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AN INTRODUCTION TO NON-COMMUTATIVE

arXiv:hep-th/9207084v1 26 Jul 1992CERN–TH.6565/92DFTT-22/92AN INTRODUCTION TO NON-COMMUTATIVEDIFFERENTIAL GEOMETRY ON QUANTUM GROUPSPaolo Aschieri⋄∗and Leonardo Castellani∗⋄CERN, CH-1211 Geneva 23, Switzerland.∗Istituto Nazionale di Fisica Nucleare, Sezione di TorinoandDipartimento di Fisica TeoricaVia P. Giuria 1, 10125 Torino, Italy.AbstractWe give a pedagogical introduction to the differential calculus on quan-tum groups by stressing at all stages the connection with the classical case(q →1 limit). The Lie derivative and the contraction operator on forms andtensor fields are found.

A new, explicit form of the Cartan–Maurer equationsis presented. The example of a bicovariant differential calculus on the quan-tum group GLq(2) is given in detail.

The softening of a quantum group isconsidered, and we introduce q-curvatures satisfying q-Bianchi identities, abasic ingredient for the construction of q-gravity and q-gauge theories.CERN–TH.6565/92DFTT-22/92July 1992email addresses: Decnet=(39163::ASCHIERI, CASTELLANI; VXCERN::ASCHIERI)Bitnet=(ASCHIERI, CASTELLANI@TORINO.INFN.IT )

1IntroductionQuantum groups [1]–[4] have emerged as interesting non-trivial generalizations ofLie groups. The latter are recovered in the limit q →1, where q is a continuousdeformation parameter (or set of parameters).In the q ̸= 1 case, the q-groupmay have, and in general does have, more than one corresponding q-algebra, theq-analogue of the Lie algebra.

This can be rephrased by saying that the differentialcalculus on q-groups is not unique [5].On a classical Lie group one can definea left and right action of the group on itself, and these commute. By imposingthis “bicovariance” to hold also in the q-deformation, one restricts the possibledifferential calculi (still to a number > 1 in general).Our motivations for studying the differential geometry of quantum groups aretwofold:– the q-differential calculus offers a natural scenario for q-deformations of grav-ity theories, based on the quantum Poincar´e group [6].

Space-time becomes non-commutative, a fact that does not contradict (Gedanken) experiments under thePlanck length, and that could possibly provide a regularization mechanism [7, 8].– the quantum Cartan-Maurer equations define q-curvatures, and these can beused for constructing q-gauge theories [9]. Here space-time can be taken to be theordinary commutative Minkowski spacetime, while the q-structure resides on thefibre, the gauge potentials being non-commuting.

These theories could offer inter-esting examples of a novel way of breaking symmetry by q-deforming the classicalone.In this paper we intend to give an introductory review of the q-differential calcu-lus. A discussion on Hopf structures will not be omitted: in Section 2 we recognizethese structures in ordinary Lie groups and Lie algebras.

Quantum groups andtheir non-commutative geometry are discussed in Sections 3 and 4, and Section5 describes an explicit construction of a bicovariant calculus on quantum groups.Some new results of Section 5 include a formula that gives the commutations of theleft-invariant one-forms, and (as a consequence) a new expression for the Cartan-Maurer equations.The example of GLq(2) [and its restrictions to Uq(2) and SUq(2) ] is systemati-cally used to illustrate the general concepts. In Section 6 we study the quantum Liederivative, an essential tool for the definition of q-variations.

Section 7 extends thenotion of “soft” group manifolds (see for example [10]) to q-groups, by introducingq-curvatures.After refs. [5], there have been a number of papers treating the differentialcalculus on q-groups from various points of view [11]–[20].In this paper we use the formalism of refs.

[5] and [17].1

2Hopf structures in ordinary Lie groups and LiealgebrasLet us begin by considering Fun(G) , the set of differentiable functions from aLie group G into the complex numbers C. Fun(G) is an algebra with the usualpointwise sum and product (f + h)(g) = f(g) + h(g), (f · h) = f(g)h(g), (λf)(g) =λf(g), for f, h ∈Fun(G), g ∈G, λ ∈C. The unit of this algebra is I, defined byI(g) = 1, ∀g ∈G.Using the group structure of g, we can introduce on Fun(G) three other linearmappings, the coproduct ∆, the counit ε, and the coinverse (or antipode) κ:∆(f)(g, g′)≡f(gg′),∆: Fun(G) →Fun(G) ⊗Fun(G)(2.1)ε(f)≡f(e),ε : Fun(G) →C(2.2)(κf)(g)≡f(g−1),κ : Fun(G) →Fun(G)(2.3)where e is the unit of G. It Xis not difficult to verify the following properties of theco-structures:(id ⊗∆)∆= (∆⊗id)∆(coassociativity of ∆)(2.4)(id ⊗ε)∆(a) = (ε ⊗id)∆(a) = a(2.5)m(κ ⊗id)∆(a) = m(id ⊗κ)∆(a) = ε(a)I(2.6)and∆(ab) = ∆(a)∆(b),∆(I) = I ⊗I(2.7)ε(ab) = ε(a)ε(b),ε(I) = 1(2.8)κ(ab) = κ(b)κ(a),κ(I) = I(2.9)where a, b ∈A = Fun(G) and m is the multiplication map m(a ⊗b) ≡ab.

Theproduct in ∆(a)∆(b) is the product in A ⊗A: (a ⊗b)(c ⊗d) = ab ⊗cd.In general a coproduct can be expanded on A ⊗A as:∆(a) =Xiai1 ⊗ai2 ≡a1 ⊗a2,(2.10)where ai1, ai2 ∈A and a1⊗a2 is a short-hand notation we will often use in the sequel.For example for A = Fun(G) we have:∆(f)(g, g′) = (f1 ⊗f2)(g, g′) = f1(g)f2(g′) = f(gg′). (2.11)Using (2.11), the proof of (2.4)-(2.6) is immediate.An algebra A endowed with the homomorphisms ∆: A →A⊗A and ε : A →C,and the antimorphism κ : A →A satisfying the properties (2.4)-(2.9) is a Hopf2

algebra. Thus Fun(G) is a Hopf algebra.1Note that the properties (2.4)-(2.9)imply the relations:∆(κ(a)) = κ(a2) ⊗κ(a1)(2.12)ε(κ(a)) = ε(a).

(2.13)Consider now the algebra A of polynomials in the matrix elements T ab of thefundamental representation of G. The algebra A is said to be freely generated bythe T ab.It is clear that A ⊂Fun(G), since T ab(g) are functions on G. In fact everyfunction on G can be expressed as a polynomial in the T ab (the reason is that thematrix elements of all irreducible representations of G form a complete basis ofFun(G), and these matrix elements can be constructed out of appropriate productsof T ab(g)), so that A = Fun(G). The group manifold G can be completely charac-terized by Fun(G), the co-structures on Fun(G) carrying the information about thegroup structure of G. Thus a classical Lie group can be “defined” as the algebra Afreely generated by the (commuting) matrix elements T ab of the fundamental rep-resentation of G, seen as functions on G. This definition admits non-commutativegeneralizations, i.e.

the quantum groups discussed in the next Section.Using the elements T ab we can write an explicit formula for the expansion (2.10)or (2.11): indeed (2.1) becomes∆(T ab)(g, g′) = T ab(gg′) = T ac(g)T cb(g′),(2.14)since T is a matrix representation of G. Therefore:∆(T ab) = T ac ⊗T cb. (2.15)Moreover, using (2.2) and (2.3), one finds:ε(T ab) = δab(2.16)κ(T ab) = (T −1)ab.

(2.17)Thus the algebra A = Fun(G) of polynomials in the elements T ab is a Hopf algebrawith co-structures defined by (2.15)-(2.17) and (2.7)-(2.9).Another example of Hopf algebra is given by any ordinary Lie algebra, or moreprecisely by the universal enveloping algebra of a Lie algebra, i.e. the algebra (withunit I) of polynomials in the generators Ti modulo the commutation relations[Ti, Tj] = CkijTk.

(2.18)1To be precise, Fun(G) is a Hopf algebra when Fun(G × G) can be identified with Fun(G) ⊗Fun(G), since only then can one define a coproduct as in (2.1). This is possible for compact G.3

Here we define the co-structures as:∆(Ti) = Ti ⊗I + I ⊗Ti∆(I) = I ⊗I(2.19)ε(Ti) = 0ε(I) = 1(2.20)κ(Ti) = −Tiκ(I) = I(2.21)The reader can check that (2.4)-(2.6) are satisfied.In general the dual of a (finite-dimensional) Hopf algebra A is a Hopf algebraA′, whose structures and co-structures are given, respectively, by the co-structuresand structures of A, i.e. :χ1χ2(a) ≡(χ1 ⊗χ2)∆(a),χ1, χ2 ∈A′(2.22)I′(a) ≡ε(a)I′ = unit of A′(2.23)and:∆′(χ)(a ⊗b) ≡χ(ab)(2.24)ε′(χ) ≡χ(I)(2.25)κ′(χ)(a) ≡χ(κ(a))(2.26)3Quantum groups.

The example of GLq(2)Quantum groups are introduced as non-commutative deformations of the algebraA = Fun(G) of the previous section [more precisely as non-commutative Hopfalgebras obtained by continuous deformations of the Hopf algebra A = Fun(G)].In the following we consider quantum groups defined as the associative algebras Afreely generated by non-commuting matrix entries T ab satisfying the relationRabefT ecT fd = T bfT aeRefcd(3.1)and some other conditions depending on which classical group we are deforming(see later).The matrix R controls the non-commutativity of the T ab, and itselements depend continuously on a (in general complex) parameter q, or even a setof parameters. For q →1, the so-called “classical limit”, we haveRabcdq→1−→δac δbd,(3.2)i.e.the matrix entries T ab commute for q = 1, and one recovers the ordinaryFun(G).The associativity of A implies a consistency condition on the R matrix, thequantum Yang–Baxter equation:Ra1b1a2b2Ra2c1a3c2Rb2c2b3c3 = Rb1c1b2c2Ra1c2a2c3Ra2b2a3b3.

(3.3)4

For simplicity we rewrite the “RTT” equation (3.1) and the quantum Yang–Baxterequation asR12T1T2 = T2T1R12(3.4)R12R13R23 = R23R13R12,(3.5)where the subscripts 1, 2 and 3 refer to different couples of indices. Thus T1 in-dicates the matrix T ab, T1T1 indicates T acT cb, R12T2 indicates RabcdT de and soon, repeated subscripts meaning matrix multiplication.

The quantum Yang–Baxterequation (3.5) is a condition sufficient for the consistency of the RTT equation(3.4). Indeed the product of three distinct elements T ab, T cd and T ef, indicatedby T1T2T3, can be reordered as T3T2T1 via two differents paths:T1T2T3 րցT1T3T2 →T3T1T2T2T1T3 →T2T3T1ցր T3T2T1(3.6)by repeated use of the RTT equation.

The relation (3.5) ensures that the two pathslead to the same result.The algebra A (“the quantum group”) is a non-commutative Hopf algebrawhose co-structures are the same of those defined for the commutative Hopf al-gebra Fun(G) of the previous section, eqs. (2.15)-(2.17), (2.7)-(2.9).Let us give the example of SLq(2), the algebra freely generated by the elementsα, β, γ and δ of the 2 × 2 matrixT ab = αβγδ!

(3.7)satisfying the commutationsαβ = qβα,αγ = qγα,βδ = qδβ,γδ = qδγβγ = γβ,αδ −δα = (q −q−1)βγ,q ∈C(3.8)anddetqT ≡αδ −qβγ = I. (3.9)The commutations (3.8) can be obtained from (3.1) via the R matrixRabcd =q00001000q −q−110000q(3.10)where the rows and columns are numbered in the order 11, 12, 21, 22.5

It is easy to verify that the “quantum determinant” defined in (3.9) commuteswith α, β, γ and δ, so that the requirement detq T = I is consistent. The matrixinverse of T ab is(T −1)ab = (detqT)−1 δ−q−1β−qγα!

(3.11)The coproduct, counit and coinverse of α, β, γ and δ are determined via formulas(2.15)-(2.17) to be:∆(α) = α ⊗α + β ⊗γ,∆(β) = α ⊗β + β ⊗δ∆(γ) = γ ⊗α + δ ⊗γ,∆(δ) = γ ⊗β + δ ⊗δ(3.12)ε(α) = ε(δ) = 1,ε(β) = ε(γ) = 0(3.13)κ(α) = δ,κ(β) = q−1β,κ(γ) = −qγ,κ(δ) = α. (3.14)Note 1: In general κ2 ̸= 1, as can be seen from (3.14).

The following usefulrelation holds [3]:κ2(T ab) = DacT cd(D−1)db = dad−1b T ab,(3.15)where D is a diagonal matrix, Dab = daδab, given by da = q2a−1 for the q-groupsAn−1.Note 2: The commutations (3.8) are compatible with the coproduct ∆, in thesense that ∆(αβ) = q∆(βα) and so on. In general we must have∆(R12T1T2) = ∆(T2T1R12),(3.16)which is easily verified using ∆(R12T1T2) = R12∆(T1)∆(T2) and ∆(T1) = T1 ⊗T1.This is equivalent to proving that the matrix elements of the matrix product T1T ′1,where T ′ is a matrix [satisfying (3.1)] whose elements commute with those of T ab,still obey the commutations (3.4).Note 3: ∆(detq T) = detq T ⊗detq T so that the coproduct property ∆(I) = I⊗Iis compatible with detq T = I.Note 4: Other conditions compatible with the RTT relation can be imposed onT ab:i) Unitarity condition: T † = T −1 ⇒¯α = δ, ¯β = −qγ, ¯γ = −q−1β, ¯δ = α, whereq is a real number and the bar denotes an involution, the q-analogue of complexconjugation, satisfying (αβ) = ¯β ¯α etc.

Restricts SLq(2) to SUq(2).ii) Reality condition: ¯T = T ⇒¯α = α, ¯β = β, ¯γ = γ, ¯δ = δ, |q| = 1. RestrictsSLq(2) to SLq(2, R).iii) The q-analogue of orthogonal and symplectic groups can also be defined, see[3].6

Note 5: The condition (3.9) can be relaxed. Then we have to include the centralelement ζ = (detq T)−1 in A, so as to be able to define the inverse of the q-matrixT ab as in (3.11), and the coinverse of the element T ab as in (2.17).

The q-groupis then GLq(2), and the unitarity condition restricts it to Uq(2). The reader candeduce the co-structures on ζ: ∆(ζ) = ζ ⊗ζ, ε(ζ) = 1, κ(ζ) = detq T.Note 6: More generally, the quantum determinant of n × n q-matrices is de-fined by detq T = Pσ(−q)l(σ)T 1σ(1) · · · T nσ(n), where l(σ) is the minimal number ofinversions in the permutation σ.

Then detq T = 1 restricts GLq(n) to SLq(n).Note 7: We recall the important relations [3] for the ˆR matrix defined by ˆRabcd ≡Rbacd, whose q = 1 limit is the permutation operator δadδbc:ˆR2 = (q −q−1) ˆR + I,for An−1(Hecke condition)(3.17)( ˆR −qI)( ˆR + q−1I)( ˆR −q1−NI) = 0,for Bn, Cn, Dn,(3.18)with N = 2n + 1 for the series Bn and N = 2n for Cn and Dn. Moreover for allA, B, C, D q-groups the R matrix is lower triangular and satisfies:(R−1)abcd(q) = Rabcd(q−1)(3.19)Rabcd = Rdcba.

(3.20)4Differential calculus on quantum groupsIn this section we give a short review of the bicovariant differential calculus on q-groups as developed by Woronowicz [5]. The q →1 limit will constantly appear inour discussion, so as to make clear which classical structure is being q-generalized.Consider the algebra A of the preceding section, i.e.

the algebra freely generatedby the matrix entries T ab, modulo the relations (3.1) and possibly some reality ororthogonality conditions.A first-order differential calculus on A is then defined byi) a linear map d: A →Γ, satisfying the Leibniz ruled(ab) = (da)b + a(db),∀a, b ∈A;(4.1)Γ is an appropriate bimodule (see for example [21]) on A, which essentially meansthat its elements can be multiplied on the left and on the right by elements of A,and q-generalizes the space of 1-forms on a Lie group;ii) the possibility of expressing any ρ ∈Γ asρ = akdbk(4.2)7

for some ak, bk belonging to A.Left- and right-covarianceThe first-order differential calculus (Γ, d) is said to be left- and right-covariantif we can consistently define a left and right action of the q-group on Γ as follows∆L(adb) = ∆(a)(id ⊗d)∆(b),∆L : Γ →A ⊗Γ(leftcovariance)(4.3)∆R(adb) = ∆(a)(d ⊗id)∆(b),∆R : Γ →Γ ⊗A(rightcovariance) (4.4)How can we understand these left and right actions on Γ in the q →1 limit ?The first observation is that the coproduct ∆on A is directly related, for q = 1, tothe pullback induced by left multiplication of the group on itselfLxy ≡xy,∀x, y ∈G. (4.5)This induces the left action (pullback) L∗x on the functions on G:L∗xf(y) ≡f(xy)|y,L∗x : Fun(G) →Fun(G)(4.6)where f(xy)|y means f(xy) seen as a function of y.

Let us introduce the mappingL∗defined by(L∗f)(x, y) ≡(L∗xf)(y) = f(xy)|yL∗: Fun(G) →Fun(G × G) ≈Fun(G) ⊗Fun(G). (4.7)The coproduct ∆on A, when q = 1, reduces to the mapping L∗.

Indeed, consideringT ab(y) as a function on G, we have:L∗(T ab)(x, y) = L∗xT ab(y) = T ab(xy) = T ac(x)T cb(y),(4.8)since T ab is a representation of G. ThereforeL∗(T ab) = T ac ⊗T cb(4.9)and L∗is seen to coincide with ∆, cf. (2.15).The pullback L∗x can also be defined on 1-forms ρ as(L∗xρ)(y) ≡ρ(xy)|y(4.10)and here too we can define L∗as(L∗ρ)(x, y) ≡(L∗xρ)(y) = ρ(xy)|y.

(4.11)In the q = 1 case we are now discussing, the left action ∆L coincides with thismapping L∗for 1-forms. Indeed for q = 1∆L(adb)(x, y) = [∆(a)(id ⊗d)∆(b)](x, y) = [(a1 ⊗a2)(id ⊗d)(b1 ⊗b2)](x, y)= [a1b1 ⊗a2db2](x, y) = a1(x)b1(x)a2(y)db2(y) = a1(x)a2(y)dy[b1(x)b2(y)]= L∗(a)(x, y)dy[L∗(b)(x, y)] = a(xy)db(xy)|y.

(4.12)8

On the other hand:L∗(adb)(x, y) = a(xy)db(xy)|y,(4.13)so that ∆L →L∗when q →1. In the last equation we have used the well-knownproperty L∗x(adb) = L∗x(a)L∗x(db) = L∗x(a)dL∗x(b) of the classical pullback.

A similardiscussion holds for ∆R, and we have ∆R →R∗when q →1 , where R∗is definedvia the pullback R∗x on functions (0-forms) or on 1-forms induced by the rightmultiplication:Rxy = yx,∀x, y ∈G(4.14)(R∗xρ)(y) = ρ(yx)|y(4.15)(R∗ρ)(y, x) ≡(R∗xρ)(y). (4.16)These observations explain why ∆L and ∆R are called left and right actions of thequantum group on Γ when q ̸= 1.From the definitions (4.3) and (4.4) one deduces the following properties [5]:(ε ⊗id)∆L(ρ) = ρ,(id ⊗ε)∆R(ρ) = ρ(4.17)(∆⊗id)∆L = (id ⊗∆L)∆L,(id ⊗∆)∆R = (∆R ⊗id)∆R(4.18)BicovarianceThe left- and right-covariant calculus is said to be bicovariant when(id ⊗∆R)∆L = (∆L ⊗id)∆R,(4.19)which is the q-analogue of the fact that left and right actions commute for q = 1(L∗xR∗y = R∗yL∗x).Left- and right-invariant ωAn element ω of Γ is said to be left-invariant if∆L(ω) = I ⊗ω(4.20)and right-invariant if∆R(ω) = ω ⊗I.

(4.21)This terminology is easily understood: in the classical limit,L∗ω = I ⊗ω(4.22)R∗ω = ω ⊗I(4.23)indeed define respectively left- and right-invariant 1-forms.9

Proof: the classical definition of left-invariance is(L∗xω)(y) = ω(y)(4.24)or, in terms of L∗,(L∗ω)(x, y) = L∗xω(y) = ω(y). (4.25)But(I ⊗ω)(x, y) = I(x)ω(y) = ω(y),(4.26)so thatL∗ω = I ⊗ω(4.27)for left-invariant ω.

A similar argument holds for right-invariant ω.ConsequencesFor any bicovariant first-order calculus one can prove the following [5]:i) Any ρ ∈Γ can be uniquely written in the form:ρ = aiωi(4.28)ρ = ωibi(4.29)with ai, bi ∈A, and ωi a basis of invΓ, the linear subspace of all left-invariantelements of Γ. Thus, as in the classical case, the whole of Γ is generated by a basisof left invariant ωi.

An analogous theorem holds with a basis of right invariantelements ηi ∈Γinv. Note that in the quantum case we have aωi ̸= ωia in general,the bimodule structure of Γ being non-trivial for q ̸= 1.ii) There exist linear functionals f ij on A such thatωib = (f ij ∗b)ωj ≡(id ⊗f ij)∆(b)ωj(4.30)aωi = ωj[(f ij ◦κ−1) ∗a](4.31)for any a, b ∈A.

In particular,ωiT ab = (id ⊗f ij)(T ac ⊗T cb)ωj = T acf ij(T cb)ωj. (4.32)Once we have the functionals f ij, we know how to commute elements of A throughelements of Γ.

The f ij are uniquely determined by (4.30) and for consistency mustsatisfy the conditions:f ij(ab) = f ik(a)f kj(b)(4.33)f ij(I) = δij(4.34)(f kj ◦κ)f ji = δki ε;f kj(f ji ◦κ) = δki ε,(4.35)10

so that their coproduct, counit and coinverse are given by:∆′(f ij) = f ik ⊗f kj(4.36)ε′(f ij) = δij(4.37)κ′(f kj)f ji = δki ε = f kjκ′(f ji)(4.38)cf. (2.24)-(2.26).Note that in the q = 1 limit f ij →δijε, i.e.f ij becomesproportional to the identity functional ε(a) = a(e), and formulas (4.30), (4.31)become trivial, e.g.

ωib = bωi [use ε ∗a = a from (2.5)].iii) There exists an adjoint representation M ijof the quantum group, definedby the right action on the (left-invariant) ωi:∆R(ωi) = ωj ⊗M ij ,M ij ∈A. (4.39)It is easy to show that ∆R(ωi) belongs to invΓ ⊗A, which proves the existence ofM ij .

In the classical case, M ijis indeed the adjoint representation of the group.We recall that in this limit the left-invariant 1-form ωi can be constructed asωi(y)Ti = (y−1dy)iTi,y ∈G. (4.40)Under right multiplication by a (constant) element x ∈G : y →yx we have, 2ωi(yx)Ti = [x−1y−1d(yx)]iTi = [x−1(y−1dy)x]iTi(4.41)= [x−1Tjx]i(y−1dy)jTi = M ij (x)ωj(y)Ti,(4.42)so thatωi(yx) = ωj(y)M ij (x)(4.43)orR∗ωi(y, x) = ωj ⊗M ij (y, x),(4.44)which reproduces (4.39) for q = 1.The co-structures on the M ij can be deduced [5]:∆(M ij ) = M lj ⊗M il(4.45)ε(M ij ) = δij(4.46)κ(M li )M jl= δji = M li κ(M jl ).

(4.47)For example, in order to find the coproduct (4.45) it is sufficient to apply (id ⊗∆)to both members of (4.39) and use the second of eqs. (4.18).The elements M ij can be used to build a right-invariant basis of Γ.

Indeed theηi defined byηi ≡ωjκ(M ij )(4.48)2Recall the q = 1 definition of the adjoint representation x−1Tjx ≡M ij (x)Ti.11

are a basis of Γ (every element of Γ can be uniquely written as ρ = ηibi) and theirright-invariance can be checked directly:∆R(ηi) = ∆R(ωj)∆[κ(M ij )] =[ωk ⊗M jk ][κ(M is ) ⊗κ(M sj )] = ωkκ(M is ) ⊗δskI = ηi ⊗I(4.49)It can be shown that the functionals f ij previously defined satisfy:ηib = (b ∗f ij ◦κ−2)ηj(4.50)aηi = ηj[a ∗(f ij ◦κ−1)],(4.51)where a ∗f ≡(f ⊗id)∆(a), f ∈A′.Moreover, from the last of these relations, using (4.48) and (4.31) one can provethe relationM ji (a ∗f ik) = (f ji ∗a)M ik ,(4.52)with a ∗f ij ≡(f ik ⊗id)∆(a).iv) An exterior product, compatible with the left and right actions of the q-group, can be defined by a bimodule automorphism Λ in Γ ⊗Γ that generalizes theordinary permutation operator:Λ(ωi ⊗ηj) = ηj ⊗ωi,(4.53)where ωi and ηj are respectively left and right invariant elements of Γ. Bimoduleautomorphism means thatΛ(aτ) = aΛ(τ)(4.54)Λ(τb) = Λ(τ)b(4.55)for any τ ∈Γ ⊗Γ and a, b ∈A. The tensor product between elements ρ, ρ′ ∈Γis defined to have the properties ρa ⊗ρ′ = ρ ⊗aρ′, a(ρ ⊗ρ′) = (aρ) ⊗ρ′ and(ρ ⊗ρ′)a = ρ ⊗(ρ′a).

Left and right actions on Γ ⊗Γ are defined by:∆L(ρ ⊗ρ′) ≡ρ1ρ′1 ⊗ρ2 ⊗ρ′2,∆L : Γ ⊗Γ →A ⊗Γ ⊗Γ(4.56)∆R(ρ ⊗ρ′) ≡ρ1 ⊗ρ′1 ⊗ρ2ρ′2,∆R : Γ ⊗Γ →Γ ⊗Γ ⊗A(4.57)where as usual ρ1, ρ2, etc., are defined by∆L(ρ) = ρ1 ⊗ρ2,ρ1 ∈A, ρ2 ∈Γ(4.58)∆R(ρ) = ρ1 ⊗ρ2,ρ1 ∈Γ, ρ2 ∈A. (4.59)More generally, we can define the action of ∆L on Γ ⊗Γ ⊗· · · ⊗Γ as∆L(ρ ⊗ρ′ ⊗· · · ⊗ρ′′) ≡ρ1ρ′1 · · · ρ′′1 ⊗ρ2 ⊗ρ′2 ⊗· · · ⊗ρ′′2∆L : Γ ⊗Γ ⊗· · · ⊗Γ →A ⊗Γ ⊗Γ ⊗· · · ⊗Γ(4.60)12

∆R(ρ ⊗ρ′ ⊗· · · ⊗ρ′′) ≡ρ1 ⊗ρ′1 · · · ⊗ρ′′1 ⊗ρ2ρ′2 · · · ρ′′2∆R : Γ ⊗Γ ⊗· · · ⊗Γ →Γ ⊗Γ ⊗· · · ⊗Γ ⊗A. (4.61)Left-invariance on Γ ⊗Γ is naturally defined as ∆L(ρ ⊗ρ′) = I ⊗ρ ⊗ρ′ (similardefinition for right-invariance), so that, for example, ωi ⊗ωj is left-invariant, andis in fact a left-invariant basis for Γ ⊗Γ.– In general Λ2 ̸= 1, since Λ(ηj ⊗ωi) is not necessarily equal to ωi ⊗ηj.

Bylinearity, Λ can be extended to the whole of Γ ⊗Γ.– Λ is invertible and commutes with the left and right action of q-group G,i.e. ∆LΛ(ρ ⊗ρ′) = (id ⊗Λ)∆L(ρ ⊗ρ′) = ρ1ρ′1 ⊗Λ(ρ2 ⊗ρ′2), and similar for ∆R.Then we see that Λ(ωi ⊗ωj) is left-invariant, and therefore can be expanded on theleft-invariant basis ωk ⊗ωl:Λ(ωi ⊗ωj) = Λijklωk ⊗ωl.

(4.62)– From the definition (4.53) one can prove that [5]:Λijkl = f il(M jk );(4.63)thus the functionals f il and the elements M jk ∈A characterizing the bimodule Γare dual in the sense of eq. (4.63) and determine the exterior product:ρ ∧ρ′ ≡ρ ⊗ρ′ −Λ(ρ ⊗ρ′)(4.64)ωi ∧ωj ≡ωi ⊗ωj −Λijklωk ⊗ωl.

(4.65)Notice that, given the tensor Λijkl, we can compute the exterior product of anyρ, ρ′ ∈Γ, since any ρ ∈Γ is expressible in terms of ωi [cf. (4.28), (4.29)].

Theclassical limit of Λijkl isΛijklq→1−→δilδjk(4.66)since f ijq→1−→δilε and ε(M kj ) = δjk. Thus in the q = 1 limit the product defined in(4.65) coincides with the usual exterior product.From the property (4.54) applied to the case τ = ωi ⊗ωj, one can derive therelationΛnmijf ipf jq = f nif mjΛijpq.

(4.67)Applying both members of this equation to the element M sryields the quantumYang–Baxter equation for Λ:ΛnmijΛikrpΛjskq = ΛnkriΛmskjΛijpq,(4.68)which is sufficient for the consistency of (4.67).Taking a = M qp in (4.52), and using (4.63), we find the relationM ji M qr Λirpk = ΛjqriM rp M ik(4.69)13

and, definingRjikl ≡Λijkl,(4.70)we see that M jisatisfies a relation identical to the “RTT” equation (3.1) for T ab,and that Rijkl satisfies the quantum Yang–Baxter equation (3.3), sufficient for theconsistency of (4.69). The range of the indices is different, since i, j, ... are adjointindices whereas a, b ... are in the fundamental representation of Gq.v) Having the exterior product we can define the exterior differentiald : Γ →Γ ∧Γ(4.71)d(akdbk) = dak ∧dbk,(4.72)which can easily be extended to Γ∧n (d : Γ∧n →Γ∧(n+1), Γ∧n being defined as inthe classical case but with the quantum permutation operator Λ [5]) and has thefollowing properties:d(θ ∧θ′) = dθ ∧θ′ + (−1)kθ ∧dθ′(4.73)d(dθ) = 0(4.74)∆L(dθ) = (id ⊗d)∆L(θ)(4.75)∆R(dθ) = (d ⊗id)∆R(θ),(4.76)where θ ∈Γ∧k, θ′ ∈Γ∧n.

The last two properties express the fact that d commuteswith the left and right action of the quantum group, as in the classical case.vi) The space dual to the left-invariant subspace invΓ can be introduced as alinear subspace of A′, whose basis elements χi ∈A′ are defined byda = (χi ∗a)ωi,∀a ∈A. (4.77)In order to reproduce the classical limitda =∂∂yµa(y)dyµ = ∂∂yµa!eµi(y)eiν(y)dyν = ∂∂yµa!eµi(y)ωi(y),(4.78)where eiν(y) is the vielbein of the group manifold (and eµi is its inverse), we mustrequireχi(a)q→1−→∂∂xi a(x)|x=e.

(4.79)Indeed, for q = 1 we have(χi ∗a)(y) = (id ⊗χi)L∗(a)(y) = (id ⊗χi)(a1 ⊗a2)(y) = a1(y)χi(a2) =a1(y)[ ∂∂xia2(x)]|x=e =∂∂xi [a1(y)a2(x)]|x=e = ∂∂xi [(L∗a)(y, x)]|x=e =∂∂xi [a(yx)]|x=e =∂∂(yx)µa(yx)|x=e∂∂xi (yx)µ|x=e = ( ∂∂yµa(y))eµi(y) (4.80)14

and we recover (4.78). In other wordsχi ∗aq→1−→∂∂yµa(y)eµi ≡∂ia(y),(4.81)so that χi∗is the q-analogue of left-invariant vector fields, while χi is the q-analogueof the tangent vector at the origin e of G.vii) The χi functionals close on the “quantum Lie algebra”:χiχj −Λklijχkχl = Ckijχk,(4.82)with Λklij as given in (4.63).

The product χiχj is defined byχiχj ≡(χi ⊗χj)∆(4.83)and sometimes indicated by χi∗χj. Note that this ∗product (called also convolutionproduct) is associative:χi ∗(χj ∗χk) = (χi ∗χj) ∗χk(4.84)χi ∗(χj ∗a) = (χi ∗χj) ∗a,a ∈A.

(4.85)We leave the easy proof to the reader. The q-structure constants Ckijare given byCkij= χj(M ki ).

(4.86)This last equation is easily seen to hold in the q = 1 limit, since the (χj) ki ≡Ckijare indeed in this case the infinitesimal generators of the adjoint representation:M ki= δki + Ckijxj + 0(x2). (4.87)Using χjq→1−→∂∂xj |x=e indeed yields (4.86).By applying both sides of (4.82) to M sr ∈A, we find the q-Jacobi identities:CnriCsnj−ΛklijCnrk Csnl= CkijCsrk ,(4.88)which give an explicit matrix realization (the adjoint representation) of the gener-ators χi:(χi) lk = χi(M lk ) = Clki .

(4.89)Note that the q-Jacobi identities (4.88) can also be given in terms of the q-Liealgebra generators χi as :[[χr, χi], χj] −Λklij[[χr, χk], χl] = [χr, [χi, χj]],(4.90)where[χi, χj] ≡χiχj −Λklijχkχl(4.91)is the deformed commutator of eq. (4.82).15

viii) The left-invariant ωi satisfy the q-analogue of the Cartan-Maurer equations:dωi + Cijk ωj ∧ωk = 0,(4.92)whereCijk≡χjχk(xi)(4.93)χi(xk) ≡δki . (4.94)The xk ∈A defined in (4.94) are called the “coordinates of Gq” and satisfy ε(xi) = 0,which is the q-analogue of the fact that classically they vanish at the origin of G[recall that ε(xi)q→1−→xi(e)].

Such xi can always be found [5]. Note that χi ∗xj isthe q-analogue of the inverse vielbein.The structure constants C satisfy the Jacobi identities obtained by taking theexterior derivative of (4.92):(Cijk Cjrs−Cirj Cjsk )ωr ∧ωs ∧ωk = 0.

(4.95)In the q = 1 limit, ωj ∧ωk becomes antisymmetric in j and k, and we haveCijkq→1−→= 12(χjχk −χkχj)(xi) = 12Cljkχl(xi) = 12Cijk ,(4.96)where Cljkare now the classical structure constants. Thus when q = 1 we haveCijk= 12Cijkand (4.92) reproduces the classical Cartan-Maurer equations.For q ̸= 1, we find the following relation:Cijk= Cijk−ΛrsjkCirs(4.97)after applying both members of eq.

(4.82) to xi.Note that, using (4.97), theCartan-Maurer equations (4.92) can also be written as:dωi + Cijk ωj ⊗ωk = 0. (4.98)ix) Finally, we derive two operatorial identities that become trivial in the limitq →1.

From the formulad(h ∗θ) = h ∗dθ,h ∈A′, θ ∈Γ∧n(4.99)[a direct consequence of (4.76)] with h = f nl, we findχkf nl = Λijklf niχj(4.100)By requiring consistency between the external derivative and the bimodule struc-ture of Γ, i.e. requiring thatd(ωia) = d[(f ij ∗a)ωj],(4.101)16

one finds the identityCimn f mjf nk + f ijχk = Λpqjkχpf iq + Cljk f il. (4.102)See Appendix A for the derivation of (4.100) and (4.102).In summary, a bicovariant calculus on a Gq is characterized by functionals χiand f ij on A (“the algebra of functions on the quantum group”) satisfyingχiχj −Λklijχkχl = Ckijχk(4.103)Λnmijf ipf jq = f nif mjΛijpq(4.104)Cimn f mjf nk + f ijχk = Λpqjkχpf iq + Cljk f il(4.105)χkf nl = Λijklf niχj,(4.106)where the q-structure constants are given by Cijk= χk(M ij ) and the braidingmatrix by Λijkl = f il(M jk ).

In fact, these four relations seem to be also suffi-cient to define a bicovariant differential calculus on A (see e.g. [12]).

By applyingthem to the element M sr we express these relations (henceforth called bicovarianceconditions) in the adjoint representation:CnriCsnj−RklijCnrk Csnl= CkijCsrk(q-Jacobi identities) (4.107)ΛnmijΛikrpΛjskq = ΛnkriΛmskjΛijpq(Yang–Baxter)(4.108)Cimn ΛmlrjΛnslk + ΛilrjCslk= ΛpqjkΛilrqCslp+ CmjkΛisrm (4.109)CmrkΛnsml = ΛijklΛnmriCsmj(4.110)We conclude this section by giving the co-structures on the quantum Lie algebragenerators χi [those on the functionals f ij have been given in (4.36)-(4.38)]:∆′(χi) = χj ⊗f ji + I′ ⊗χi(4.111)ε′(χi) = 0(4.112)κ′(χi) = −χjκ′(f ji),(4.113)which q-generalize the ones given in (2.19)-(2.21). These co-structures derive from(2.24)-(2.26).

For example, using (4.73) and (4.30), eq. (2.24) yields (4.111).

Theyare consistent with the bicovariance conditions (4.103)-(4.106).In the next section, we describe a constructive procedure due to Jurˇco [17] for abicovariant differential calculus on any q-group of the A, B, C, D series consideredin [3]. The procedure is illustrated on the example of GLq(2), for which all theobjects f ij, M sr , Λijkl, Cijkand Cijkare explicitly computed.17

5Constructive procedure and the example of GLq(2)The generic q-group discussed in Section 3 is characterized by the matrix Rabcd. Interms of this matrix, it is possible to construct a bicovariant differential calculus onthe q-group.

The general procedure is described in this section, and the results forthe specific case of GLq(2) are collected in the table.The L± functionalsWe start by introducing the linear functionals (L±)ab, defined by their value onthe elements T ab:(L±)ab(T cd) = (R±)acbd,(5.1)where(R+)acbd ≡c+Rcadb(5.2)(R−)acbd ≡c−(R−1)acbd,(5.3)where c+, c−are free parameters (see later). The inverse matrix R−1 is defined by(R−1)abcdRcdef ≡δaeδbf ≡Rabcd(R−1)cdef.

(5.4)We see that the (L±)ab functionals are dual to the T ab elements (fundamentalrepresentation) in the same way the f ij functionals are dual to the M jielementsof the adjoint representation. To extend the definition (5.1) to the whole algebraA, we set:(L±)ab(ab) = (L±)ag(a)(L±)gb(b),∀a, b ∈A(5.5)so that, for example,(L±)ab(T cdT ef) = (R±)acgd(R±)gebf.

(5.6)In general, using the compact notation introduced in Section 3,L±1 (T2T3...Tn) = R±12R±13...R±1n. (5.7)Finally, the value of L± on the unit I is defined by(L±)ab(I) = δab .

(5.8)Thus the functionals (L±)ab have the same properties as their adjoint counterpartf ij, and not surprisingly the latter will be constructed in terms of the former.From (5.7) we can also find the action of (L±)ab on a ∈A, i.e. (L±)ab ∗a.Indeed(L±)ab ∗(T c1d1T c2d2 · · ·T cndn) = [id ⊗(L±)ab]∆(T c1d1T c2d2 · · · T cndn) =[id ⊗(L±)ab]∆(T c1d1) · · ·∆(T cndn) =[id ⊗(L±)ab](T c1e1 · · · T cnen ⊗T e1d1 · · · T endn)T c1e1 · · · T cnen(L±)ab(T e1d1 · · · T endn) =T c1e1 · · · T cnen(R±)ae1g1d1(R±)g1e2g2d2 · · · (R±)gn−1enbdn(5.9)18

or, more compactly,L±1 ∗T2...Tn = T2...TnR±12R±13...R±1n,(5.10)which can also be written asL±1 ∗T2 = T2R±12L±1 . (5.11)It is not difficult to find the commutations between (L±)ab and (L±)cd:R12L±2 L±1 = L±1 L±2 R12(5.12)R12L+2 L−1 = L−1 L+2 R12,(5.13)where as usual the product L±2 L±1 is the convolution product L±2 L±1 ≡(L±2 ⊗L±1 )∆.ConsiderR12(L+2 L+1 )(T3) = R12(L+2 ⊗L+1 )∆(T3) = R12(L+2 ⊗L+1 )(T3 ⊗T3) = (c+)2 R12R32R31(5.14)andL+1 L+2 (T3)R12 = (c+)2 R31R32R12(5.15)so that the equation (5.12) is proven for L+ by virtue of the quantum Yang–Baxterequation (3.3), where the indices have been renamed 2 →1, 3 →2, 1 →3.

Similarly,one proves the remaining “RLL” relations.Note 1 : As mentioned in [3], L+ is upper triangular, L−is lower triangular(this is due to the upper and lower triangularity of R+ and R−, respectively).Note 2 : When detq T = 1, we have detq−1L± = (L±)11(L±)22 · · · (L±)nn = ε,and (L+)ii(L−)jj = (L−)jj(L+)ii (no sum on repeated indices). Then (detq−1L±)(detq T) =detq R± implies detR± = 1, which requires c+ = q−1/n, c−= q1/n for the An−1 se-ries (and c± = 1 for the remaining B, C, D series) in (5.2) and (5.3).

In the moregeneral case of GLq(n), c± are extra free parameters, cf. [20].

In fact, they appearonly in the combination s = (c+)−1c−. They do not enter in the Λ matrix, nor inthe structure constants or the Cartan-Maurer equations (see the table).

Differentvalues of s lead to isomorphic differential calculi (in the sense of ref. [5]), so that sis not really an essential parameter.The co-structures are defined by the duality (5.1):∆′((L±)ab)(T cd ⊗T ef) ≡(L±)ab(T cdT ef) = (L±)ag(T cd)(L±)gb(T ef)(5.16)ε′((L±)ab) ≡(L±)ab(I)(5.17)κ′((L±)ab)(T cd) ≡(L±)ab(κ(T cd))(5.18)19

cf. [(2.24)-(2.26)], so that∆′((L±)ab) = (L±)ag ⊗(L±)gb(5.19)ε′((L±)ab) = δab(5.20)κ′((L±)ab) = (L±)ab ◦κ(5.21)and the (L±)ab generate the Hopf algebra dual to the quantum group.

Note that(L±)ab(κ(T cd)) = ((R±)−1)acbd,(5.22)since(L±)ab(κ(T cd)T de) = δcd(L±)ab(I) = δceδab(5.23)and(L±)ab(κ(T cd)T de)=(L±)af(κ(T cd))(L±)fb(T de)=(L±)af(κ(T cd))(R±)fdbe. (5.24)The space of quantum 1-formsThe bimodule Γ (“space of quantum 1-forms”) can be constructed as follows.First we define ω ba to be a basis of left-invariant quantum 1-forms.

The index pairsba or ab will replace in the sequel the indices i or i of the previous section. Thedimension of invΓ is therefore N2 at this stage.

The existence of this basis can beproven by considering Γ to be the tensor product of two fundamental bimodules,see refs. [17, 16].

Here we just assume that it exists. Since the ω ba are left-invariant,we have:∆L(ω ba ) = I ⊗ω ba ,a, b = 1, ..., N.(5.25)The left action ∆L on the whole of Γ is then defined by (5.25), since ω ba is a basisfor Γ.

The bimodule Γ is further characterized by the commutations between ω baand a ∈A [cf. eq.

(4.30)]:ω a2a1 b = (f a2b1a1b2 ∗b)ω b2b1 ,(5.26)wheref a2b1a1b2 ≡κ′((L+)b1a1)(L−)a2b2. (5.27)Finally, the right action ∆R on Γ is defined by∆R(ω a2a1 ) = ω b2b1 ⊗Mb1a2b2a1,(5.28)where Mb1a2b2a1, the adjoint representation, is given byMb1a2b2a1≡T b1a1κ(T a2b2).

(5.29)20

It is easy to check that f a2b1a1b2 fulfill the consistency conditions (4.33)-(4.35), wherethe i,j,... indices stand for pairs of a,b,...indices. Also, the co-structures are asgiven in (4.36)-(4.38).The Λ tensor and the exterior productThe Λ tensor defined in (4.70) can now be computed:Λ a2 d2a1 d1 |c1 b1c2 b2 ≡f a2b1a1b2(Mc1d2c2d1) = κ′((L+)b1a1)(L−)a2b2(T c1d1κ(T d2c2))=[κ′((L+)b1a1) ⊗(L−)a2b2]∆(T c1d1κ(T d2c2))=[κ′((L+)b1a1) ⊗(L−)a2b2](T c1e1 ⊗T e1d1)(κ(T f2c2) ⊗κ(T d2f2))=[κ′((L+)b1a1) ⊗(L−)a2b2][T c1e1κ(T f2c2) ⊗T e1d1κ(T d2f2)]=(L+)b1a1(κ2(T f2c2)κ(T c1e1)) (L−)a2b2(T e1d1κ(T d2f2))=df2d−1c2 (L+)b1a1(T f2c2κ(T c1e1)) (L−)a2b2(T e1d1κ(T d2f2)=df2d−1c2 (L+)b1g1(T f2c2) (L+)g1a1(κ(T c1e1)) (L−)a2g2(T e1d1) (L−)g2b2(κ(T d2f2))=df2d−1c2 Rf2b1c2g1(R−1)c1g1e1a1(R−1)a2e1g2d1Rg2d2b2f2(5.30)where we made use of relations (2.12), (2.26), (3.15), (5.1) and (5.22).

The Λ tensorallows the definition of the exterior product as in (4.65). For future use we givehere also the inverse Λ−1 of the Λ tensor, defined by:(Λ−1) a2 d2a1 d1 |b1 c1b2 c2Λ b2 c2b1 c1 |e1 f1e2 f2 = δa2e2 δe1a1δf1d1δd2f2.

(5.31)It is not difficult to see that(Λ−1) a2 d2a1 d1 |b1 c1b2 c2 = f d2b1d1b2(T a2c2κ−1(T c1a1)) =Rf1b1a1g1(R−1)a2g1e2d1(R−1)d2e2g2c2Rg2c1b2f1(d−1)c1df1(5.32)does the trick. Another useful relation gives a particular trace of the Λ matrix:Λ c2 bc1 b |a1 b1a2 b2 = δa1a2δb1c1δc2b2.

(5.33)This identity is simply proven. Indeed:Λ c2 bc1 b |a1 b1a2 b2 ≡f c2b1c1b2(Ma1ba2b) =κ′((L+)b1c1)(L−)c2b2(T a1bκ(T ba2)) = κ′((L+)b1c1)(L−)c2b2(δa1a2I) =δa1a2[κ′((L+)b1c1) ⊗(L−)c2b2](I ⊗I) = δa1a2δb1c1δc2b2.

(5.34)The relations (3.17), (3.18) for the R matrix reflect themselves in relations forthe Λ matrix (5.30). For example, the Hecke condition (3.17) implies:(Λ + q2)(Λ + q−2)(Λ −I) = 0(5.35)21

for the An−1 q-groups, and replaces the classical relation (Λ −1)(Λ + 1) = 0, Λbeing for q = 1 the ordinary permutation operator, cf. (4.66).With the help of (5.35) we can give explicitly the commutations of the left-invariant forms ω.

Indeed, reverting to the i,j... indices, relation (5.35) implies:(Λijkl + q2δikδjl )(Λklmn + q−2δkmδln)(Λmnrs −δmr δns )ωr ⊗ωs =(Λijkl + q2δikδjl )(Λklmn + q−2δkmδln)ωm ∧ωn = 0(5.36)and it is easy to see that the last equality can be rewritten asωi ∧ωj = −Zijklωk ∧ωl(5.37)Zijkl ≡1q2 + q−2[Λijkl + (Λ−1)ijkl]. (5.38)The exterior differentialThe exterior differential on Γ∧k is defined by means of the bi-invariant (i.e.

left-and right-invariant) element τ =Pa ω aa ∈Γ as follows:dθ ≡1λ[τ ∧θ −(−1)kθ ∧τ],(5.39)where θ ∈Γ∧k, and λ is a normalization factor depending on q, necessary in orderto obtain the correct classical limit. It will be later determined to be λ = q −q−1.Here we can only see that it has to vanish for q = 1, since otherwise dθ would vanishin the classical limit.

For a ∈A we haveda = 1λ[τa −aτ]. (5.40)This linear map satisfies the Leibniz rule (4.1), and properties (4.73)-(4.76), as thereader can easily check (use the definition of exterior product and the bi-invarianceof τ).

A proof that also the property (4.2) holds can be obtained by consideringthe exterior differential of the adjoint representation:dM ij = (χk ∗M ij )ωk = M lj Cikl ωk(5.41)orκ(M jl )dM ij = Cikl ωk. (5.42)Multiplying by Clni , we have:Clni κ(M jl )dM ij = Cikl Clni ωk ≡gnkωk,(5.43)where gnk is the q-Killing metric.

The explicit example of this paper being GLq(2),one may wonder what happens to the invertibility of the q-Killing metric, sinceits classical limit is no more invertible [GL(2) being nonsemisimple]. The answer22

is that for q ̸= 1 the q-Killing metric of GLq(2) is invertible, as can be checkedexplicitly from the values of the structure constants given in the table. ThereforeGLq(2) could be said to be “q-semisimple”.

With an analogous procedure (usingT ab instead of M ij ) we have derived in the table the explicit expression of the ωiin terms of the dT ab for GLq(2).The q-Lie algebraThe “quantum generators” χa1a2 are introduced as in (4.77):da = 1λ[τa −aτ] = (χa1a2 ∗a)ω a2a1 . (5.44)Using (5.26) we can find an explicit expression for the χa1a2 in terms of the L±functionals.

Indeedτa = ω bb a = (f bc1bc2 ∗a)ω c2c1 = ([κ′((L+)c1b)(L−)bc2] ∗a)ω c2c1 . (5.45)Thereforeda = 1λ[(κ′((L+)c1b)(L−)bc2 −δc1c2ε) ∗a]ω c2c1(5.46)(recall ε ∗a = a), so that the q-generators take the explicit formχc1c2 = 1λ[κ′((L+)c1b)(L−)bc2 −δc1c2ε].

(5.47)The commutations between the χ’s can now be obtained by taking the exteriorderivative of eq. (5.46).

We findd2(a) = 0 = d[(χc1c2 ∗a)ω c2c1 ] = (χd1d2 ∗χc1c2 ∗a)ω d2d1 ∧ω c2c1 + (χc1c2 ∗a)dω c2c1= (χd1d2 ∗χc1c2 ∗a)(ω d2d1 ⊗ω c2c1 −Λ d2 c2d1 c1 |e1 f1e2 f2ω e2e1 ⊗ω f2f1 )+1λ(χc1c2 ∗a)(ω bb ∧ω c2c1 + ω c2c1 ∧ω bb ). (5.48)Now we use the fact that τ = ω bb is bi-invariant, and therefore also right-invariant,so that we can writeω bb ∧ω c2c1 + ω c2c1 ∧ω bb ≡ω bb ⊗ω c2c1 −Λ(ω bb ⊗ω c2c1 ) + ω c2c1 ⊗ω bb −Λ(ω c2c1 ⊗ω bb ) =ω c2c1 ⊗ω bb −Λ(ω bb ⊗ω c2c1 ) =ω c2c1 ⊗ω bb −Λ b c2b c1 |e1 f1e2 f2ω e2e1 ⊗ω f2f1 ,(5.49)where we have used Λ(ω c2c1 ⊗τ) = τ ⊗ω c2c1 , cf.

(4.53). After substituting (5.49) in(5.48), and factorizing ω d2d1 ⊗ω c2c1 , we arrive at the q-Lie algebra relations:χd1d2χc1c2 −Λ e2 f2e1 f1 |d1 c1d2 c2 χe1e2χf1f2 = 1λ[−δc1c2χd1d2 + Λ b e2b e1 |d1 c1d2 c2 χe1e2].

(5.50)23

The structure constants are then explicitly given by:Ca1 b1a2 b2| c2c1 = 1λ[−δb1b2δa1c1 δc2a2 + Λ b c2b c1 |a1 b1a2 b2]. (5.51)Here we determine λ.

Indeed we first observe thatΛ a2 d2a1 d1 |c1 b1c2 b2 = δb1a1δa2b2 δc1d1δd2c2 + (q −q−1)U a2 d2a1 d1 |c1 b1c2 b2,(5.52)where the matrix U is finite and different from zero in the limit q = 1. This can beproven by considering the explicit form of the R and R−1 matrices.

In the case ofthe An−1 q-groups, for example, these matrices have the form [3]:Rabcd = δac δbd + (q −q−1)" q −1q −q−1δac δbdδab + δbcδadθ(a −b)#(5.53)(R−1)abcd = δac δbd −(q −q−1)"1 −q−1q −q−1δacδbdδab + δbcδadθ(a −b)#,(5.54)where θ(x) = 1 for x > 0 and vanishes for x ≤0. Substituting these expressionsin the formula for Λ (5.30) we find (5.52).

Using (5.52) in the expression (5.51) forthe q-structure constants C, we find that the terms proportional to1λ do cancel,and we are left with:Ca1 b1a2 b2| c2c1 = −1λ(q −q−1)U b c2b c1 |a1 b1a2 b2. (5.55)A simple choice for λ is therefore λ = q −q−1, ensuring that C remains finite in thelimit q →1 .The Cartan-Maurer equationsThe Cartan-Maurer equations are found as follows:dω c2c1 = 1λ(ω bb ∧ω c2c1 + ω c2c1 ∧ω bb ) ≡−Ca1 b1a2 b2| c2c1 ω a2a1 ∧ω b2b1 .

(5.56)In order to obtain an explicit expression for the C structure constants in (5.56), wemust use the relation (5.37) for the commutations of ω a2a1 with ω b2b1 . Then the termω c2c1 ∧ω bbin (5.56) can be written as −Zωω via formula (5.37), and we find theC-structure constants to be:Ca1 b1a2 b2| c2c1=−1λ(δa1a2δb1c1δc2b2 −1q2 + q−2[Λ c2 bc1 b |a1 b1a2 b2 + (Λ−1) c2 bc1 b |a1 b1a2 b2])=−1λ(δa1a2δb1c1δc2b2 −1q2 + q−2[δa1a2δb1c1δc2b2 + (Λ−1) c2 bc1 b |a1 b1a2 b2]),(5.57)where we have also used eq.

(5.33). By considering the analogue of (5.52) for Λ−1,it is not difficult to see that the terms proportional to 1λ cancel, and the q →1 limit24

of (5.57) is well defined. For a more detailed discussion, including also the Bn, Cnand Dn q-groups, we refer to [22].In the table we summarize the results of this section for the case of GLq(2).

Thecomposite indicesba are translated into the corresponding indices i, i = 1, +, −, 2,according to the convention:11 →1,21 →+,12 →−,22 →2. (5.58)A similar convention holds for ab →i.25

TableThe bicovariant GLq(2) algebraR and D-matrices:Rabcd =q00001000q −q−110000q(R−)abcd ≡c−(R−1)abcd = c−q−100001000−(q −q−1)10000q−1(R+)abcd ≡c+Rbadc = c+q00001q −q−100010000q,Dab = q00q3!Non-vanishing components of the Λ matrix:Λ1111 = 1Λ1++1 = q−2Λ1−−1 = q2Λ1221 = 1Λ+11+ = 1Λ+1+1 = 1 −q−2Λ++++ = 1Λ+−11 = 1 −q2Λ+−−+ = 1Λ+−21 = 1 −q−2Λ+2+1 = −1 + q−2Λ+22+ = 1Λ−11−= 1Λ−1−1 = 1 −q2Λ−+11 = −1 + q2Λ−++−= 1Λ−+21 = −1 + q−2Λ−−−−= 1Λ−2−1 = −1 + q2Λ−22−= 1Λ2111 = (q2 −1)2Λ2112 = 1Λ21+−= q2 −1Λ21−+ = 1 −q2Λ2121 = 2 −q2 −q−2Λ2+1+ = −q2 + q4Λ2++2 = q2Λ2+2+ = 1 −q2Λ2−1−= 1 −q2Λ2−−1 = q−2 −1 −q2 + q4Λ2−−2 = q−2Λ2−2−= 1 −q−2Λ2211 = −(q2 −1)2Λ22+−= 1 −q2Λ22−+ = q2 −1Λ2221 = (q−1 −q)2Λ2222 = 1Non-vanishing components of the C structure constants:C111= q(q2 −1)C211= −q(q2 −1)C+1+= q3C−1−= −qC121= q−1 −qC221= q −q−1C+2+= −qC−2−= q−1C++1= −q−1C++2= qC1+−= qC2+−= −qC−−1= q(q2 + 1) −q−1C−−2= −q−1C1−+ = −qC2−+ = q26

Non-vanishing components of the C structure constants:C111= q(q2−1)21+q4C211= q3(1−q2)1+q4C+1+=q51+q4C−1−=−q31+q4C112= q(1−q2)1+q4C++1=−q31+q4C1+−=q31+q4C2+−=−q31+q4C++2=q1+q4C−−1=q51+q4C1−+ =−q31+q4C2−+ =q31+q4C−−2=−q31+q4C121= q(1−q2)1+q4C+2+=−q31+q4C−2−=q1+q4C222= q(1−q2)1+q4Cartan-Maurer equations:dω1 + qω+ ∧ω−= 0dω+ + qω+(−q2ω1 + ω2) = 0dω−+ q(−q2ω1 + ω2)ω−= 0dω2 −qω+ ∧ω−= 0The q-Lie algebra:χ1χ+ −χ+χ1 −(q4 −q2)χ2χ+ = q3χ+χ1χ−−χ−χ1 + (q2 −1)χ2χ−= −qχ−χ1χ2 −χ2χ1 = 0χ+χ−−χ−χ+ + (1 −q2)χ2χ1 −(1 −q2)χ2χ2 = q(χ1 −χ2)χ+χ2 −q2χ2χ+ = qχ+χ−χ2 −q−2χ2χ−= −q−1χ−Commutation relations between left-invariant ωi and ωj:ω1 ∧ω+ + ω+ ∧ω1 = 0ω1 ∧ω−+ ω−∧ω1 = 0ω1 ∧ω2 + ω2 ∧ω1 = (1 −q2)ω+ ∧ω−ω+ ∧ω−+ ω−∧ω+ = 0ω2 ∧ω+ + q2ω+ ∧ω2 = q2(q2 −1)ω+ ∧ω1ω2 ∧ω−+ q−2ω−∧ω2 = (1 −q2)ω−∧ω1ω2 ∧ω2 = (q2 −1)ω+ ∧ω−ω1 ∧ω1 = ω+ ∧ω+ = ω−∧ω−= 0Commutation relations between ωi and the basic elements of A (s = (c+)−1c−):27

ω1α = sq−2αω1ω+α = sq−1αω+ω1β = sβω1ω+β = sq−1βω+ + s(q−2 −1)αω1ω1γ = sq−2γω1ω+γ = sq−1γω+ω1δ = sδω1ω+δ = sq−1δω+ + s(q−2 −1)γω1ω−α = sq−1αω−+ s(q−2 −1)βω1ω2α = sαω2 + s(q−1 −q)βω+ω−β = sq−1βω−ω2β = sq−2βω2 + s(q−1 −q)αω−+ s(q−1 −q)2βω1ω−γ = sq−1γω−+ s(q−2 −1)δω1ω2γ = sγω2 + s(q−1 −q)δω+ω−δ = sq−1δω−ω2δ = sq−2δω2 + s(q−1 −q)γω−+ s(q−1 −q)2δω1Values and action of the generators on the q-group elements:χ1(α) = s−q2q3−qχ+(α) = 0χ−(α) = 0χ2(α) =s−1q−q−1χ1(β) = 0χ+(β) = 0χ−(β) = −sχ2(β) = 0χ1(γ) = 0χ+(γ) = −sχ−(γ) = 0χ2(γ) = 0χ1(δ) = −q2+s(1−q2+q4)q3−qχ+(δ) = 0χ−(δ) = 0χ2(δ) = s−q2q3−qχ1 ∗α = s−q2q3−q αχ+ ∗α = −sβχ−∗α = 0χ2 ∗α =s−1q−q−1 αχ1 ∗β = −q2+s(1−q2+q4)q3−qβχ+ ∗β = 0χ−∗β = −sαχ2 ∗β = (s−q2)q3−q βχ1 ∗γ = s−q2q3−q γχ+ ∗γ = −sδχ−∗γ = 0χ2 ∗γ =s−1q−q−1 γχ1 ∗δ = −q2+s(1−q2+q4)q3−qδχ+ ∗δ = 0χ−∗δ = −sγχ2 ∗δ = s−q2q3−q δExterior derivatives of the basic elements of A:dα = s−q2q3−qαω1 −sβω+ +s−1q−q−1αω2dβ = −q2+s(1−q2+q4)q3−qβω1 −sαω−+ s−q2q3−qβω2dγ = s−q2q3−qγω1 −sδω+ +s−1q−q−1γω2dδ = −q2+s(1−q2+q4)q3−qδω1 −sγω−+ s−q2q3−qδω2The ωi in terms of the exterior derivatives on α, β, γ, δ:ω1 =qs(−q2−q4+s+sq4)[(q2 −s)(κ(α)da + κ(β)dγ) + q2(s −1)(κ(γ)dβ + κ(δ)dδ)]ω+ = −1s[κ(γ)dα + κ(δ)dγ]ω−= −1s[κ(α)dβ + κ(β)dδ]ω2 =qs(−q2−q4+s+sq4)[(s −q2 −sq2 + sq4)(κ(α)dα + κ(β)dγ) + (q2 −s)(κ(γ)dβ + κ(δ)dδ)]28

Lie derivative on ωi:χ1 ∗ω1 = q(q2 −1)ω1 + (q−1 −q)ω2χ+ ∗ω1 = −qω−χ1 ∗ω+ = −q−1ω+χ+ ∗ω+ = −qω2 + q3ω1χ1 ∗ω−= [q(q2 + 1) −q−1]ω−χ+ ∗ω−= 0χ1 ∗ω2 = −q(q2 −1)ω1 −(q−1 −q)ω2χ+ ∗ω2 = qω−χ−∗ω1 = qω+χ2 ∗ω1 = 0χ−∗ω+ = 0χ2 ∗ω+ = qω+χ−∗ω−= q−1ω2 −qω1χ2 ∗ω−= −q−1ω−χ−∗ω2 = −qω+χ2 ∗ω2 = 029

6More q-geometry: the contraction operator andthe Lie derivativeIn section 4 we have seen that the χi defined by da = (χi ∗a) ωi are the quantumanalogues of the tangent vectors at the origin of the group :χiq→1−→∂∂xi |x=0(6.1)and that the left-invariant vector fields ti constructed from the χi are :ti = χi∗= (id ⊗χi)∆(6.2)tiq→1−→e µi∂∂xµ . (6.3)There is a one-to-one correspondence χi ↔ti = χi∗.

In order to obtain χi from χi∗we simply apply ε :(ε ◦ti)(a) = ε(id ⊗χi)∆(a) = ε(a1χi(a2)) = ε(a1)χi(a2) = χi(ε ⊗id)∆(a) = χi(a)(6.4)[recall (2.5)].Note 1: The vector space T can also be defined intrinsically as the space of alllinear functionals from A to C such that χ(I) = 0 and χ(a) = 0 if da = 0; indeedfrom 0 = da = (χi ∗a) ωi we have χi ∗a = 0 and applying ε we get χi(a) = 0.Note 2: The vector space T is a quantum Lie algebra with Lie bracket [χ, χ′] asgiven in (4.82); the vector space invΞ spanned by the left-invariant vector fields ti isalso a Lie algebra with the induced Lie bracket [t, t′] ≡[χ, χ′] ∗.The ∗product of a functional with any τ ∈Γ⊗n may be defined asχ ∗τ ≡(id ⊗χ)∆R(τ),(6.5)where the ∆R acts on a generic element τ = ρ1 ⊗ρ2 ⊗· · · ρn∈Γ⊗n as in (4.61).DefinitionWe call quantum Lie derivative along the left-invariant vector field t = (id ⊗χ)∆the operator:ℓt ≡χ ∗,(6.6)that isℓt(τ) ≡(id ⊗χ)∆R(τ) = χ ∗τℓt :Γ⊗n −→Γ⊗n .For example:ℓt(a) = t(a),a ∈A,(6.7)30

ℓti(ωj) = (id ⊗χi)∆R(ωj) = ωkχi(Mkj) = Cjki ωk,(6.8)the classical limit being evident.The quantum Lie derivative has properties analogous to that of the ordinary Liederivative:i) it is linear in τ:ℓt(λτ + τ ′) = λℓt(τ) + ℓt(τ ′);(6.9)ii) it is linear in t:ℓλt+t′ = λℓt + ℓt′,λ ∈C. (6.10)By virtue of this last property we can just study ℓti, where {ti} is a basis of invΞ.TheoremThe following relation holds:ℓti(τ ⊗τ ′) = ℓtj(τ) ⊗f ji ∗τ ′ + τ ⊗ℓti(τ ′)(6.11)Proofℓti(τ ⊗τ ′)==(id ⊗χi)∆R(τ ⊗τ ′)=(id ⊗χi)(τ1 ⊗τ ′1 ⊗τ2τ ′2)=(τ1 ⊗τ ′1)χi(τ2τ ′2) = (τ1 ⊗τ ′1)[χj(τ2)f ji (τ ′2) + ε(τ2)χi(τ ′2)]=τ1χj(τ2) ⊗τ ′1f ji (τ ′2) + τ1ε(τ2) ⊗τ ′1χi(τ ′2)=ℓtj(τ) ⊗(id ⊗f ji ) ∗τ ′ + τ ⊗ℓti(τ ′)[remember that χj(a) and f ji(a) are C numbers].

The same argument leads to:ℓti(aωj) = ℓtk(a)(f ki ∗ωj) + aℓti(ωj)(6.12)ℓti(ωja) = ℓtk(ωj)(f ki ∗a) + ωjℓti(a). (6.13)The classical limit of (6.11) is easy to recover if we remember that ε ∗τ = τ.Formulas (6.11), (6.7) and (6.8) uniquely define the quantum ℓt, which reduces, forq →1 , to the classical Lie derivative.Theorem:The Lie derivative commutes with the exterior derivative:ℓti(dϑ) = d(ℓtiϑ),ϑ :generic form.

(6.14)Proof:ℓti(dϑ) = (id ⊗χi)∆R(dϑ) = (id ⊗χi)(d ⊗id))∆R(ϑ) =(d ⊗χi)∆R(ϑ) = dϑ1 χi(ϑ2)| {z }∈C= d[ϑ1χi(ϑ2)] = d(ℓtiϑ),31

where in the second equality we have used property (4.76).Theorem:The Lie derivative commutes with the left and right actions ∆L and ∆R:(id ⊗ℓt)∆L(θ) = ∆L(ℓtθ)(6.15)(id ⊗ℓt)∆R(θ) = ∆R(ℓtθ),θ ∈Γ⊗n. (6.16)The proof is easy and relies on the fact that left and right actions commute, cf.

eq.(4.19). In the classical limit, eq.

(6.15) becomes :ℓt(L∗xθ) = L∗x(ℓtθ). (6.17)Note 3: It is not difficult to prove the associativity of the generalized ∗product,for example that (χ ∗χ′) ∗τ = χ ∗(χ′ ∗τ).

From this property it follows that theq-Lie derivative is a representation of the q-Lie algebra:[ℓt, ℓt′](τ) = ℓ[t,t′](τ),where [ℓt, ℓt′](τ) ≡[χ, χ′] ∗τ.We now come to the construction of the contraction operator it along the left-invariant vector field t.DefinitionThe operator it is characterized by:α)iti(a) = 0a ∈Aβ)iti(ωj) = δji Iγ)iti(ωi1 ∧. .

. ωin) = itj(ωi1)f ji ∗(ωi2 ∧.

. .

ωin) −ωi1 ∧iti(ωi2 ∧. .

. ωin)δ)iti(aϑ + ϑ′) = aiti(ϑ) + iti(ϑ′)ϑ, ϑ′ generic formsε)iλiti = λiitiλi ∈CThese relations uniquely define it, and its existence is ensured by the unicityof the expansion of a generic n-form on a basis of left-invariant 1-forms : ϑ =ai1i2...inωi1 ∧.

. .

ωin.Relation δ) expresses the A-linearity of iti (not just the C-linearity).Relation γ) in the commutative limit reduces to the analog property of the classicalcontraction. This relation can be generalized by substituting ⊗to ∧.32

Theorem:With the above-defined contraction operator it, the Lie derivative can be ex-pressed as:ℓti = itid + diti. (6.18)A proof of this theorem is given in Appendix B, together with the proof of thepropertyiti(ωi1 ∧.

. .

ωin)=itj(ωi1 ∧. .

. ωis) ∧f ji ∗(ωis+1 ∧.

. .

ωin)+(−1)sωi1 ∧. .

. ωis ∧iti(ωis+1 ∧.

. .

ωin),(6.19)where s and n are integers such that 1 ≤s < n.By induction on n one can also prove that(id ⊗it)∆L = ∆Lit(6.20)holds on any n-form. This formula q-generalizes the classical commutativity of itwith the left action ∆L, when t is a left- invariant vector field.33

7Softening the quantum groupAs in the classical case, we may consider the softening of the q-group structure.The idea is to allow the right-hand side of the Cartan-Maurer equations (4.92) tobe nonvanishing, i.e. to consider “deformations” µi of ωi that are no longer left-invariant.

The amount of “deformation” is measured, as in the classical case, by aq-curvature two-form Ri:Ri = dµi + Cijk µj ∧µk(7.1)For this definition to be consistent with d2 = 0, the following q-Bianchi identitiesmust hold:dRi −Cijk Rj ∧µk + Cijk µj ∧Rk = 0;(7.2)these are easily obtained by taking the exterior derivative of (7.1) and using theq-Jacobi identities for the C structure constants given in (4.95).The bimodule structure of the deformed Γ is assumed to be unchanged, i.e.the commutations between elements of A and elements of the deformed Γ are un-changed. Also, the definition (4.65) for the wedge product is still kept unaltered,so that the commutations between the µi are identical to those for the ωi given in(5.37):µi ∧µj = −Zijklµk ∧µl,(7.3)with Z given by (5.38).

Note that by taking the exterior derivative of (7.3) we caninfer the commutations of Ri with µj:Ri ∧µj −µi ∧Rj = −Zijkl(Rk ∧µl −µk ∧Rl). (7.4)Indeed the terms trilinear in µ that arise after using dµ = R−Cµµ do cancel, sincethey cancel in the case Ri = 0, and the wedge products are unaltered.In the constructive procedure of Section 5, we have defined the exterior derivativeto act as:da = 1λ[τa −aτ](7.5)dθ = 1λ[τ ∧θ −(−1)kθ ∧τ](7.6)with θ ∈Γk.

It is now clear that eq. (7.6) must be modified.

Indeed this equation,with τ = µ bb , leads to the Cartan-Maurer equations (5.56), since the commutationsbetween the µi just mimic those between the left-invariant ωi. Then we define theexterior differential as:da = 1λ[sa −as](7.7)dθ = 1λ[s ∧θ −(−1)kθ ∧s],(7.8)withs = τ + φ.

(7.9)34

It is not difficult to see that this d still satisfies the usual properties of the exteriorderivative, provideds ∧s = 0,(7.10)and that it can be extended over the whole “soft” exterior algebra in the same wayas in the undeformed case.From s ∧s = 0 we find:τ ∧φ + φ ∧τ + φ ∧φ = 0(7.11)since τ ∧τ = 0 still holds. It is easy to compute the curvatures, as defined in (7.1),in terms of φ:Ri = 1λ[s ∧µi + µi ∧s] + Cijk µj ∧µk = 1λ[φ ∧µi + µi ∧φ].

(7.12)where the last equality is due to the fact that if s = τ the Cartan-Maurer equationshold (⇒Ri = 0). Similarly we find the curvature of τ:R(τ) = 1λ[φ ∧τ + τ ∧φ] = −1λ[φ ∧φ],(7.13)the last equality being due to (7.11).A more detailed discussion on the differential calculus corresponding to this“soft” exterior derivative will be given in a later publication.

Here we mention thatthe soft calculus allows the definition of quantum “diffeomorphisms”:δtµk ≡ℓtµk = (itd + dit)µk = (∇t)k + itRk,(7.14)where ∇is the quantum covariant derivative whose definition can be read offtheBianchi identities (7.2) ∇Rk = 0. The construction of an action, invariant underthese diffeomorphisms, proceeds as in the classical case.

We refer to [6, 9] for somepreliminary applications of this formalism to the construction of q-gravity and q-gauge theories.35

AThe derivation of two equationsIn this Appendix we derive the two equations (4.100) and (4.102). Consider theexterior derivative of eq.

(4.30):d(ωia) = d[(f ij ∗a)ωj]. (A.1)The left-hand side is equal to:d(ωia) == dωi ∧a −ωi ∧da = −Cijk ωj ⊗ωka −ωi ∧(χj ∗a)ωj == −Cijk ωj ⊗ωka −(f is ∗χj ∗a)ωs ∧ωj == −Cijk (f jp ∗f kq ∗a)ωp ⊗ωq −(f is ∗χj ∗a)(ωs ⊗ωj −Λsjpqωp ⊗ωq) == [(−Cijk f jpf kq −f ipχq + Λsjpqf isχj) ∗a](ωp ⊗ωq)(A.2)The right-hand side reads:d[(f ij ∗a)ωj] == d(f ij ∗a)ωj + (f ij ∗a)dωj == (χk ∗f ij ∗a)ωk ∧ωj −(f ij ∗a)Cjpq ωp ⊗ωq == (χk ∗f ij ∗a)(ωk ⊗ωj −Λkjpqωp ⊗ωq) −(f ij ∗a)Cjpq ωp ⊗ωq == [(χpf iq −Λkjpqχkf ij −Cjpq f ij) ∗a](ωp ⊗ωq),(A.3)so that we deduce the equation−Cijk f jpf kq −f ipχq + Λsjpqf isχj == χpf iq −Λkjpqχkf ij −Cjpq f ij.

(A.4)We now need two lemmas.Lemma 1f nl ∗aθ = (f nr ∗a)(f rl ∗θ),a ∈A, θ ∈Γ⊗n. (A.5)Proof:f nl ∗aθ =(id ⊗f nl)∆(a)∆R(θ) = a1θ1f nl(a2θ2) =a1θ1f nr(a2)f rl(θ2) = a1f nr(a2)θ1f rl(θ2) =(f nr ∗a)θ1f rl(θ2) = (f nr ∗a)(f rl ∗θ).

(A.6)36

Lemma 2f rl ∗ωj = Λrjklωk. (A.7)Proof:f rl ∗ωj =(id ⊗f rl)∆R(ωj) = (id ⊗f rl)[ωk ⊗M jk ] == ωkf rl(M jk ) = Λrjkl.

(A.8)Consider now eq. (4.99) with h = f nl:d(f nl ∗a) = f nl ∗da.

(A.9)The first member is equal to (χk ∗f nl ∗a)ωk, while the second member is:f nl ∗da=f nl ∗[(χj ∗a)ωj] = (f nr ∗χj ∗a)(f rl ∗ωj)=(f nr ∗χj ∗a)(Λrjklωk)(A.10)We have used here the two lemmas (A.5) and (A.7). Therefore the following equa-tion holds:χk ∗f nl = Λrjkl f nr ∗χj,(A.11)which is just eq.

(4.100). Equation (4.102) is obtained simply by subtracting (A.11)from eq.

(A.4).37

BTwo theorems on it and ℓtTheoremThe contraction operator it satisfies:∀n, ∀s : 1 ≤s < n,iti(ωi1 ∧ωi2 ∧. .

. ωin)=itj(ωi1 ∧ωi2 ∧.

. .

ωis) ∧f ji ∗(ωis+1 ∧. .

. ωin)+(−1)sωi1 ∧.

. .

ωis ∧iti(ωis+1 ∧. .

. ωin).

(B.1)ProofFor all n, when s = 1 (B.1) is just property γ) of the definition of iti (see Section6). We prove the theorem by induction on s. Suppose that (B.1) be true for s −1;then it is true for s. Indeed :iti(ωi1∧.

. .

ωin) ==itj(ωi1 ∧. .

. ωis−1) ∧f ji ∗(ωis ∧.

. .

ωin)++(−1)s−1ωi1 ∧. .

. ωisp−1 ∧iti(ωis ∧.

. .

ωin) ==itj(ωi1 ∧. .

. ωis−1) ∧f jk ∗ωis ∧f ki ∗(ωis+1 ∧.

. .

ωin)++(−1)s−1ωi1 ∧. .

. ωis−1 ∧itj(ωis)f ji ∗(ωis+1 ∧.

. .

ωin)+−(−1)s−1ωi1 ∧. .

. ωis−1 ∧ωis ∧iti(ωis+1 ∧.

. .

ωin) ==[itj(ωi1 ∧. .

. ωis−1) ∧f jk ∗ωis++(−1)s−1ωi1 ∧.

. .

ωis−1 ∧itk(ωis)] ∧f ki ∗(ωis+1 ∧. .

. ωin)++(−1)sωi1 ∧.

. .

ωis ∧iti(ωis+1 ∧. .

. ωin) ==itk(ωi1 ∧.

. .

ωis−1 ∧ωis) ∧f ki ∗(ωis+1 ∧. .

. ωin)++(−1)sωi1 ∧.

. .

ωis ∧iti(ωis+1 ∧. .

. ωin);in the last equality we have used the inductive hypothesis.We can conclude that (B.1) is true for all s : 1 ≤s < n.Q.E.D.Remembering the A-linearity of iti the subsequent generalization is straightfor-ward :ita(ai1...inωi1 ∧.

. .

ωin)=itj(ai1...inωi1 ∧. .

. ωis) ∧f ji ∗(ωis+1 ∧.

. .

ωin)++(−1)sai1...inωi1 ∧. .

. ωis ∧iti(ωis+1 ∧.

. .

ωin)with ai1...in ∈A.38

Theoremℓti = itid + ditithat is∀ai1...inωi1 ∧. .

. ωin ∈Γ∧n,ℓti(ai1...inωi1 ∧.

. .

ωin)=(itid + diti)(ai1...inωi1 ∧. .

. ωin).

(B.2)We will show this theorem by induction on the integer n. To do this, we needthe following:LemmaIf n = 1, the theorem is true, i.e.ℓti(bkωk) = (itid + diti)(bkωk). (B.3)First we show that:ℓti(ωk) = (itid + diti)ωk.

(B.4)We already know that ℓti(ωk) = ωjCkji. The right-hand side of (B.4) yields:(itid + diti)(ωk)=itidωk + d(itiωk) ==−Cnjkiti(ωn ∧ωj) ==−Cnjk f ni ∗ωj −ωnδji==−Cnjk h(id ⊗f ni) ∆R(ωj) −δji ωni==−Cnjk h(id ⊗f ni)ωℓ⊗Mℓj−δji ωni==−Cnjk hωℓΛnjℓi −δji ωni==+Cnjk hδnℓδji −Λnjℓiiωℓ==Ckℓiωℓand (B.4) is thus proved.The right-hand side of (B.3) gives:(itid + diti)(bkωk)=itidbk ∧ωk + bkdωk+ dbkiti(ωk)==itj(dbk)f ji ∗ωk −(dbk)iti(ωk)++bkiti(dωk) + (dbk)iti(ωk) ==itj((χn ∗bk)ωn)f ji ∗ωk + bkiti(dωk) ==(χn ∗bk)δnj f ji ∗ωk + bk(itid + diti)ωk ==(χn ∗bk)f ni ∗ωk + bkℓti(ωk) ==ℓtn(bk)f ni ∗ωk + bkℓti(ωk) ==ℓti(bkωk),and the lemma is proved.

We now finally prove the theorem.Let us suppose it to be true for a (n −1)-form:ℓta(ai2...inωi2 ∧. .

. ωin) = (itid + diti)(ai2...inωi2 ∧.

. .

ωin). (B.5)39

Then it holds also for an n-form. Indeed, the left-hand side of (B.2) yieldsℓti(ai1...inωi1 ∧.

. .

ωin) == ℓtj(ai1...inωi1) ∧f ji ∗(ωi2 ∧. .

. ωin) + ai1...inωi1 ∧ℓti(ωi2 ∧.

. .

ωin)while the right-hand side of (B.2) is given by :(itid + diti)(ai1...inωi1 ∧. .

. ωin) ==iti[d(ai1...inωi1) ∧ωi2 ∧.

. .

ωin −(ai1...inωi1) ∧d(ωi2 ∧. .

. ωin)] +d[itj(ai1...inωi1)f ji ∗(ωi2 ∧.

. .

ωin) −(ai1...inωi1) ∧iti(ωi2 ∧. .

. ωin)] ==itj(dai1...inωi1) ∧f ji ∗(ωi2 ∧.

. .

ωin) + d(ai1...inωi1) ∧iti(ωi2 ∧. .

. ωin) +−itj(ai1...inωi1)f ji ∗d(ωi2 ∧.

. .

ωin) + ai1...inωi1 ∧itid(ωi2 ∧. .

. ωin) ++ditj(ai1...inωi1)f ji ∗(ωi2 ∧.

. .

ωin) + itj(ai1...inωi1) ∧f ji ∗d(ωi2 ∧. .

. ωin) +−d(ai1...inωi1) ∧iti(ωi2 ∧.

. .

ωin) + ai1...inωi1 ∧diti(ωi2 ∧. .

. ωin) ==[(itjd + ditj)(ai1...inωi1)] ∧f ji ∗(ωi2 ∧.

. .

ωin) ++ai1...inωi1(itid + diti)(ωi2 ∧. .

. ωin) ==ℓtj(ai1...inωi1) ∧f ji ∗(ωi2 ∧.

. .

ωin) + ai1...inωi1 ∧ℓti(ωi2 ∧. .

. ωin)and the theorem is proved.40

CA collection of formulasWe list here some useful formulas. Most have been derived in the paper, or areparticular cases of those.dωi = −Cijk ωj ∧ωk = −Cijk ωj ⊗ωkCijk= (χj ∗χk)(xi),Cijk= [χj, χk](xi) = χk(M ij )f ij(M kl ) = Λikljχi(ab) = χj(a)f ji(b) + ε(a)χi(b)xj ∈A :χi(xj) = δjiε(xj) = 0χi(xjb) = f ji(b)(χi ∗ab) = (χj ∗a)(f ji ∗b) + a(χi ∗b)χi ∗ωj = Cjki ωk,f ij ∗ωk = Λikljωlχi ∗M jk = CjliM lk ,f ij ∗M lk = ΛilmjM mkχi ∗(aθ) = (χj ∗a)(f ji ∗θ) + a(χi ∗θ)χi ∗(θa) = (χj ∗θ)(f ji ∗a) + θ(χi ∗a)f ij ∗(aθ) = (f ik ∗a)(f kj ∗θ)f ij ∗(θa) = (f ik ∗θ)(f kj ∗a)ℓti(τ ⊗τ ′) = ℓtj(τ) ⊗f ji ∗τ ′ + τ ⊗ℓti(τ ′)M ji (a ∗f ik) = (f ji ∗a)M ik[χi, χj] = Ckijχk[[χr, χi], χj] −Λklij[[χr, χk], χl] = [χr, [χi, χj]]∆′(χi) = χj ⊗f ji + ε ⊗χi,ε′(χi) = 0Cimn f mjf nk + f ijχk = Λpqjkχpf iq + Cljk f ilχkf nl = Λijklf niχj41

Λnmijf ipf jq = f nif mjΛijpq∆′(f ij) = f ik ⊗f kj,ε′(f ij) = δijκ′(f il)f lj = f ilκ′(f lj) = δijε,κ′−1(f li)f jl = f liκ′−1(f jl) = δijεM ji M qr Λirpk = ΛjqriM rp M ik∆(M ji ) = M ki⊗M jk ,ε(M ji ) = δijκ(M ji )(M lj ) = M ji κ(M lj ) = δliI,κ−1(M lj )(M ji ) = M lj κ−1(M ji ) = δliIℓt(dτ) = d(ℓtτ)(id ⊗ℓt)∆L(τ) = ∆L(ℓtτ)(id ⊗ℓt)∆R(τ) = ∆R(ℓtτ)iti(τ ⊗τ ′) = itj(τ) ⊗f ji(τ ′) + τ ⊗iti(τ ′)iti(τ ∧τ ′) = itj(τ) ∧f ji(τ ′) + τ ∧iti(τ ′)(id ⊗it)∆L = ∆Litℓt(θ) = [itd + dit](θ)42

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