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Rings of skew polynomials and

arXiv:hep-th/9306138v1 26 Jun 1993Rings of skew polynomials andGel’fand-Kirillov conjecture forquantum groupsKenji Iohara∗, Feodor Malikov†‡Department of Mathematics, Kyoto University ,Kyoto 606 JapanReceived:AbstractWe introduce and study action of quantum groups on skew polynomialrings and related rings of quotients. This leads to a “q-deformation” ofthe Gel’fand-Kirillov conjecture which we partially prove.

We propose aconstruction of automorphisms of certain non-commutaive rings of quo-tients coming from complex powers of quantum group generators; this isapplied to explicit calculation of singular vectors in Verma modules overUq(sln+1). We finally give a definition of a q−connection with coefficientsin a ring of skew polynomials and study the structure of quantum groupmodules twisted by a q−connection.1IntroductionThis work was mainly inspired by the Feigin’s construction which associates toan element of the Weyl group w ∈W an associative algebra homomorphismof a “nilpotent part” of a quantum group to an appropriate algebra of skewpolynomials:Φ(w) : U −q (g) →C[X],(1)where X stands for X1, .

. .

, Xl, l is a length of w and XjXi = qαijXiXj forsome αij ∈Z, 1 ≤i, j ≤l. The main topics treated in the work are as follows.∗e-mail address: iohara@kurims.kyoto-u.ac.jp†Supported by the Japan Society for the Promotion of Science Post -Doctoral Fellowshipfor Foreign Researchers in Japan.‡Address after August, 1, 1993: Mathematics Department Yale University New Haven CT06520 USA; e-mail address: malikov@kusm.kyoto-u.ac.jp1

Skew polynomials and quantum groups21.Realizations of Lie algebras and quantum groups and Gel’fand-Kirillov con-jecture. The fact that a Lie algebra of an algebraic group (“algebraic Lie alge-bra”) can be realized in differential operators acting on a suitable manifold is,probably, more fundamental than the notion of a Lie algebra.

Explicit formulasfor such a realization in the case when the algebra is simple, the manifold is abig cell of a flag manifold have become especially popular recently because oftheir relation to the free field approach to 2-dimesional conformal field theory( [F-F, B-McC-P1]). An important property of the realization was discoveredby Gel’fand and Kirillov [G-K1] long ago and in a remarkable generality.

Theirobservation is that however complicated classification of algebraic Lie algebrasmay be, equivalence classes of rings of quotients of universal enveloping alge-bras are labelled by pairs of positive integers: a ring of quotients of a universalenveloping algebra is isomorphic with a ring of quotients of a ring of differ-ential operators on n variables with polynomial coefficients trivially extendedby a k−dimensional center, where k is a dimension of a generic orbit in thecoadjoint representation and 2n + k is equal to the dimension of the algebra. (This conjecture has been proven by themselves and others in many cases[G-K1, J, McC].

)A natural class of rings suitable for formation of rings of quotients is providedby the so-called Ore domains. Besides above mentioned universal enveloping al-gebras of finite dimensional Lie algebras and rings of differential operators theclass of Ore domains comprises (deformed) enveloping algebras of affine Lie alge-bras, rings of skew polynomials and q−difference operators.

We prove that theFeigin’s morphism Φ(w0) associated to the longest Weyl group element providesan isomorphism of rings of quotients Q(U −q (sln+1)) ≈Q(C[X]). This isomor-phism allows to equip Q(C[X]) with a structure of Uq(sln+1)-module.

Moreprecisely we define an n−parameter family of associative algebra homomor-phisms from Uq(sln+1) to an algebra of q−difference operators with coefficientsin Q(C[X]). We conjecture that this provides an isomorphism of Q(Uq(sln+1))with an n−dimensional central extension of the algebra of q−difference opera-tors.

We prove this conjecture for Uq(sl2), Uq(sl3) in a slightly weaker form.2.Complex powers, automorphisms and screening operators.A remark-able observation made in early works on Kac-Moody algebras[L-W, F-K] isthat affine Lie algebras, like finite dimensional simple ones, are also realizedin differential operators, though on infinite many variables. A family of suchrealizations depending on a highest weight λ was constructed by Wakimoto [W]for csl2 and by Feigin and Frenkel [F-F] for all non-twisted affine Lie algebras.Thus obtained modules are now known as Wakimoto modules F(λ).

The mainingredient of the 2-dimensional conformal field theory associated to an affinealgebra ˆg is a 2-sided complex consisting of direct sums of Wakimoto modules· · · →F (−1) →F (0) →F (1) →· · ·such that its homology is concentrated in the 0-th dimension and is equal to anirreducible highest weight module over ˆg ( BRST resolution).

Skew polynomials and quantum groups3Bowknegt, McCarthy and Pilch revealed a quantum group structure hiddenin the differential of the BRST resolution. Recall that Uq(g)-morphisms of aVerma module M(λ) into a Verma module M(µ) are in 1-1 correspondencewith singular vectorsof the weight λ in the latter ( the correspondence isestablished by assigning to a morphism an image of the vacuum vector underthis morphism).Denote by Singλ(M(µ)) the set of singular vectors of theweight λ in M(µ).

It is argued in [B-McC-P1, B-McC-P2] that there is a linearmapSingλ(M(µ)) →Homˆg(F(λ), F(µ))(2)and that conjecturally this map is an isomorphism.Singular vectors in Verma modules over quantum groups related to an arbi-trary Kac-Moody algebra were found in the form [M-F-F, M]F slil · · · F s1i1 F Ni0 F t1i1 · · · F tlil ,(3)where si, ti, 1 ≤i ≤l are appropriate complex numbers, Fi, 1 ≤i ≤n arecanonical Cartan generators of U −q (g), N ∈N. Here we carry out an explicitcalculation of ( 3), i.e.

rewrite it in the form containing only natural powers ofF ′s. Observe that the map ( 2) is determined by assigning to each Fi what isknown as a screening operator.One of the consequences of the prescription how to choose powers in ( 3) isthat si + ti, 1 ≤i ≤l are all non-negative integers.

More generally, one mayconsider a mapU −q (g) ∋p 7→F βi pF −βi, β ∈C.A simple calculation using the notion of the q−commutator shows that thismap extends to an automorphism of the quotient ring Q(U −q (g)). Therefore asingular vector is obtained by, roughly speaking, a sequence of automorphismsapplied to a Cartan generator.Similarly one may consider an operator of conjugation by a complex powerof a linear formC[x1, .

. .

, xk] ∋p 7→(xi1 + · · · xil)βp(xi1 + · · · xil)−β,acting on a certain completion of a ring of skew polynomials C[x1, . .

. , xk].

Sim-ple but nice calculation based on the q−binomial theorem shows that this mapactually determines an automorphism of the ring of quotients Q(C[x1, . .

. , xk]).This construction may be interesting in its own right: unlike the things arein commutative realm, the very existence of ( non-trivial ) automorphisms ofQ(C[x1, .

. .

, xk]) is not quite obvious. Combined with the Feigin’s morphismthis construction answers an informal question: “how does it happen that com-plex powers in the singular vector formula cancel out?” Another application ofthese automorphisms is that they produce natural examples of q−connectionswith coefficients in skew polynomials.

Skew polynomials and quantum groups43. Quantum group modules twisted by q−connections.

It has been realized[A-Y, F-G-P-P] that the “singular vector decoupling condition” makes it nec-essary to consider non-bounded – neither highest nor lowest weight – modulesin 2-dimensional conformal field theory at a rational level. On the other handthe singular vector formula ( 3) makes it natural to consider an extension of aVerma module by complex powers of generators, which transparently producesnon-bounded modules.

It was shown in [F-M] that the duals to such modulesare realized in multi-valued functions on a flag manifold or, in other words, inmodules twisted by connections; in particular a family of integral intertwiningoperators acting among such modules was constructed. ( One may also findin [F-M] and in the forthcoming paper [I-M] integral formulas for solutions toKnizhnik-Zamolodchikov equations with coefficients in non-bounded modules.

)Here we adjust the definition of a q−connection given by Aomoto and Kato[A-K] in the commutative case to the case of skew polynomials. This definitionidentifies q−connections with the cohomology group H1(Zk, Q(C[x1, .

. .

, xk])).We also produce a family of elements of H1(Zk, Q(C[x1, . .

. , xk])) associatedwith complex powers of linear forms all this being independent of quantumgroups.

In the case when the ring of skew polynomials is the one coming fromthe Feigin’s morphism, the twisting by such a q−connection is nothing but apassage from a Verma module to its extension by complex powers of generators.This allows to construct q−analogues of the intertwiners of [F-M], which in thequantum case may be thought of as right multiplications by certain complexpowers of linear forms. We also find out what the Uq(g)−module structure ofa Verma module extended by a complex power of only 1 generator is.

Such amodule can be viewed as a module induced from a non-bounded module overa parabolic subalgebra. It turns out that its structure is formally close to thatof a Verma module.

In particular, the notion of a singular vector is naturallyreplaced with that of a singular chain and a singular chain encodes informationon a family of singular vectors.Acknowledgments. Our thanks go to B.Feigin who explained to us hisunpublished results, to N.Reshetikhin who brought to our attention the paper[A-K] and to M.Jimbo for his interest in the work.

Some results of the workwere reported at Mie University in Tsu. F.M.

is indebted to M.Wakimoto forhis heartiest hospitality during the visit.2Main definitions related to quantum groupsThe material of this secton is fairly standard. Usually the reference is the work[DC-K].1.Let, as usual, A = (aij), 1 ≤i, j ≤n stand for a generalized sym-metrizable Cartan matrix , symmetrized by non-zero relatively prime integersd1, .

. .

, dn such that diaij = djaji for all i, j. A Kac-Moody Lie algebra g at-tached to A is an algebra on generators Ei, Fi, Hi, 1 ≤i ≤n and well-known

Skew polynomials and quantum groups5relations explicitly depending on entries of A ( see [K] ). Among the structuresrelated to g we shall use the following:the triangular decomposition g = n−⊕h ⊕n+;the dual space h∗; elements of h∗will be referred to as weights;the root space decomposition n± = ⊕α∈∆±gα, gαi = CEi;the root lattice Q ∈h∗, {α1, .

. .

, αn} ⊂∆+ ⊂h∗being the set of simpleroots;the invariant bilinear form Q × Q →Z defined by (αi, αj) = diaij.2.For q ∈C, d ∈Z set:[n]d = 1 −q2nd1 −q2d ,[n]d! = [n]d · · · [1]d, njd= [n]d · · · [n −j + 1]d[j]d!,omitting the subscript if d = 1.Suppose g is a Kac-Moody Lie algebra attached to A.

The Drinfeld -Jimboquantum group Uq(g), q ∈C is said to be a Hopf algebra with antipode S,comultiplication ∆and 1 on generators Ei, Fi, Ki, K−1i, 1 ≤i ≤n and definingrelationsKiK−1i= K−1iKi = 1, KiKj = KjKi,(4)KiEjK−1i= qaijiEj, KiFjK−1i= q−aijiFj, qi = qdi,(5)EiFj −FjEi = δijKi −K−1iqi −qi, qi = qdi,(6)1−aijXν=0(−1)νqν(ν−1+aij)i1 −aijνdiE1−aij−νiEjEνi=0,1−aijXν=0(−1)νqν(ν−1+aij)i1 −aijνdiF 1−aij−νiFjF νi=0 (i ̸= j),(7)the comultiplication being given by∆Ei = Ei ⊗1 + Ki ⊗Ei, ∆Fi = Fi ⊗K−1i+ 1 ⊗Fi, ∆Ki = Ki ⊗Ki,(8)and antipode - bySEi = −K−1iEi, SFi = −FiKi, SKi = K−1i. (9)The relations admit the C−algebra anti-automorphism ωωEi = Fi, ωFi = Ei, ωKi = Ki(10)

Skew polynomials and quantum groups6Set U +q (g) (U −q (g)) equal to the subalgebra, generated by Ei (Fi resp.) (1 ≤i ≤n) and U 0q (g) = C[K±11 , .

. .

, K±1n ]. We will sometimes reduce these nota-tions to U +q , U −q , U 0q if this does leads to confusion.

One may check that themultiplication induces an isomorphism of linear spacesUq(g) ≈U −q (g) ⊗U 0q (g) ⊗U +q (g). (11)Set U ≥q = U 0q U +q .

From now on unless otherwise stated A is assumed to beof finite type.3. For any Q−graded associative algebra A = ⊕β∈QAβ define a q−commutator,which associates to any homogeneous b ∈Aβ a mappingadqb : A →Aof degree β determined byadqb(c) = bc −q(β,γ)cb, if c ∈Aγ.

(12)One deduces that the q−bracket is a q−derivation of A, meaning thatadqa(bc) = (adqa(b))c + q(α,β)b adqa(c) if a ∈Aα, b ∈Aβ, c ∈Aγ. (13)Using (13) one proves the following useful formulabnc = qn(β,γ)cbn +nXj=1q(n−j)(β,γ)nj(β,β)/2(adqb)j(c)bn−j,(14)its Lie algebra analogue beingbnc = cbn +nXj=1 nj(ad b)j(c)bn−j.

(15)In the case A = U ±q (g) one realizes that the relations (7) simply mean that(adqEi)−aij+1(Ej) = (adqFi)−aij+1(Fj) = 0, if i ̸= j. (16)The following observation will be used below in the discussion of rings ofquotients:the relations (13,14,16) imply that for any Fi, b ∈U −q (g) one hasF Ni b ∈U −q (g)Fi,(17)for all sufficiently large N.

Skew polynomials and quantum groups74. Following Lusztig[L] introduce the following automorphisms Ri, 1 ≤i ≤n of the algebra U(g):RiEi = −FiKi, RiEj=−aijXs=0(−1)s−aijq−siE−aij−si[−aij −s]di!EjEsi[s]di!

(18)if i̸=j,RiFi = −K−1iEi, RiFj=−aijXs=0(−1)s−aijqsiF si[s]di!EjE−aij−si[−aij −s]di! (19)if i̸= j,RiKj=KjK−aiji.

(20)Fix a reduced decomposition ri1ri2 · · · riN of the longest element of the Weylgroup W. This gives an ordering of the set of positive roots:β1 = αi1, β2 = ri1αi2, . .

. , βN = ri1 · · · riN−1αiN .One introduces root vectors [L]Eβs = Ri1 · · · Ris−1Eis,(21)Fβs = Ri1 · · · Ris−1Fis.

(22)For k = (k1, . .

. , kN) ∈ZN+ set Ek = Ek1β1 · · · EkNβN , F k = ωEk.Proposition 2.1 (i)[L] Elements Ek (F k resp.

), k ∈ZN+, form a basis ofU +q (g) (U −q (g) resp.) over C.(ii)[DC-K] The algebra Uq(g) affords a structure of a Z2N+1+−filtered al-gebra, so that the associated graded algebra Gr(Uq(g)) is an associative algebraover C on generators Eα, Fα (α ∈∆+) K±i (0 ≤i ≤n) subject to the followingrelationsKiKj = KjKi, KiK−1i= 1, EαFβ = FβEα,KiEα = q(α,αi)EαKi, KiFα = q−(α,αi)FαKi,EαEβ = q(α,β)EβEα, FαFβ = q(α,β)FβFα, if α > β.Recall that an algebra Cs[x1, .

. .

, xk] on generators x1, . .

. , xk and defining re-lations xixj = λijxjxi for i > j, where λij ∈C∗, is called an algebra of skewpolynomials.

Therefore, the item (ii) of Proposition 2.1 asserts that Gr(Uq(g))is a skew polynomial algebra. The “classical” analogue of this is the fact thatGr(U(g)) is a symmetric algebra S(g).An algebra of skew polynomials has no zero divisors, therefore, the same istrue for Uq(g) [DC-K].

Skew polynomials and quantum groups83Rings of quotients associated to (deformed)enveloping algebras3.1Gelgand - Kirillov’ conjecture and Feigin’s construc-tion.1. A ( non-commutative ) ring A with no zero divisors is called an Ore domainif any 2 elements of A have a common right and a common left multiple.

A classof examples of Ore domains is provided by the rings of polynomial growth. Weshall be calling an N−filtered ring A = ∪i≥1A(i) a ring of polynomial growthif dimA(i) is equivalent to a certain polynomial as i →∞.Lemma 3.1 A ring of polynomial growth with no zero divisors is an Ore do-main.Proof.

Assume that dimA(i) ∼a0ik, as i →∞. Then for any ideal I on1 generator one has: dimI(i) ∼a0ik, I(i) = I ∩A(i).

If 2 ideals I1, I2 on 1generator have zero intersection then (I1 + I2)(i) ∼2a0ik, contradicting theassumption. ✷The simplest examples of rings of polynomial growth are, therefore, algebrasof (skew) polynomials.

Further, affine and finite-dimensional Lie algebras aredistinguished among Kac-Moody algebras as algebras of polynomial growth [K].This combined with Proposition 2.1 implies that U(g), if g is of either affine orfinite dimensional, and Uq(g), if g is finite dimensional, are Ore domains, as wellas the corresponding U ±(g) (U ±q (g).2. An Ore domain A is a suitable object for formation of a ring of quotients.Consider expressions of the form ab−1, b−1a, a, b ∈A called right and left (resp.) quotients.

Introduce a relation ≈by saying that(i) ab−1 ≈c−1d ⇔ca = db,(ii) 2 right (left) quotients are in relation ≈if and only if they are in relation ≈to one and the same left (right) quotient.The Ore domain conditions imply that ≈is an equivalence relation. De-note the set of equivalence classes of ≈by Q(A).

One more application of theOre domain conditions gives that each equivalence class contains left and rightquotients and that any 2 left ( right ) quotients are eqivalent to left (right)quotients with one and the same denominator. This allows to define operationsof addition and multiplication ( in the most natural way ), which completes thedefinition of the ring of quotients Q(A).A definition of a ring of quotients A[S−1] with respect to S ⊂A is a moresubtle matter because due to the noncommutativity of A it is not clear what canreally appear as a denominator.

However in the case when A is a (quantized)

Skew polynomials and quantum groups9enveloping algebra one can say more. It follows from (14) thatF −niFj = q−n(αi,αj)FjF −ni+∞Xj=1q(−n−j)(αi,αj)−nj(αi,αi)/2(adqFi)j(Fj)F −n−ji,if i ̸= j, n > 0.

(Observe that (16) implies that only finite number of terms in the right-handside of the above formula can be non-zero and, therefore, it makes sense as anelement of U −q (g).) Therefore, the result of commuting negative powers of aCartan generator to the right is negative powers of the same generator on theright.

One also deduces from (17) that 2 words F s1i1 · · · F smim , F t1j1 · · · F tlil have acommon right multiple of the formF N1i1 · · · F Nmim F t1j1 · · · F tlil ,if N1, . .

. , Nm are sufficiently large.

Now, if S ⊂U −q (g) is a multiplicativelyclosed subset multiplicatively generated by Fi1, . .

. , Fik, one defines U −q (g)[S−1]as a subset of Q(U −q (g)) consisting of classes of quotients of the form ab−1, b ∈S.The above discussion shows that U −q (g)[S−1] is a subring.

We shall sometimesdenote U −q (g)[S−1] by U −q (g)[F −1i1 , F −1i2 , . .

. , F −1ik ].It is easy to see that the same goes through with F ′s replaced with E′s orU −q (g) replaced with Uq(g), as well as with everything replaced with its classical(q →1) analogues.

Further, even though a ring of quotients is not defined foran arbitrary Kac-Moody algebra, this discussion shows that a ring of quotientsU(g)[S−1] is well-defined if S is multiplicatively generated by a (sub)set of realroot vectors. Actually, formulas (13,15 ) provide an algorithm of carrying outoperations of multiplication and addition on elements of U(g)[S−1].3.

It often happens that rings of quotients of universal enveloping algebras ofdifferent finite-dimensional Lie algebras are isomorphic with each other. Denoteby Dnk an algebra on generators a1, .

. .

, an, a∗1, . .

. , a∗n, c1, .

. .

, ck and definingrelations[ai, a∗j] = δij, [ci, aj] = [ci, a∗j] = [ai, aj] = [a∗i , a∗j] = 0, for all i, j.Dnk can, of course, be viewed as an algebra of differential operators on n vari-ables trivially extended by k−dimensional center.Conjecture 3.2 ( Gelfand - Kirillov [G-K1] ) If g is an algebraic Lie algebrathen Q(U(g)) is isomorphic with Q(Dnk) for k equal to the dimension of ageneric g−orbit in the coadjoint representation and n = (dimg −k)/2.This conjecture has been proven in many cases [G-K1, J, McC].4. It seems that the following construction ( due to Feigin [F] ) is relevantto a proper q−deformation of the Gelfand-Kirillov’s conjecture.

For a pair ofQ−graded associative algebras A, B define a q−twisted tensor product as an

Skew polynomials and quantum groups10algebra A ⊗q B isomorphic with A ⊗B as a linear space and with the multi-plication given by a1 ⊗b1 · a2 ⊗b2 = q(α,β)a1a2 ⊗b1b2 if a2 ∈A(α), b1 ∈B(β).Evidently, A ⊗q B is a Q−graded algebra.Proposition 3.1[F] For any Kac-Moody algebra g the map˜∆: U ±q (g) →U ±q (g) ⊗q U ±q (g),˜∆: 1 7→1 ⊗1,˜∆: Ei 7→Ei ⊗1 + 1 ⊗Ei (Fi →Fi ⊗1 + 1 ⊗Fi resp. ),1 ≤i ≤nis a homomorphism of associative algebras.Remark.

It is known that the map U ±q (g) →U ±q (g) ⊗U ±q (g) does not existin the category of associative algebras.Iterating ˜∆one obtains a sequence of maps˜∆m : U −q (g) →U −q (g)⊗m, m = 2, 3 . .

.,determined by ˜∆2 = ˜∆, ˜∆m = ( ˜∆⊗id) ◦˜∆m−1.For any simple root αi consider a ring of polynomials on 1 variable C[Xi],which we regard as a Q−graded by setting deg Xi = αi. There arises a morphismof Q−graded associative algebrasρi : U −q (g) →C[Xi],Fj 7→δijxi.Now, for any sequence of simple roots αi1, .

. .

, αik there arises a morphismof Q−graded associative algebras:(ρi1 ⊗· · · ⊗ρik) ◦˜∆k : U −q (g) →C[X1i1] ⊗q · · · ⊗q C[Xkik]. ( The double indexation of X′s is necessary because some number can appearin the sequence i1, .

. .

, ik more than once but the corresponding indeterminateshave to be regarded as different. )Evidently, C[X1i1] ⊗q · · · ⊗q C[Xkik] is an algebra of skew polynomialsC[X1i1 .

. .

Xkik], satisfying the relations XsisXtit = q(αis,αit)XtitXsis, s > t.Therefore, we have constructed a family of morphisms of a “ maximal nilpo-tent subalgebra ” of a quantum group associated to an arbitrary Kac-Moodyalgebra to algebras of skew polynomials. It is interesting that a proper classicalanalogue of this construction is not so obvious and is best understood in theframework of rings of quotients ( see below).We now assume that g is a simple finite-dimensional Lie algebra.

Let w0 =ri1 · · · riN ∈W be a reduced decomposition of the element of maximal length.Set Φ(i1, . .

. , iN) = (ρi1 ⊗· · · ⊗ρiN ) ◦˜∆N.

Skew polynomials and quantum groups11Conjecture 3.3[F](i) Φ(i1, . .

. , iN) is an embedding.

(ii) Φ(i1, . .

. , iN) extends – at least for a special choice of a reduced decompo-sition w0 = ri1 · · · riN – to an isomorhism of Q(U −q (g)) with Q(C[X1i1 .

. .

XNiN]).5. Example: g = sln+1.

From the abstract point of view g = sln+1 is analgebra related to the Cartan matrix (aij), whereaij =2ifi = j−1if| i −j |= 10if| i −j |> 1Choose a reduced decomposition of the longest Weyl group element to bew0 = r1r2 · · · rnr1r2 · · · rn−1 · · · · · · r1r2r1. Denote by C[X] the skew polynomialring on generators Xij labelled by all pairs i, j satisfying 1 ≤j ≤n, 1 ≤i ≤n −j + 1 and defining relationsXijXrs = pijrsXrsXij,wherepijrs =q2ifi > r, j = sqifi ≤r, j = s −1q−1ifi > r, j = s −11ifj < s −1.In this case the map Φ = Φ(w0) ( here w0 stands for the reduced decompo-sitionw0 = r1r2 · · · rnr1r2 · · · rn−1 · · · · · · r1r2r1 ) acts as followsΦ(w0)(Fi) = X1i + X2i + · · · Xn+1−i i 1 ≤i ≤n.

(23)One solves (23) as a system of equations on Xij, 1 ≤i < j ≤n withcoefficients in Q(Φ(U −q (sln+1))).Lemma 3.4 The following formulas holdX1 n−1 =qq −q−1 [Φ(Fn−1), Φ(Fn)]qΦ(Fn)−1,X2 n−1 =1q −q−1 [Φ(Fn), Φ(Fn−1)]qΦ(Fn)−1,X1i =qq −q−1 [Φ(Fi), X1i+1]qX−11i+1, 1 ≤i ≤n −2,Xn−i+1 i =1q −q−1 [Xn−i i+1, Φ(Fi)]qX−1n−i i+1, 1 ≤i ≤n −2,Xji =1q −q−1 (Xj−1 i+1Φ(Fi)X−1j−1 i+1 −Xj i+1Φ(Fi)X−1j i+1), 2 ≤j ≤n −i.

Skew polynomials and quantum groups12One uses this lemma to prove that the Conjecture 3.3 is true.Theorem 3.5(i) The map Φ is an embedding. (ii) The embedding Φ : U −q (sln+1) →C[X] induces an isomorphismQ(U −q (sln+1)) ≈Q(C[X]).Proof.

Lemma3.4 actually shows that Q(Φ(U −q (sln+1))) ≈Q(C[X]). Itis, therefore, enough to prove that Φ is injective.

In [G-K1] Gel’fand and Kir-illov associated a number to an arbitrary algebra A which is now known as theGel’fand-Kirillov dimension dimG−KA. For example, Gel’fand Kirillov dimen-sion of a polynomial ring on n variables, as well as that of the correspondingring of quotients, is equal to n. One of results of [G-K1] is that if an algebraA has a filtration such that the associated graded algebra GrA is isomorphicwith a polynomial ring on n variables then dimG−KA = dimG−KQ(A) = n.Regarding q as an indeterminate and introducing filtration by powers of q −1one derives from the mentioned results of [G-K1] their “q−analogues”: dimen-sion of a ring of skew polynomials on n indeterminates is equal to n and ifGrA is isomorphic with a ring of skew polynomials on n indeterminates thendimG−KA = n. It follows from Proposition2.1 that dimG−KU −q (sln+1) =dimG−KΦ(U −q (sln+1)) = n(n + 1)/2 and, therefore, Φ is injective.✷3.2Complex powers, automorphisms and singular vectors3.2.1Construction of automorphisms of quotient rings of (deformed) univer-sal enveloping algebras and algebras of skew polynomials1.

For any k ∈N the mapCki : Q(U −q (g)) →Q(U −q (g)), x 7→F ki xF −kiis an automorphism. Clearly, Ck1+k2i= Ck1i ◦Ck2i .

Formulas (14, 16) imply thatCki (x) is a polynomial function of k. For example, in the sln+1−case one hasCki+1(Fi) = F ki+1FiF −ki+1 = {k}Fi+1FiF −1i+1 + {1 −k}Fi,(24)where we have used “symmetric” q−numbers: {k} =qk−q−kq−q−1 . Using this wedefine an automorphism Cki by analytic continuation.Therefore, with everyword F βlil · · · F β1i1 we have associated an automorphism Cβlil · · · Cβ1i1 of Q(U −q (g)).2.

Since in the case g = sln+1 the rings Q(U −q (g)) and Q(C[X]) are isomor-phic with each other ( Theorem3.5 ) the above provides the family of automor-phisms – also denoted by Cβlil · · · Cβ1i1 – of Q(C[X]). Moreover, the last assertionis valid for any g regardless of Conjecture 3.3.

In reality, there is a constructionof automorphisms of a ring of skew polynomials which has nothing to do withquantum groups.

Skew polynomials and quantum groups13Consider for simplicity the ring C[x] = C[x1, . .

. , xm], xjxi = q2xixj if j > i.To proceed we need a q−commutative version of the q−binomial theorem:(x1 + · · · + xm)n =Xi1+···+im=n[n]![i1]!

· · · [im]!xi11 xi22 · · · ximm n ∈N(25)For β ∈C we set(x1 + · · · + xm)β =∞Xj=0Xj1+···+jm−1=j[β][β −1] · · · [β −j + 1][j1]! · · · [jm−1]!xj11 · · · xjr−1r−1 xβ−jrxjrr+1 · · · xjm−1m,(26)for some 1 ≤r ≤m, thus making sense out of (x1 + · · · + xm)β as an element ofa certain completion of C[x] consisting basically of formal power series (thereare exactly m different ways to do that).Obviously the map p 7→(x1+· · ·+xm)βp(x1+· · ·+xm)−β is an automorphismof the above-mentioned completion.

An explicit calculation ( see below ) showsthatQ(C[x]) ∋p =⇒(x1 + · · · + xm)βp(x1 + · · · + xm)−β ∈Q(C[x])(27)Note that the same is true for (x1 + · · · + xm) replaced with (xi1 + · · · +xik), 1 ≤i1 < · · · < ik and – with minor restrictions – for C[x] replaced with anarbitrary ring of skew polynomials. In particular, in the case of the ring C[X]related to Uq(g) by the Feigin’s construction one obtains automorphismsCβi p = (Φ(Fi))βp(Φ(Fi))−β.Remarks.(i).

It is natural to set log Fi =ddk |k=0 Cki . By definition log Fi is a differen-tiation of Q(U −q (g)) as well as of Q(C[X]).

It is easy to see that, moreover, thisis an exterior differentiation. Problem: classify non-trivial ( exterior moduloinner ) automorphisms of Q(U −q (g)), Q(C[X]).(ii).

The set of words F βlil · · · F β1i1 , β1, . .

. βl ∈C is naturally equipped witha group structure.

With each such a word one may associate an infinite series:its expansion over a “Poincare-Birkhof-Witt type basis” F k, where complexpowers of Fi are allowed. (For details in classical setting see [M-F-F].) Thuswe have identified this group with a subgroup of a certain infinite-dimensionalgroup with a non-trivial topology.

In [Kh-Z] similar group was considered inthe classical case of differential operators on the line. In particular, it was shownthat this is a Poisson-Lie group.

Skew polynomials and quantum groups143. Calculation of (x1 + x2)βx2(x1 + x2)−β.

It is easy to see that the proofof (27) reduces to the case m = 2, p = x2. One has(x1 + x2)βx2(x1 + x2)−β = x2(q−2x1 + x2)β(x1 + x2)−β.The q−commutative version of the binomial theorem gives(q−2x1 + x2)β = q−2βxβ1∞Xi=0(q−2β)i(q2)i(−q2(β+1)x−11 x2)i(28)(x1 + x2)−β = {∞Xi=0(q2β)i(q2)i(−x−11 x2)i}x−β1 ,(29)where as usual (a)i = (1 −a)(1 −aq2) · · · (1 −aq2(i−1)).In order to show that in the product of the left hand sides of (28- 29) almosteverything cancels out we employ a commutative version of the q−binomialtheorem [G-R] which reads as follows:∞Xi=0(a)i(q2)izi = (az)∞(z)∞, z ∈C,(30)where (a)∞= Qi≥0(1 −aq2i).

( Although we are in the non-commutative realm the usage of (30) makessense for the right hand sides of (28- 29) basically involve only one “variable”x−11 x2. )By (30) the equalities (28- 29) are rewritten as follows:(q−2x1 + x2)β=q−2βxβ1(−q2x−11 x2)∞(−q2(β+1)x−11 x2)∞(31)(x1 + x2)−β=(−q2βx−11 x2)∞(x−11 x2)∞x−β1(32)Carrying out the multiplication one observes that almost all factors of infiniteproducts cancel out:(x1 + x2)βx2(x1 + x2)−β = q−2βx2xβ1 (1 + q2βx−11 x2)(1 + x−11 x2)−1x−β1=q−2βx2(1 + x−11 x2)(1 + q−2βx−11 x2)−1, (33)which completes the proof.

Skew polynomials and quantum groups153.2.2Application to singular vectors in Verma modulesIt follows from sect. 3.2.1 that elements of the formF slil · · · F s1i1 F Ni0 F t1i1 · · · F tlilbelong to Q(U −q (g)) if N ∈Z, si+ti ∈Z, 1 ≤i ≤l.

It was shown in [M-F-F, M]that such expressions are relevant to singular vectors in Verma modules. Herewe explicitly calculate them in the case g = sln+1.Recall that a Verma module M(λ), λ = (λ1, .

. .

, λn) ∈Cn is said to be aUq(g)−module on one generator vλ and the following defining relationsU +q (g)vλ = 0, Kivλ = qλii vλ i = 1, . .

. , n.It is easy to see that M(λ) is reducible if and only if it contains a singularvector, i.e.

a non-zero vector different from vλ and annihilated by U +q (g). Thereducibility criterion is the same as in the classical case [K-K, DC-K] and forUq(sln+1) reads as follows:M(λ) is reducible if and only if for some 1 ≤i < j ≤n, N ∈Nλi + λi+1 + · · · λj + j −i + 1 = N.(34)It is known that for a generic point λ on the hyperplane determined by(34) there is a unique (up to proportionality) singular vector in M(λ).

Thismeans that there is a function sending a point λ on the hyperplane to SNij (λ) ∈U −q (sln+1) so that the vector SNij (λ)vλ is singular. We are going to evaluateSNij (λ).

(34) can be rewritten in the following parametric formλj = N −tj −1,λj−1 = tj −tj−1 −1,λj−2 = tj−1 −tj−2 −1,. .

.λi+1 = ti+2 −ti+1 −1,λi = ti+1 −1.It follows from [M] thatSNij (t) = F tjj · · · F ti+1i+1 F Ni F N−ti+1i+1· · · F N−tjj. (35)Though (35) is not quite explicit it is sometimes most convenient for derivationof properties of singular vectors.

For example, playing with complex powers one

Skew polynomials and quantum groups16proves that singular vectors related to N > 1 are expressed in terms of singularvectors related to N = 1. One hasSNii+1(t) = F ti+1F Ni F N−ti+1=F ti+1FiF 1−ti+1 F t−1i+1 FiF 2−ti+1 · · · F t−N+1i+1FiF N−ti+1=S1ii+1(t)S1ii+1(t −1) · · · S1ii+1(t −N + 1).

(36)Arguing by induction one proves that likewiseSNij (t) = S1ij(t)S1ij(t −1) · · · S1ij(t −N + 1). (37)Therfore, it is enough to calculate S1ij(t).

It follows from (24) thatS1ii+1(t) = {t}Fi+1Fi + {1 −t}FiFi+1. (38)Using (38) several times one reduces (35) to a form containing only naturalpowers of generators.

Denote by P the set of all sequences ǫ = (ǫi+1, . .

. , ǫj),where each ǫm is either 0 or 1.

For each ǫ ∈P fix a bijection kǫ : {i+1, . .

. , j} →{i + 1, .

. .

, j} satisfyingk−1ǫ (m) < k−1ǫ (m −1)ifǫm = 1,k−1ǫ (m) > k−1ǫ (m −1)ifǫm = 0. ( Such a bijection obviously exists, though is not unique.

However, the finalresult is independent of a choice.) Further, with each ǫ ∈P associate a numberAǫ, given byAǫ =j−iYm=1{tm,ǫ},wheretm,ǫ =tmifǫm+i = 11 −tmifǫm+i = 0Theorem 3.6S1ij(t) =Xǫ∈PAǫFkǫ(i)Fkǫ(i+1) · · · Fkǫ(j),SNij (t) = S1ij(t)S1ii+1(t −1) · · · S1ij(t −N + 1).

Skew polynomials and quantum groups174Uq(g)−modules and q−connections4.1Modules U−q [S−1]vλLet S ⊂U −qconsist of homogeneous elements, and such that Uq[S−1] is well-defined. A typical example of S is a multiplicative span of an arbitrary subsetof {F1, .

. .

, Fn}. The following isomorphism of vector spaces is an analogue ofthe triangular decomposition:Uq[S−1] ≈U −q [S−1] ⊗U ≥q .

(Existence of this isomorphism follows from the relation[Ei, S−1j] = −S−1j[Ei, Sj]S−1j, which allows to commute E′s to the right.) De-note by Cλ a character of U ≥qdefined by Ei →0, Ki →qλi; 1 ≤i ≤n.A Verma module over Uq[S−1] is said to be Uq[S−1] ⊗U≥q Cλ.

Denote by vλthe image of 1 ⊗1 in Uq[S−1] ⊗U≥q Cλ.Clearly, Uq[S−1] ⊗U≥q Cλ is a freeU −q [S−1]−module generated by vλ. We shall be interested in the restriction ofUq[S−1] ⊗U≥q Cλ to Uq(g).

Due to the lack of better notation U −q [S−1]vλ willstand for this restriction.Note that the module U −q [S−1]vλ is always reducible for it contains a Vermamodule M(λ) = U −q vλ. Though its structure is unknown in general, we are ableto describe it in the simplest case when S is multiplicatively generated by oneof F ′s, say, Fi.It is easy to see thatEjF mi vλ = δij{m}F m−1iqλi−m+1i−q−λi+m−1iqi −q−1ivλ, m ∈Z.

(39)One realizes that U −q [F −1i]vλ is a module induced from the representaion ofthe parabolic subalgebra generated by E1, . .

. , En, Fi in the space ⊕m∈ZF mi vλ.Further, if λi is not in {−2, −3, .

. .} then (39) implies that Ei acts freely on thequotient module U −q [F −1i]vλ/M(λ).

It is now easy to show that U −q [F −1i]vλ/M(λ)is a Verma module related to a Borel subalgebra RiU ≥q twisted by the Lusztig’sautomorphism ( see (18,19,20) ) and the highest weight λ + αi.If, how-ever, λi does belong to {−2, −3, . .

.} then, as (39) implies, the vector F λi+1iis singular.This means that there arises a chain of submodules M(λ) ⊂M(λ + (λi + 1)αi) ⊂U −q [F −1i]vλ.

As above one shows that the quotient moduleU −q [F −1i]vλ/M(λ + (λi + 1)αi) is a Verma module related to a twisted Borelsubalgebra and the highest weight λ + (λi + 2)αi).For the sake of breavity, denote by RiMq(λ) an Uq(g)−module, isomorphicto M(λ) as a vector space with the action being twisted by Ri:Uq(g) ∋x 7→Rix 7→End(M(λ)).We have obtained

Skew polynomials and quantum groups18Proposition 4.1 If λi is not in {−2, −3, . .

.} thenU −q [F −1i]vλ/M(λ) ≈RiM(λ + αi).If λi ∈{−2, −3, .

. .} then there exists a chain of submodules M(λ) ⊂M(λ+(λi + 1)αi) ⊂U −q [F −1i]vλ andU −q [F −1i]vλ/M(λ + (λi + 1)αi) ≈RiM(λ + (λi + 2)αi).Observe that Proposition 4.1 along with its proof carries over to the case ofa quantum group attached to an arbitrary symmetrizable Cartan matrix A.4.2Modules realized in skew polynomials1.The Feigin’s embedding U −q→Cs[X] makes the latter into a U −q −module,action being defined by means of the left multiplication.One may want toextend this to an action of the entire Uq.

It is straightforward in view of theresults of the previous section if g = sln+1 for in this case Q(C[X]) ≈Q(Uq)(Theorem 3.5) and one obtains a family of modules Q(C[X])vλ (= Q(Uq)vλ).The module Q(C[X])vλ is definitely too big and it is natural to confine to thesmallest submodule containing C[X]vλ. This module is still always reducible,for example, it contains a Verma module Mq(λ) – the one generated by X1i +· · · + Xn−i+1,i, 1 ≤i ≤n – and U −q [F −1n ]vλ – the one generated by X1i + · · · +Xn−i+1,i, 1 ≤i ≤n −1, X±11n Though we do not have an explicit descriptionof this module in general we are able to consider the case of sl3 in full detail.Proposition 4.2 For generic λ Uq(sl3)·C[X11, X21, X12]vλ = C[X11, X21, X±112 ]vλ.For any λ C[X11, X21, X±112 ]vλ ≈U −q (sl3)[F −12]vλ.Proposition 4.1, therefore, determines the structure of C[X11, X21, X±12]vλ.As to the general case, the module in question should also be isomorphicto a module U −q [S−1]vλ for an appropriate set S determined by formulas ofLemma 3.4.2.

The above is relevant to the Gel’fand-Kirillov conjecture for Uq(sln+1).Denote by D[X] an algebra of q-difference operators acting on Q(C[X]). Inother words, D[X] is an algebra generated by Q(C[X]) viewed as operators ofleft multiplication and Tij, 1 ≤j ≤n, 1 ≤i ≤n −j + 1, whereTij : Xrs 7→ qXrsif(i, j) = (rs)Xrsif(i, j) ̸= (rs).Denote by D[X, λ] the trivial central extension of D[X] by commuting vari-ables qλ1, .

. .

, qλn, where λ = (λ1, . .

. , λn) is understood as a highest weight.The definition of the module Q(C[X])vλ implies that there exists a family ofembeddings – parametrized by λ – of Uq(sln+1) into D[X] or, equivalently, anembeddingρ : Uq(sln+1) →D[X, λ].

(40)

Skew polynomials and quantum groups19Conjecture 4.1 ρ provides an isomorphism of a quotient field a certain alge-braic extension ˆUq(sln+1) of Uq(sln+1) with a quotient field of a certain subalge-bra of D[X, λ].Construction of ˆUq(sln+1), which goes back to Gelfand and Kirillov [G-K2],is as follows:Identify Uq(sln+1) with ρ(Uq(sln+1)) ⊂D[X, λ] and define ˆUq(sln+1) to bethe subalgebra of D[X, λ] generated by Uq(sln+1), qλ1, . .

. , qλn and Kω1, .

. .

, Kωn,where ωi, 1 ≤i ≤n are dual fundamental weights, i.e. αj(ωi) = δji.

( It ismeant that KωEi = qαi(ω)iEiKω.) Note that elements qλ1, .

. .

, qλn generate acertain finite algebraic extension of the center of Uq(sln+1). ( Description of thecenter of Uq(sln+1) may be found in [DC-K].

)We have been able to verify the conjecture in the cases of sl2, sl3 by straight-forward calculation of ρ−1, which is simple in the sl2-case and rather tiresomein the sl3-case.Proposition 4.3(i) Q( ˆUq(sl2)) ≈D[X, λ]. (In this case X stands for X11.

)(ii)Q( ˆUq(sl3)) is isomorphic with the quotient field of the subalgebra ofD[X, λ] generated by T 211, T 221, T11T21, T12; X11, X21, X12; qλ1, qλ2.4.3Uq(g)−modules and q−connections.4.3.1A q−connection with coefficients in a ring of skew polynomials.Let C[x] := C[x1, . .

. , xn], xjxi = q2xixj, i < j be a ring of skew polynomials(as yet it has nothing to do with quantum groups) and D[x] be the correspondingring of q−difference operators.By a quantum line bundle we mean a free rank 1 module over C[x] or, moregenerally, C[x][S−1 for a suitable S. Sections of a quantum line bundle, i.e.elements of C[x][S−1], therefore become a D[x]-module.

In other words therearises an inclusion∇triv : D[x] ֒→End(C[x][S−1]).By a q−connection with coefficients in a quantum line bundle we mean anassociative algebra homomorphism ∇:D[x] →End(C[x][S−1]) such that∇(xi) = L(xi), ∇(Ti) = R(bi(x))Ti, 1 ≤i ≤n, where bi(x) ∈Q(C[x]) andL(xi), R(bi(x)) stand for the operator of the left or right (resp.) multiplicationby xi or bi(x) (resp.

).The same can be equivalently described in terms of cohomology. For χ =(χ1, .

. .

, χn) ∈Zn set T χ = T χ11◦· · · ◦T χnnand ∇χ = ∇(T χ). Obviously, ∇χ =R(bχ(x))T χ for some bχ(x) ∈Q(C[x]).

The associative algebra homomorphismcondition reads asbχ1+χ2(x) = (T χ1bχ2(x))bχ1(x).

Skew polynomials and quantum groups20The last equality simply means that the map Zn ∋χ 7→bχ(x) ∈Q(C[x]) is a1-cocycle of an abelian group Zn with coefficients in Q(C[x]). In this languagethe above ∇triv is related to the cocycle χ 7→1.It is natural to say that a cocycle is trivial if it is given by bχ(x) = (T χr(x))r−1(x)for some r(x) ∈Q(C[x]).Indeed the cocycle χ 7→(T χr(x))r−1(x) makes into thecocycle χ 7→1 by “the change of trivialization”: f(x) 7→f(x)r(x)−1.

The co-cycle χ 7→(T χr(x))r−1(x) is a coboundary of the 0-cocycle r(x). Therefore wehave established a 1-1 correspondence between non-trivial q−connections andelements of H1(Zn, Q(C[x])).To produce a construction of some elements of H1(Zn, Q(C[x])) fix arbitrarysubsets J1, .

. .

, Jl of {1, . .

. , m} and setri =Xj∈Jixj.Let Ψ = rβ11 rβ22 · · · rβllfor some β1, .

. .

, βl ∈C ( see sect.3.2.1 ).Lemma 4.2 The correspondence χ 7→Ψχ = (T χΨ)Ψ−1 represents an elementof H1(Zn, Q(C[X])).Proof. The fact that Ψχ = (T χΨ)Ψ−1 is a 1-cocycle is obvious for this is acoboundary of Ψ.

(One may also think of it as the “local” change of trivializationf(x) 7→f(x)Ψ(x) in the bundle with the trivial q−connection.) What has to beproven is that Ψχ ∈Q(C[X]) for any χ.

To do this observe thatx−1i1 −T 2i1 −q2 (x1 + · · · + xm)β = [β](q−2x1 + · · · + q−2xi−1 + xi + · · · + xm)β−1.The calculation as in (31,32,33) shows that (q−2x1 + · · · + q−2xi−1 + xi + · · · +xm)β−1 = p(x1 + · · · + xm)β, p ∈Q(C[x]). It implies thatT 2i (x1 + · · · + xm)β = {1 −(1 −q2)[β]xip}(x1 + · · · + xm)β.Therefore, a q−difference operator makes Ψ into p1rβ11 · · · plrβllfor somep1, .

. .

, pl ∈Q(C[x]). To rewrite the latter in the form pΨ, p ∈Q(C[x]) onewants to move each pi to the left.

This can be done by using the automorphismsQ(C[x]) ∋q 7→rβi qr−βi,see (27). One hasp1rβ11 · · · plrβll= p1 ˜p2 · · · ˜plΨ,where ˜pj = rβ11 · · · rβj−1j−1 pjr−βj−1j−1· · · p−β11, 2 ≤j ≤l.

✷We will denote by ∇(Ψ) the connection χ 7→Ψχ = (T χΨ)Ψ−1 given byLemma 4.2.

Skew polynomials and quantum groups21In the classical case tensor product of a pair of trivial line bundles with flatconnections is a trivial line bundle equipped with a canonical flat connection.This gives an operation on connections. In classical case connections are alsoidentified with a certain 1st cohomology group and this operation happens tobe simply an addition.

Though we are unable to carry out the same in fullgenerality in the q−commutative realm, we can produce the following non-commutative operation on the q-connections of the form ∇(Ψ):∇(Ψ1) q⊗∇(Ψ2) = ∇(Ψ1Ψ2).This operation is obviously a q−analogue of an addition of 1-cocycles in theclasscal setting.Remark. Our approach here is a q−commuative version of that of Aomotoand Kato in [A-K].

In particular the construction of cocycles in Lemma 4.2 hasits commuative counterpart, which is claimed to possess a sort of universalityproperty. The same may be true – with minor modifications – in our case.4.3.2Uq(g)-modules twisted by a q−connection.1.

Intertwining operators. Of course everything written in the previous sectionapplies to more general rings of skew polynomials provided one takes more careabout the choice of elements r1, .

. .

, rl. For example, in the case of an algebraC[X], coming from the Feigin’s morphism Φ : U −q→C[X] ( see sect.

3.1 ),a natural choice is rj = Φ(Fij) for an arbitrary sequence i1, . .

. , il.

( Thereare some others which one can easily think of.) In the case of g = sln+1 theUq(sln+1)−module structure on Q(C[X]) implies a homomorphism ( see (40) ):ρλ : Uq(sln+1) →D[X].

(41)Given a q−connection ∇one twists this Uq(sln+1)−module structure by∇◦ρλ : Uq(sln+1) →D[X]. (42)As above denote by ∇(Ψ) the q−connection coming from(T χΨ)Ψ−1 ∈H1(ZN, Q(C[X])) , Ψ = (Φ(Fi1))β1 · · · (Φ(Fil))βl for some β1, .

. .

, βl ∈C. Let ri ∈W be a reflection at the simple root αi.

Setβj = 2(ril+2−j · · · ril · λ, αil+1−j)(αil+1−j, αil+1−j)+ 1,(43)where ri·λ stands for the shifted action of the Weyl group. Set ∇(ri1 · · · ril; λ) =∇(Ψ) if β1, .

. .

, βl are given by (43).Proposition 4.4 There is a Uq(sln+1)−linear map of the module related to∇(Ψ) ◦ρw·λ into the one related to ∇(Ψ) q ⊗∇(ri1 · · · ril; λ) ◦ρλ, where w =ri1 · · · ril.

Skew polynomials and quantum groups22Proof Passage from ρµ to ∇(Ψ) ◦ρµ means that one replaces vµ (i.e. unitof C[X])) with F β1i1 · · · F βlil vµ.

(See the proof of Lemma4.2.) Formula (39)implies that under the choice (43) the vector F β1i1 · · · F βlil vλ is singular of theweight ri1 · · · ril · λ and, therefore, satisfies all the conditions imposed on vw·λ.✷Observe that having looked over the definitions one can make a precise senseout of the statement:the module related to ∇(Ψ) ◦ρw·λ embeds into the one related to ∇(Ψ) q⊗∇(ri1 · · · ril; λ)◦ρλ as a space of sections satisfying a certain regularity condition.2.

Structure of modules U −q [F −1i]F βi vλ. Here we obtain the structure de-scription of the modules twisted by a q−connection in the simplest case of∇= ∇((Φ(Fi))β), β ∈C.

This module always contains a submodule generatedby 1 ∈C[X] and, as one easily sees, isomorphic with the following extension ofa Verma module: U −q [F −1i]F βi vλ. The Uq(g)−module structure on the latter isdefined as follows:(i) F1, .

. .

, Fn act by left multiplication;(ii) action of E1, . .

. , En is determined by setting ( c.f.

( 39) )EjF βi vλ = δij{β}F β−1iqλi−β+1i−q−λi+β−1iqi −q−1ivλ, β ∈C. (44)Denote by iU +q the parabolic subalgebra of Uq(g) generated byE1, .

. .

, En; K±11 , . .

. K±1n ; Fi.

The equation ( 44) determines a structure of iU +q -module on the space spanned by F β+ki, k ∈Z. Denote a module obtained inthis way by Viλ.

Clearly, U −q [F −1i]F βi vλ is isomorphic with the unduced moduleIndUqiU+q Viλ. This isomorphism provides a precise analogy between U −q [F −1i]F βi vλand a Verma module U −q vλ: one is obtained from another by replacing thevacuum vector vλ with the vacuum chain Viλ.This analogy can be pushedfurther by remarking that U −q [F −1i]F βi vλ is reducible if and only if it contains asingular chain in much the same way as a Verma module is reducible if and onlyif it contains a singular vector.

Here by a singular chain we naturally mean a non-zero iU +q -linear map Viµ ⊂U −q [F −1i]F βi vλ different from Viλ ⊂U −q [F −1i]F βi vλ.A weight lattice of a singular chain Viµ is of the form µ + Zαi. By the weightof a singular chain Viµ we mean an element ¯µ ∈h∗/Zαi, where ¯µ stands for animage of µ under the natural projection h∗→h∗/Zαi.

The following partiallyrelies on sect. 4.1.Theorem 4.3 (i)If β ∈Z or β ∈λ(Hi) + Z then U −q [F −1i]F βi vλ is isomorphicwith either U −q [F −1i]vλ or U −q [F −1i]vri·λ (resp.

); see Proposition 4.1. (ii) Otherwise U −q [F −1i]F βi vλ is reducible ⇐⇒it contains a singular chain ofthe weight λ −Nα for some α ∈∆+, α ̸= αi, N ∈N ⇐⇒there is j ∈Z, N ∈

Skew polynomials and quantum groups23N such that(λ + ρ, α + jαi) = N2 (α + jαi, α + jαi), α + jαi ∈∆+,where ρ ∈h∗is determined by ρ(Hk) = 1, 1 ≤k ≤n.Remark. The Verma module M(λ) contains a singular vector of the weightλ −Nα, α ∈∆+ if and only if λ belongs to the Kac-Kazhdan hyperplane( [K-K], see also sect.

3.2.2) related to the pair (α, N):(λ + ρ, α) = N2 (α, α), α ∈∆+. (45)Item (ii) of Theorem 4.3 claims that U −q [F −1i]F βi vλ contains a singular chainof the weight λ −Nα if and only if λ belongs to the union of Kac-Kazhdanhyperplanes related to all pairs (α+jαi, N), α+jαi ∈∆+.

Therefore a singularchain encodes information on a collection of singular vectors in a Verma module.Proof of Theorem4.3. Item (i) immediately follows from definitions and( 44).

As to (ii), fix root vectors Eα, α ∈∆+ as in (21). It is clear that thespace spanned by all singular chains coincides with the space of solutions of thefollowing system of linear equationEαw = 0 for all α ̸= αi(46)(This system should be regarded as resticted to each weight space of the mod-ule.

)One deduces from Proposition2.1 the space of solutions to ( 46) is a −module from which one easily extracts a singular chain.All the vector spaces U −q [F −1i]F βi vλ parametrized by β ∈C are naturallyisomorphic with each other and with the space U −q [F −1i]. Therefore ( 46) maybe regarded as a family of systems of linear equations on U −q [F −1i] polynomiallydepending on β.

For a fixed weight space of U −q [F −1i] existence of solutionsto ( 46) lying in this space is equivalent to vanishing of a certain polynomialdepending on λ and β. It is easy to deduce, however, that under the assumptionsof (ii) once there is a solution for β = β0 then there are solutions for infinite manyvalues β ∈β0 + Z.

Therefore the mentioned polynomial is actually independentof β. Now we may set β = 0 without lack of generality.

But then it is easyto see that – again under the assumptions of (ii) – the same system ( 46) givesreducibility criterion for the submodule M(λ) of U −q [F −1i]vλ, see Proposition4.1. The proof now follows from the Kac-Kazhdan equations, see the aboveRemark.

✷The Kac-Kazhdan reducibility criterion ( 45) was carried over to the case ofa quantum group attached to an arbitrary symmetrizable Cartan matrix A in[M]. It follows that Theorem 4.3 remains valid in this general setting.

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