1990), volume 1510 of Lec.

Notes. in Math., pages 79–89.

Springer. [14] S. Majid.

More examples of bicrossproduct and double cross product Hopf algebras. Isr.J.

Math, 72:133–148, 1990. [15] S. Majid.Quasitriangular Hopf algebras and Yang-Baxter equations.Int.

J. ModernPhysics A, 5(1):1–91, 1990. [16] S. Majid.

Braided groups and algebraic quantum field theories. Lett.

Math. Phys., 22:167–176, 1991.

[17] S. Majid. Examples of braided groups and braided matrices.

J. Math.

Phys., 32:3246–3253,1991. [18] S. Majid.

Transmutation theory and rank for quantum braided groups. Math.

Proc. Camb.Phil.

Soc., 113:45–70, 1993. [19] N.Yu.

Reshetikhin and M.A. Semenov-Tian-Shansky.

Central extensions of quantum cur-rent groups. Lett.

Math. Phys., 19:133–142, 1990.

[20] N.Yu. Reshetikhin and M.A.

Semenov-Tian-Shansky. Quantum R-matrices and factoriza-tion problems.

J. Geom. Phys., 1990.55

[21] M. Scheunert. Generalized lie algebras.

J. Math.

Phys., 20:712–720, 1979. [22] P. Schupp, P. Watts, and B. Zumino.Bicovariant quantum alegbras and quantum liealgebras.

Preprint, September, 1992. [23] A. Sudbery.

Quantum differential calculus and lie algebras. In Proc.

of XX1 DGM, Tianjin,China May, 1992. World Sci.

To appear. [24] M.E.

Sweedler. Hopf Algebras.

Benjamin, 1969.56

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