Poisson orders on large quantum groups
aa r X i v : . [ m a t h . QA ] A ug POISSON ORDERS ON LARGE QUANTUM GROUPS
NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO, AND MILEN YAKIMOV
Abstract.
We bring forward the notions of large quantum groups and their relatives.The starting point is the concept of distinguished pre-Nichols algebra [An3] belongingto a one-parameter family; we call such an object a large quantum unipotent subalge-bra. By standard constructions we introduce large quantum groups and large quantumBorel subalgebras. We first show that each of these three large quantum algebras hasa central Hopf subalgebra giving rise to a Poisson order in the sense of [BG]. Wedescribe explicitly the underlying Poisson algebraic groups and Poisson homogeneousspaces in terms of Borel subgroups of complex semisimple algebraic groups of adjointtype. The geometry of the Poisson algebraic groups and Poisson homogeneous spacesthat are involved and its applications to the irreducible representations of the algebras U q ⊃ U > q ⊃ U + q are also described. Multiparameter quantum super groups at roots ofunity fit in ou context as well as quantizations in characteristic 0 of the 34-dimensionalKac-Weisfeler Lie algebras in characteristic 2 and the 10-dimensional Brown Lie alge-bras in characteristic 3. All steps of our approach are applicable in wider generalityand are carried out using general constructions with restricted and non-restricted in-tegral forms and Weyl groupoid actions. Our approach provides new proofs to resultsin the literature without reductions to rank two cases. Contents
1. Introduction 22. Poisson orders and restrictions to central subalgebras 73. Hopf algebras 94. Large quantum groups 145. The specialization setting for large quantum groups 206. Poisson orders on large quantum groups 237. The associated Poisson algebraic groups 258. Poisson geometry and representations 34Appendix A. Families of finite-dimensional Nichols algebras 38Appendix B. Lie bialgebras and Poisson algebraic groups 41References 43
Key words and phrases.
Hopf algebras, Nichols algebras, integral forms of quantum groups, Poissonorders, Poisson algebraic groups and homogeneous spaces.MSC2020: 16T05, 16T20, 17B37, 17B62.This material is based upon work supported by the National Science Foundation under Grant No.DMS-1440140 while N. A. was in residence at the Mathematical Sciences Research Institute in Berkeley,California, in the Spring 2020 semester. The work of N. A. and I. A. was partially supported byCONICET and Secyt (UNC). The work of M.Y was partially supported by NSF grant DMS-1901830and Bulgarian Science Fund grant DN02/05. Introduction
Quantum groups and Poisson orders.
Let g be a complex finite-dimensionalsimple Lie algebra and let ξ ∈ C be a root of 1 with some restrictions on its orderdepending on g . In the papers [DK, DKP, DP] a quantized enveloping algebra U ξ ( g ) at ξ was introduced and studied; it is a version of the Drinfeld-Jimbo quantized universalenveloping algebra different from the one defined in [L1, L2].The algebra U ξ ( g ) is module-finite over a central Hopf subalgebra Z ξ ( g ) and the corre-sponding small quantum group of Lusztig [L1, L2] arises as the quotient U ξ ( g ) //Z ξ ( g ) inthe sense of Hopf algebras. A geometric approach to the representation theory of U ξ ( g ) was proposed in [DP], based on these facts. The key ingredients of this approach are: ◦ The existence of a Poisson structure on Z ξ ( g ) so that the algebraic group M corre-sponding to this algebra is a Poisson algebraic group, whose Lie bialgebra is dual tothe standard Lie bialgebra structure on g . ◦ The Hamiltonian vector fields on M extend to (explicit) derivations of U ξ ( g ) .The approach consists in packing the irreducible finite-dimensional representations of U ξ ( g ) along the symplectic leaves of M and predicting their dimensions. These ideaswere distilled in the notion of Poisson order in [BG], see Section 2. The construction of aPoisson order structure on an algebra has substantial applications to the representationtheory of the algebra: using this route the irreducible representations of quantum func-tion algebras were studied in [DL], the Azumaya loci of symplectic reflection algebras wasdescribed in [BG], the irreducible representations of the 3 and 4-dimensional PI Sklyaninalgebras were fully classified in [WWY1, WWY2], the Azumaya loci of the multiplicativequiver varieties and quantum character varieties were studied in [GJS]. See [BGo, PartIII] for a comprehensive exposition of the applications the notion of Poisson order to therepresentation theory of quantum algebras at roots of unity.1.2. Large quantum groups and pre-Nichols algebras.
The main goal of this paperis to study by means of Poisson orders the representation theory of a larger class of Hopfalgebras introduced by the second author in [An2] and studied in [An3]. The keystoneof the definition of these Hopf algebras is the notion of distinguished pre-Nichols algebra .Nichols algebras of diagonal type are essential for various classification problems ofHopf algebras. Those with finite dimension were classified in the celebrated paper [H2]while the defining relations were provided in [An1, An2]. Let q be a braiding matrix asin the list of [H2] and let B q be the corresponding finite-dimensional Nichols algebra ofdiagonal type. The distinguished pre-Nichols algebra e B q of B q is a covering of the latterdefined by excluding the powers of the root vectors of Cartan type from its definingideal. The Hopf algebras dealt with in the present paper are Drinfeld doubles of thebosonizations of the distinguished pre-Nichols algebras; they are denoted U q , see §4.3and are module-finite over the central Hopf subalgebra Z q defined in [An3], see §4.5. Onthe other hand, the graded dual of e B q gives rise to a Lie algebra n q which is either 0 orthe nilpotent part of a semisimple Lie algebra g q that is explicitly determined [AAR3].We focus on Hopf algebras U q with a further restriction: the related Nichols algebra B q is deformable, i.e. belongs to a one-parameter family of Nichols algebras. We callthem large quantum groups . By inspection, the matrix q is of one of three types:(a) Cartan type (multiparameter versions of the quantum groups from [DKP] withoutrestrictions on ξ ); OISSON ORDERS ON LARGE QUANTUM GROUPS 3 (b) super type (multiparameter quantum groups associated to finite dimensional simplecontragredient Lie superalgebras at roots of unity);(c) modular types wk (4) or br (2) (quantizations at root of unity of some simple Liealgebras in characteristics 2 and 3 respectively).But it stems from the list in [H2] that there are finite-dimensional Nichols algebras ofdiagonal type that do not belong to such one-parameter families. Remark 1.1.
To be precise we need three technical assumptions:(i) The base field is C to have on hand symplectic leaves, although this is not essential,see for details [BG].(ii) Condition (4.23) is needed for the centrality of Z q in U q .(iii) The Non-degeneracy Assumption 7.5 is used to identify some dual vector spacesin order to compute some Lie bialgebras.See the Appendix A and the survey [AA] for full details on these algebras. We considerthe chain of subalgebras U + q ⊂ U > q ⊂ U q where ◦ U > q , the large quantum Borel subalgebra , is identified with the bosonization of e B q ; ◦ U + q , the large quantum unipotent subalgebra , is identified with e B q .Intersecting the central subalgebra Z q of U q gives the chain of central Hopf subalgebras(1.1) Z + q ⊂ Z > q ⊂ Z q . Each of these central Hopf subalgebras is actually isomorphic to a tensor product ofa polynomial algebra and a Laurent polynomial algebra. The maximal spectra of theHopf algebras Z q , Z > q and Z + q are the complex algebraic groups M q , M > q and M + q ,respectively. We shall also need the opposite Borel U q and its central Hopf subalgebra Z q with maximal spectrum M q and correspondingly U − q , Z − q and M − q .1.3. Main results.
As said this paper deals with the geometry of the Poisson algebraicgroup M q towards understanding the representation theory of large quantum groups.This last question contain the description of the irreducible representations of quantumsupergroups at roots of unity, an important problem which is wide open even in thesimplest case of U q ( sl ( m | n )) . We present a foundation for a thorough investigation ofthese representations. We first summarize the main results in the following statement. Theorem A.
Let U q be a large quantum group as above. Then (a) The pair ( U q , Z q ) has the structure of a Poisson order in the sense of [BG] . (b) The algebraic Poisson group M q is solvable . The Lie bialgebra of M q is dual to thestandard Lie bialgebra structure on g q ; hence the symplectic leaves of M q are known. (c) Every z ∈ M q with corresponding maximal ideal M z gives rise to a finite-dimensionalalgebra H z = U q /U q M z . Then H z ≃ H z ′ whenever z and z ′ belong to the samesymplectic leave S . By abuse of notation we set H S = H z . Every irreducible repre-sentation of U q is finite-dimensional and Irr U q = [ S symplectic leave of M Irr H S . Furthermore, we have analogous results for the pairs ( U > q , Z > q ) and ( U + q , Z + q ) . Nextwe make more precise the claims of Theorem A. We fix a large quantum group U q . NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO, AND MILEN YAKIMOV
Poisson orders.
We denote by Z ( A ) the center of an algebra A . Because of theassumption that B q is deformable in the class of Nichols algebras as mentioned above,we get Poisson order structures on the pairs ( U q , Z ( U q )) , ( U > q , Z ( U > q )) and ( U + q , Z ( U + q )) by specialization. As these centers are singular, it is more convenient to look at thecentral subalgebras in (1.1). Part (a) of Theorem A is included in the following result,see Theorem 6.2. Theorem B.
The pairs ( U q , Z q ) , ( U > q , Z > q ) and ( U + q , Z + q ) have Poisson order structuresin the sense of [BG] obtained from specialization. Presently it is not known whether for the remaining braiding matrices q in the listof [H2] the pair ( U q , Z q ) has the structure of a Poisson order. Indeed the other Nicholsalgebras of diagonal type with arithmetic root system in the classification given in [H2]do not admit such a one-parameter family and for instance our proof of Theorem 6.2does not generalize to them.1.3.2. Poisson algebraic groups and Lie bialgebras.
Recall the semisimple Lie algebra g q determined in [AAR3] and fix a Cartan subalgebra h q . We consider on g q the Lie bial-gebra structure with the standard Belavin–Drinfeld triple [BD] that we extend triviallyto g q ⊕ h q , see [ES, §4.4]. Let m q be the Lie bialgebra dual to g q ⊕ h q .Let G q be the semisimple algebraic group of adjoint type with Lie G q ≃ g q . Forinstance, when U q = U q ( sl ( m | n )) , G q ≃ PSL m ( C ) × PSL n ( C ) . Let B + q be a Borelsubgroup of G q , T q B + q a maximal torus and N + q B + q the unipotent radical; weidentify N + q ≃ B + q /T q . Also B − q is the opposite Borel subgroup and N − q B − q is itsunipotent radical.Here is a more precise statement of Theorem A Part (b), see Theorems 7.10 and 8.2. Theorem C. (a)
The Poisson algebraic group M q is isomorphic to the product of twoBorel subgroups of G q and Lie M q ≃ m q as Lie bialgebras. (b) The symplectic leaves of M q are in bijective correspondence with the coadjoint orbitsof G q ; each leaf is isomorphic to an open dense subset of the corresponding coadjoint orbit. Here are the promised versions for M > q and M + q . Theorem D. (a)
The Poisson algebraic group M > q is isomorphic to the Borel subgroup B + q . The Poisson structure is invariant under the left and right actions of T q . (b) The torus orbits of symplectic leaves of M > q are the double Bruhat cells of G q thatlie in B + q . (c) The algebraic group M + q is isomorphic to the unipotent radical N + q of B + q . It has aPoisson structure arising from the identification N + q ≃ B + q /T q which is invariant underthe left action of T q and is a reduction of the Poisson structure on B + q from (a) underthe right action of T q . (d) The torus orbits of symplectic leaves of M + q are the open Richardson varieties ofthe flag variety G q /B + q that lie inside an open Schubert cell identified with N + q . See Theorems 8.4 and 8.7. we refer to [FZ, KLS] for information on double Bruhatcells and open Richardson varieties respectively.
OISSON ORDERS ON LARGE QUANTUM GROUPS 5
Representations.
Since U q is a free Z q -module of finite rank, it is a PI-algebra. Let V be an irreducible representation of U q ; by the preceding V is finite-dimensional and bythe Schur Lemma, Z q acts on V by some z ∈ M q (a central character) with correspondingmaximal ideal M z . Now the algebra H z = U q /U q M z is non-zero and finite-dimensionaland V becomes a H z -module. In other words the irreps of U q with central character z are in bijective correspondence with the irreps of H z . Thus Irr U q = [ z ∈ M q Irr H z . This circle of ideas is already present in [DP]. In this way, Part (c) of Theorem A boilsdown to the following statement.
Theorem E.
For every two points z, z ′ in the same symplectic leaf of M q , the algebras H z and H z ′ are isomorphic. In particular there is a dimension preserving bijection betweenthe irreps of U q with central characters z and z ′ . See Theorem 8.2. For instance, let z = e be the identity of M q . Then its symplecticleaf is S = { e } and H S = H e is the Drinfeld double of a suitable bosonization of theNichols algebra B q . Assume that the matrix q is of Cartan type. Then H e is a variationof the small quantum group of Lusztig (with an extra copy of the finite torus), witha notoriously difficult representation theory treated intensively in the literature. Also,arguing as in [DP] one concludes that U q is a maximal order, hence for generic z , H z issemisimple. But for super and modular types, the representation theory of H e is largelyunknown, except for the somewhat standard fact that simple modules are classified byhighest weights (but there is not even a conjecture for their characters). Also, U q is nota maximal order because it has nilpotent elements.We next write down the corresponding formulations for M > q , M q , M + q and M − q . Let ⋆ ∈ { > , , + , −} . If z ∈ M ⋆ q , then we denote by M ⋆z its maximal ideal in Z ⋆ q and H ⋆z = U ⋆ q /U ⋆ q ( M ⋆z ) . (1.2)Clearly these are finite-dimensional algebras. Theorem F. (a)
For every z, z ′ in the same double Bruhat cell of B + q , the algebras H > z and H > z ′ are isomorphic. Analogously for H z and H z ′ . (b) For every z, z ′ in the same open Richardson variety, the algebras H + z and H + z ′ areisomorphic. Analogously for H − z and H − z ′ . See Theorems 8.4 and 8.7.Notice also that H z is a Hopf-Galois H e -object since U q is a cleft H ε -comodule algebra,see §3.1. Analogously, H ⋆z is a Hopf-Galois H ⋆e -object for ⋆ ∈ { > , , + , −} .1.4. Strategy and organization.
Our proofs of Theorems C–F follow a different strat-egy from that of [DK, DKP, DP]. These papers rely on direct computations of Poissonbrackets in terms of coordinates coming from Cartesian products of one-parameter unipo-tent groups and subsequent reductions to the rank 2 case. This approach does not workin the more general context of §1.2 for several reasons, the simplest of which is that thequantum Serre relations for quantum supergroups or for quantum groups at − involvemore than two Chevalley generators. NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO, AND MILEN YAKIMOV
Instead our approach is based on intrinsic properties of pairings between restricted andnon-restricted integral forms of Hopf algebras. It does not rely on reduction to low rankcases. In particular, this approach provides new proofs of results in [DK, DKP, DP]. Weexpect that these ideas could be applied to other situations not covered in this paper.Next we overview briefly the main steps of the strategy:
Step 1.
Let C ( ν ) be the field of rational functions on q and A the subalgebra defined in(5.1). Since q belongs to a family, there exists a chain of C ( ν ) -algebras U + q ⊂ U > q ⊂ U q and non-restricted integral forms over A U + q , A ⊂ U > q , A ⊂ U q , A such that the algebras U + q ⊂ U > q ⊂ U q arise as specializations from these integral forms.This provides Poisson order structures on the pairs ( U + q , Z ( U + q )) , ( U > q , Z ( U > q )) and ( U q , Z ( U q )) . This step is carried out in Section 5 in the framework of [DP, BG] evokedin Section 2. Step 2.
We use Theorem 2.3 (on the restriction of Poisson order structures obtainedfrom specialization to central subalgebras) to prove that the Poisson order structureson ( U + q , Z ( U + q )) and ( U > q , Z ( U > q )) restrict to ( U + q , Z + q ) and ( U > q , Z > q ) . To get a Poissonorder structure on ( U q , Z q ) by restriction from ( U q , Z ( U q )) , we need first to establish inTheorem 4.7 that the Weyl groupoid action preserves the central subalgebras Z q . Alongthe way we also obtain that these Poisson structures on the algebras Z q are equivariantunder the Weyl groupoid. This step is carried out in Section 6. Step 3.
This is the matter of Section 7 We introduce in §5.4 non-restricted integralforms U res ± q , A of U ± q and A -linear perfect pairings U res ± q , A × U ∓ q , A → A . We prove(i) the specializations of U res ± q , A are isomorphic to the Lusztig algebras defined in[AAR1], see Proposition 5.9;(ii) the cobrackets of the tangent Lie bialgebras to M > q and M q are linearizations ofthose specializations, see Proposition 7.1.In this way we control tangent Lie bialgebras intrinsically and consequently we computein Theorems 7.4 and 7.8 the tangent Lie bialgebras of the Poisson algebraic groups M q , M > q and M q by means of a Manin pair. Since these algebraic groups are connectedwe describe them as Poisson algebraic groups in terms of Borel subgroups of complexsemisimple algebraic groups of adjoint type. Also, M ± q are presented as Poisson homo-geneous spaces.Finally, we discuss in Section 8 the Poisson geometry of the Poisson algebraic groups M q , M > q and the Poisson homogeneous space M + q , and the applications to the irreduciblerepresentations of U q , U > q and U + q .Besides, we discuss in Section 2 Poisson orders and their restrictions to central sub-algebras, see Theorem 2.3; Section 3 is devoted to preliminaries on Hopf algebra theorywhile we present the main actors of this paper in Section 4. Acknowledgements.
This project started in visits of M. Y. to the University of Cór-doba in September 2017 and December 2018 supported by the program of guest professorsof FaMAF. It was continued during visits of I. A. and N. A. in February 2019 to theLousiana State University. Progress on this material was reported at the plenary talk of
OISSON ORDERS ON LARGE QUANTUM GROUPS 7
I. A. at the XXIII
Coloquio Latinoamericano de Álgebra , Mexico City (2019) and at thetalk of M. Y. at the International conference on Hopf algebras, Nanjing (2019).
Notations.
The base field is C ; all algebras, Hom’s and tensor products are over C . If t ∈ N , n ∈ N and t < n , then I t,n := { t, t + 1 , . . . , n } , I n := I ,n .For each integer N > , let G N be the group of N -th roots of unity in C and let G ′ N be its subset of primitive roots (of order N ). Also G ∞ = S N ∈ N G N , G ′∞ = G ∞ − { } .2. Poisson orders and restrictions to central subalgebras
This section contains background on Poisson orders, their construction from special-izations, and their relations to Hopf algebras. We prove a general result on restrictionsof Poisson orders to central subalgebras, Theorem 2.3, which plays a key role later.2.1.
Poisson orders.
Here we follow the exposition in [DP, Chapter 3, §11]. Consider ◦ a commutative C -algebra A and h ∈ A such that A/h ≃ C , ◦ an A -algebra U such that h is not a zero divisor of U . The natural map U → U/ ( h ) is denoted by x x .For any u ∈ U such that u ∈ Z ( U/ ( h )) there is a linear map D u ∈ Hom U/ ( h ) given by D u ( y ) = [ u, v ] h , if y = v. (2.1) Proposition 2.1. [DP, 11.7]
Let u ∈ U such that u ∈ Z ( U/ ( h )) . (a) D u ∈ Der U/ ( h ) . (b) Let w ∈ U . If u ′ = u + hw so that u = u ′ , then D u − D u ′ = ad w is an innerderivation. Conversely the inner derivation ad w coincides with D hw . (c) Let ϕ ∈ Aut A − alg ( U ) and let ϕ be the induced automorphism of U/ ( h ) . Then ϕ ◦ D u ◦ ϕ − = D ϕ ( u ) . (d) There is natural Poisson structure on Z := Z ( U/ ( h )) given by { x, y } = D u ( y ) = [ u, v ] h , if x = u, y = v. (2.2)(e) The map ϕ ϕ gives a group homomorphism Aut A − alg ( U ) → Aut
Poisson ( Z ) . (f) L = { D v : v ∈ U, v ∈ Z} is a Lie subalgebra of
Der U/ ( h ) . Indeed [ D u , D v ] = D [ u,v ] h , v ∈ U, v ∈ Z . (g) The Poisson structure gives rise to a Lie subalgebra L ′ of Der Z that fits into thecomplex / / Innder( U/ ( h )) / / L / / L ′ / / . (2.3) The sequence (2.3) is exact if and only if the Poisson center of Z is trivial (i.e.,there are no Casimir elements except 0). (cid:3)
Brown and Gordon [BG] axiomatized the ingredients of the above setting as follows:
Definition 2.2.
A pair of C -algebras ( R, Z ) is called a Poisson order if Z is a centralsubalgebra of R , R is a Z -module of finite rank and the following two conditions hold:(a) Z is equipped a structure of Poisson algebra {· , ·} ;(b) There exists a linear map D : Z → Der C ( R ) such that D z | Z = { z, −} for all z ∈ Z . NICOLÁS ANDRUSKIEWITSCH, IVÁN ANGIONO, AND MILEN YAKIMOV
Proposition 2.1 proves that the pair ( U/ ( h ) , Z ( U/ ( h ))) has a canonical structure ofPoisson order when U/ ( h ) is module finite over Z ( U/ ( h )) . The Poisson bracket on Z ( U/ ( h )) is given by (2.2). The linear map D is the map induced from the one in (2.1)by taking a linear section of the canonical projection U → U/ ( h ) .2.2. Restrictions of Poisson orders from specializations.
In the setting of Propo-sition 2.1 the center Z = Z ( U/ ( h )) can be singular and is more useful to work withsuitable subalgebras Z ′ . Next we prove a general fact for the construction of Poissonorders on pairs ( U/ ( h ) , Z ′ ) for subalgebras Z ′ defined from algebra automorphisms andskew-derivations. For this purpose we fix: • A -algebra endomorphisms ς i : U → U , i ∈ I . We denote by ς i the corresponding C -algebra endomorphisms of U/ ( h ) induced by ς i . • A -linear (id , ς i ) -derivations ∂ i : U → U , i ∈ I . We denote by ∂ i the corresponding C -linear (id , ς i ) -derivations induced by ∂ i . Theorem 2.3.
The Poisson order structure on ( U/ ( h ) , Z ( U/ ( h ))) from Proposition 2.1restricts to a Poisson order structure on ( U/ ( h ) , Z ′ ) , where Z ′ := Z ∩ (cid:0) ∩ i ∈ I ker ∂ i (cid:1) ∩ ( ∩ i ∈ I ker( ς i − id)) . (2.4) Proof.
We have to check that {Z ′ , Z ′ } ⊂ Z ′ . Let x j ∈ Z ′ and u j ∈ U such that x j = u j , j = 1 , . Fix i ∈ I . As ς i ( x j ) = x j and ∂ i ( x i ) = 0 , there are v j , w j ∈ U such that ς i ( u j ) = u j + h v j , ∂ i ( u j ) = h w j , j = 1 , . Now we compute ς i { x , x } = ς i [ u , u ] h ! = [ ς i ( u ) , ς i ( u )] h = [ u , u ] h + [ u , v ] + [ v , u ] = { x , x } + [ x , v ] + [ v , x ] = { x , x } ,∂ i { x , x } = ∂ i [ u , u ] h ! = ∂ i ( u ) ς i ( u ) + u ∂ i ( u ) − ∂ i ( u ) ς i ( u ) − u ∂ i ( u ) h = w ( u + h v ) + u w − w ( u + h v ) − u w = [ x , w ] + [ w , x ] = 0 . Hence { x , x } ∈ ker ∂ i ∩ ker( ς i − id) for all i ∈ I so { x, y } ∈ Z . (cid:3) Poisson-Hopf algebras.
Assume that in the above setting U is a Hopf algebraover A . Then U/ ( h ) has a canonical structure of Hopf algebra over C .Let u ∈ U such that u ∈ Z ( U/ ( h )) and furthermore ∆( u ) ∈ Z ( U/ ( h ) ⊗ U/ ( h )) . Then D ∆( u ) ∆( y ) = ∆( D u ( y )) , y ∈ U/ ( h ) . (2.5) Proposition 2.4. [DP, 11.7]
Let B be a central Hopf subalgebra of U/ ( h ) .Then T := minimal subalgebra of Z containing B and closed under the Poisson bracketis a central Hopf subalgebra of U/ ( h ) , hence a Poisson-Hopf algebra. We recall the elegant proof of [DP].
Proof.
Apply (2.5) to y ∈ Z and x = u to get ∆( { x, y } ) = { ∆( x ) , ∆( y ) } for all x, y ∈ Z .Hence e T = { t ∈ T : ∆( t ) ∈ T ⊗ T } , which is a subalgebra containing B , is also closedunder Poisson bracket; thus e T = T . (cid:3) OISSON ORDERS ON LARGE QUANTUM GROUPS 9 Hopf algebras
In this section we collect preliminaries on (braided) Hopf algebras (always with bijec-tive antipode S ), bosonizations, braided vectors spaces of diagonal type, Nichols alge-bras, Weyl groupoids, distinguished pre-Nichols algebras and Lusztig algebras. We referto [R, A] for more information on Hopf algebras, Nichols algebras, Nichols algebras ofdiagonal type, respectively.3.1. Cleft comodule algebras.
Let H be a Hopf algebra with a central Hopf subalgebra Z . Given z ∈ G = Alg( Z, C ) (the pro-algebraic group defined by Z ), let M z = ker z, I z = H M z , H z = H / I z ; thus H z is an algebra (with multiplication m z and unit u z ) and the natural projection p z : H → H z is an algebra map. Then H ε is a quotient Hopf algebra of H and thereis an exact sequence of Hopf algebras Z ֒ → H ։ H ε . Also for any z, z ′ ∈ G there arewell-defined algebra morphisms ∆ z,z ′ : H zz ′ → H z ⊗ H ′ z and in particular the maps ̺ z := ∆ z,ε : H z → H z ⊗ H ε , λ z := ∆ ε,z : H z → H ε ⊗ H z , make H z a H ε -bicomodule algebra for z ∈ G . Clearly ̺ z p z = ( p z ⊗ p ε )∆ H , λ z p z = ( p ε ⊗ p z )∆ H . (3.1)Recall that a right K -comodule algebra A (over a Hopf algebra K ) is cleft if thereexists a convolution-invertible morphism of K -comodules χ : K → A . Lemma 3.1.
If the H ε -comodule algebra H with coaction ̺ = (id ⊗ p ε )∆ H is cleft, thenso is H z for any z ∈ G . In particular H z is a Hopf-Galois object over H e .If H is a pointed Hopf algebra, then H z is H e -cleft for all z ∈ G .Proof. If χ : H ε → H is a morphism of H -comodules, then so is χ z := p z χ : H ε → H z : ( χ z ⊗ id) ̺ z = ( p z ⊗ id)( χ ⊗ id)∆ H ε = ( p z ⊗ id)(id ⊗ p ε )∆ H χ = ̺ z p z χ = ̺ z χ z . If χ is convolution-invertible, then so is χ z since p z is an algebra map.For the last statement, H is H ε -cleft by [Sc, 4.3], and then we apply the first part. (cid:3) We refer to [S] for Hopf-Galois objects. In the setting of Cayley–Hamilton Hopfalgebras, which is a refinement of the above setting for the pair ( H , Z ) , a tensor productdecomposition of the irreducible representations of H z was obtained in [DPRR].3.2. Braided Hopf algebras and bosonization.
Recall that a braided vector spaceis a pair ( V , c ) where V is a vector space and c ∈ GL ( V ⊗ V ) is a solution of the braidequation: ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ) . There are natural notions ofmorphisms of braided vector spaces and braided Hopf algebras (braided vector spaceswith compatible algebra and coalgebra structures), see [T] for details. To distinguishcomultiplications of braided Hopf algebras from those of Hopf algebras, we use a variationof the Sweedler notation for the former: ∆( r ) = r (1) ⊗ r (2) .Let H be a Hopf algebra. Then the category of (left) Yetter-Drinfeld modules HH YD isa braided tensor category and there is a forgetful functor from HH YD to the category ofbraided vector spaces, namely V ∈ HH YD goes to ( V , c ) where c ∈ GL ( V ⊗ V ) is given by c ( v ⊗ w ) = v ( − · w ⊗ v (0) in Sweedler notation. This forgetful functor sends Hopf algebrasin HH YD to braided Hopf algebras. In turn Hopf algebras in HH YD are noteworthy because of the Radford-Majid bosonization that provides a bijective correspondence between theircollection and the collection of triples ( A, π, ι ) where A π ⇄ ι H are morphisms of Hopfalgebras with πι = id H . See [R] for an exposition. More precisely, the correspondencesends the Hopf algebra R ∈ HH YD to the bosonization R H and the triple ( A, π, ι ) tothe algebra of right coinvariants R = A co π .Similar notions and results hold for the category of (right) Yetter-Drinfeld modules YD HH consisting of right H -modules and right H -comodules V satisfying the compatibility ( v · h ) (0) ⊗ ( v · h ) (1) = v (0) · h (2) ⊗ S ( h (1) ) v (1) h (3) , v ∈ V , h ∈ H. For convenience of the reader we spell out the precise definitions. First, any
V ∈ YD HH becomes a braided vector space with c ∈ GL ( V ⊗ V ) and its inverse given by c ( v ⊗ w ) = w (0) ⊗ v · w (1) , c − ( v ⊗ w ) = w · S − ( v (1) ) ⊗ v (0) , v, w ∈ V. (3.2)Let ( A, π, ι ) be a triple as before. Then the subalgebra of left coinvariants S = co π A = { s ∈ A : ( π ⊗ id)∆( s ) = 1 ⊗ s } becomes a Hopf algebra in YD HH with right action · , right coaction ρ and comultiplication ∆ given by s · h = S ( h (1) ) sh (2) , ρ ( s ) = (id ⊗ π )∆( s ) , ∆( s ) = s (1) ⊗ ϑ ( s (2) ) , s ∈ S, h ∈ H, where ϑ : A → S is given by ϑ ( a ) = π ( S ( a (1) )) a (2) , a ∈ A . Conversely, the bosonization H S of a Hopf algebra S in YD HH is the vector space H ⊗ S with the right smash productand coproduct. That is, given s, e s ∈ S and h, e h ∈ H , ( h s )( e h e s ) = h e h (1) s · e h (2) ) e s, ∆( h s ) = h (1) s (1) ) (0) ⊗ h (2) ( s (1) ) (1) s (2) . Nichols algebras.
Let
V ∈ HH YD . Then the tensor algebra T ( V ) is naturally aHopf algebra in HH YD . A pre-Nichols algebra of V is a factor of T ( V ) by a graded Hopfideal in HH YD supported in degrees > . The maximal Hopf ideal among those is denotedby J ( V ) ; the Nichols algebra of V is the quotient B ( V ) = T ( V ) / J ( V ) .The tensor algebra of a braided vector space ( V , c ) is also a braided Hopf algebra inthe sense of [T]; a pre-Nichols algebra of V is a factor of T ( V ) by a braided graded Hopfideal supported in degrees > . The maximal Hopf ideal among those is denoted e J ( V ) ;the Nichols algebra of V is the quotient B ( V ) = T ( V ) / e J ( V ) .These two structures are compatible, i.e. if V ∈ HH YD and ( V , c ) is the correspondingbraided vector space, then J ( V ) = e J ( V ) . But a pre-Nichols algebra of ( V , c ) does notnecessarily come as the forgetful functor applied to a pre-Nichols algebra of V ∈ HH YD . Remark 3.2.
Let H be cosemisimple, V ∈ HH YD and G = B ( V ) H = ⊕ n ∈ N G n , where G n = B n ( V ) H . By other characterizations of Nichols algebras, we know that(a) B ( V ) is coradically graded and generated in degree 1;(b) G is coradically graded and generated in degree 1.Since the projection π : G → H is graded, the subalgebra of left coinvariants S = co π G inherits the grading of G ; by a standard argument it is also coradically graded andgenerated in degree 1. Thus S is a Nichols algebra in YD HH . OISSON ORDERS ON LARGE QUANTUM GROUPS 11
Hopf skew-pairings of bosonizations.
Let h· , ·i : M × V → C be a bilinear formbetween two vector spaces M and V . We denote by h· , ·i : ( M ⊗ M ) × ( V ⊗ V ) → C thebilinear form determined by h m ⊗ m ′ , v ⊗ v ′ i = h m, v ′ ih m ′ , v i , m, m ′ ∈ M, v, v ′ ∈ V. (3.3)Let H and K be two Hopf algebras. A bilinear form h· , ·i : K × H → C is a Hopfskew-pairing (or skew-pairing of Hopf algebras) if for all for k, k ′ ∈ K , h, h ′ ∈ H , h k, hh ′ i = h ∆ op ( k ) , h ⊗ h ′ i , h kk ′ , h i = h k ⊗ k ′ , ∆( h ) i , h k, i = ε ( k ) , h , h i = ε ( h ) , hS ( k ) , h i = h k, S ( h ) i . (3.4)A skew-pairing of braided Hopf algebras is defined by (3.4) but with the convention ∆ op = c − ∆ . Let us fix a Hopf skew-pairing h· , ·i : K × H → C . A YD-pairing between
M ∈ YD KK and V ∈ HH YD is a bilinear form h· , ·i : M × V → C such that h m · k, v i = h k, v ( − ih m, v (0) i , h m, h · v i = h m (1) , h ih m (0) , v i , m ∈ M , k ∈ K, v ∈ V , h ∈ H. (3.5)We recall the following well-known result, whose proof is straightforward. Lemma 3.3.
Let R be a Hopf algebra in HH YD , S be a Hopf algebra in YD KK and h· , ·i :( K S ) × ( R H ) → C be a bilinear form such that h ky, xh i = h k, h ih y, x i , y ∈ S, k ∈ K, x ∈ R, h ∈ H. (3.6) Then the following are equivalent: (a) h· , ·i is a Hopf skew-pairing. (b) The restriction of h· , ·i to K × H is a Hopf skew-pairing and the restriction of h· , ·i to S × R is both a skew-pairing of braided Hopf algebras and a YD-pairing. (cid:3) A YD-pairing between
M ∈ YD KK and V ∈ HH YD extends canonically to a YD-pairing h· , ·i : T ( M ) × T ( V ) → C . This extension is actually a braided Hopf skew-pairing, i.e.,it satisfies (3.4) with respect to the braided comultiplications. The bilinear form h· , ·i :( K T ( M )) × ( T ( V ) H ) → C , h k y, x h i := h k, h ih y, x i ,y ∈ T ( M ) , k ∈ K , x ∈ T ( V ) , h ∈ H is a Hopf skew-pairing by Lemma 3.3.Assume that dim M < ∞ . Then the radical T ( M ∗ ) ⊥ with respect to h· , ·i coincideswith J ( M ) . Hence, for any V YD-paired with M we have T ( V ) ⊥ ⊇ J ( M ) . Consequently, if dim M < ∞ and dim V < ∞ , B is a pre-Nichols algebra of M in YD KK and E is a pre-Nichols algebra of V in HH YD , then h· , ·i descends to Hopf skew-pairings h· , ·i : B × E → C and h· , ·i : ( K B ) × ( E H ) → C . Nichols algebras of diagonal type.
We fix θ ∈ N and set I = I θ . Let ( V, c ) be a(complex) braided vector space of diagonal type with braiding matrix q = ( q ij ) ∈ (cid:0) C × (cid:1) I × I (3.7)with respect to a basis ( x i ) i ∈ I , i.e. c ( x i ⊗ x j ) = q ij x j ⊗ x i for all i, j ∈ I . We assume that dim B ( V ) < ∞ . These Nichols algebras are classified in [H2]. Throughout the paperwe will also assume that the Dynkin diagram of q is connected, for simplicity of theexposition.The canonical basis of Z I is denoted α , . . . , α θ . The algebra T ( V ) is Z I -graded, withgrading deg x i = α i , i ∈ I . This grading naturally specializes to the standard N -grading.Let q : Z I × Z I → C × be the Z -bilinear forms associated to the matrix q , i.e. q ( α j , α k ) := q jk , j, k ∈ I . If α, β ∈ Z I and i ∈ I , then we set q αβ = q ( α, β ) , q αα = q ( α, α ) , N α = ord q αα , N i = ord q α i α i = N α i . (3.8) Remark 3.4.
Every Z I -graded pre-Nichols algebra of V admits algebra automorphisms ς q i and (id , ς q i ) -derivations ∂ q i for each i ∈ I ; that is, ∂ q i ( xy ) = ∂ q i ( x ) ς q i ( y ) + x∂ q i ( y ) , x, y ∈ T ( V ) . Indeed the algebra automorphism ς q i : T ( V ) → T ( V ) is given by ς q i ( x ) = q ( α i , β ) x, x ∈ T ( V ) homogeneous of degree β ∈ Z I . The linear endomorphisms ∂ q i : T ( V ) → T ( V ) are defined as follows. Let ∆ m,n ( x ) bethe homogeneous component of ∆( x ) ∈ T ( V ) ⊗ T ( V ) of degree ( m, n ) ∈ N . Then ∆ n − , ( x ) = X i ∈ I ∂ q i ( x ) ⊗ x i , x ∈ T n ( V ) . It is easy to see that ∂ q i is a (id , ς q i ) -derivation. If B is a quotient of T ( V ) by a Z I -homogeneous ideal, then ς q i induces an algebra automorphism of B , also denoted by ς q i ,and ∂ q i induces a (id , ς q i ) -derivation of B , also denoted by ∂ q i .3.6. Weyl groupoids.
The notions of Weyl groupoid and generalized root systems wereintroduced in [H1, HY1]. We recall the main features needed later. Let ( c q ij ) i,j ∈ I ∈ Z I × I be the (generalized Cartan) matrix defined by c q ii := 2 and c q ij := − min { n ∈ N : ( n + 1) q ii (1 − q nii q ij q ji ) = 0 } , i = j. (3.9)Let i ∈ I . First, the reflection s q i ∈ GL ( Z I ) is given by s q i ( α j ) := α j − c q ij α i , j ∈ I . (3.10)Second, the matrix ρ i ( q ) is given by ( ρ i ( q )) jk := q ( s q i ( α j ) , s q i ( α k )) = q jk q − c q ij ik q − c q ik ji q c q ij c q ik ii , j, k ∈ I . (3.11)Finally, the braided vector space ρ i ( V ) is of diagonal type with matrix ρ i ( q ) . Set X := { ρ j . . . ρ j n ( q ) : j , . . . , j n ∈ I , n ∈ N } . The set X is called the Weyl-equivalence class of q . The set ∆ q + of positive roots consists of the Z I -degrees of the generators of a PBW-basis of B q , counted with multi-plicities. Let ∆ q := ∆ q + ∪ − ∆ q + . Then the generalized root system of q is the fibration ∆ → X , where the fiber of ρ j . . . ρ j N ( q ) is ∆ ρ j ...ρ jN ( q ) . The Weyl groupoid W q of B q acts OISSON ORDERS ON LARGE QUANTUM GROUPS 13 on this fibration, generalizing the classical Weyl group. Here is another characterizationof ∆ q + , valid because it is finite. Let ω q ∈ W q be an element of maximal length and ω q = σ q i σ i · · · σ i ℓ be a reduced expression. Then β k := s q i · · · s i k − ( α i k ) , k ∈ I ℓ (3.12)are pairwise different vectors and ∆ q + = { β k : k ∈ I ℓ } [CH, Prop. 2.12], so | ∆ q + | = ℓ .3.7. Cartan roots [An3] . This important notion is crucial for our purposes. First, i ∈ I is a Cartan vertex of q if q ij q ji = q c q ij ii , for all j = i. (3.13)Then the set of Cartan roots of q is O q = { s q i s i . . . s i k ( α i ) ∈ ∆ q : i ∈ I is a Cartan vertex of ρ i k . . . ρ i ρ i ( q ) } . Set O q + = O q ∩ N θ . Recall (3.8) and set e N β := N β , if β / ∈ O q , or else ∞ if β ∈ O q .The set of Cartan roots gives rise to a root system up to a rescaling. Set O q = { N q β β : β ∈ O q } , O q + = O q ∩ N θ , β = N q β β, β ∈ O q . (3.14) Theorem 3.5. [AAR3, Theorem 3.6]
The set O q is either empty or a root system insidethe real vector space generated by O q . The set Π q of all indecomposable elements of O q + is a basis of this root system. Here γ ∈ O q + is indecomposable if it can not be represented as a non-trivial positivelinear combination of elements of O q + . Let g q be either or the semisimple Lie algebrawith root system as in Theorem 3.5, accordingly. We fix a triangular decomposition g q = n + q ⊕ h q ⊕ n − q (3.15)and the Borel subalgebras b ± q = h q ⊕ n ± q ⊂ g q ; if g q = 0 , then n + q = h q = n − q = 0 . Wedenote the root lattice by Q q := X γ ∈ O q + Z γ = M γ ∈ Π q Z γ. (3.16)3.8. Distinguished pre-Nichols algebras.
The finite-dimensional Nichols algebras ofdiagonal type admit distinguished pre-Nichols algebras introduced in [An2, An3]. Anideal I ( V ) of T ( V ) was introduced in [An3]; it is generated by all the defining relationsof B q in [An2, Theorem 3.1], but excluding the power root vectors x N α α , α ∈ O q , andadding some quantum Serre relations. Definition 3.6. [An3] The distinguished pre-Nichols algebra e B q of V is the quotient e B q = T ( V ) / I ( V ) . Since I ( V ) is a Hopf ideal, e B q is a braided Hopf algebra.By Remark 3.4, there are automorphisms ς q i and skew-derivations ∂ q i of e B q , i ∈ I . Lusztig algebras.
The
Lusztig algebra L q associated to q is the graded dual of e B q [AAR1]. Thus L q is a braided Hopf algebra equipped with a bilinear form ( , ) : L q × e B q → C , which satisfies ( y, xx ′ ) = ( y (2) , x )( y (1) , x ′ ) and ( yy ′ , x ) = ( y, x (2) )( y ′ , x (1) ) (3.17)for all x, x ′ ∈ e B q , y, y ′ ∈ L q . Let Z q = co ̟ e B q be the subalgebra of coinvariants of thecanonical projection ̟ : e B q → B q . Then Z q is a normal Hopf subalgebra of e B q [An3, Theorems 4.10, 4.13] and we have anextension of braided Hopf algebras Z q ι ֒ → e B q ̟ ։ B q . Taking graded duals, we obtain anew extension of braided Hopf algebras, cf. [AAR2, Prop. 3.2]: B q t ̟ ∗ ֒ → L q ι ∗ ։ Z q , (3.18) Remark 3.7.
Assume that (4.23) below holds. Then the braided Hopf algebra Z q isa Hopf algebra, isomorphic to the enveloping algebra of the Lie algebra P ( Z q ) [AAR2,3.3]. Moreover P ( Z q ) ≃ n − q as in (3.15) [AAR3].4. Large quantum groups
In this section we describe the large quantum groups i.e. Drinfeld doubles of bozoniza-tions of the distinguished pre-Nichols algebras belonging to a one-parameter family; theseare the main focus of the paper. The large quantum Borel and unipotent subalgebras arealso introduced here. Throughout the rest of the paper Γ + and Γ − denote free abeliangroups of rank θ with bases denoted respectively ( K i ) i ∈ I and ( L i ) i ∈ I . Let Γ = Γ + × Γ − .4.1. Families of Nichols algebras.
From now on we assume that q belongs to a one-parameter family (except when explicitly stated otherwise). This means that there existsan indecomposable matrix q = ( q ij ) ∈ (cid:0) C [ ν ± ] × (cid:1) I × I (4.1)such that: ◦ The Nichols algebra of the C ( ν ) -braided vector space of diagonal type V C ( ν ) with basis ( x i ) i ∈ I and braiding matrix (4.1) has finite root system thus listed in [H2]. ◦ There exists an open subset ∅ 6 = O ⊆ C × such that for any x ∈ O , the matrix q ( x ) obtained by evaluation ν x has the same finite root system as q . ◦ There exists ξ ∈ G ′∞ such that q = q ( ξ ) .By inspection in [H2], all one-parameter families are listed in the Appendix A. Wedenote the Nichols algebras of V and V C ( ν ) , with braidings given by q , respectively q , by B q := B ( V ) and B q := B ( V C ( ν ) ) . The defining relations and PBW-basis of B q and B q are described in [AA] over an alge-braically closed field of characteristic 0 but the same presentation and PBW-basis arevalid over C ( ν ) . Indeed, apply to F = C ( ν ) , K = C ( ν ) the following remarks: ◦ Let K / F be a field extension and ( V, c ) a braided F -vector space. Then ( V ⊗ F K , c ⊗ id) isa braided K -vector space and B ( V ) ⊗ F K ≃ B ( V ⊗ F K ) ; use e.g. quantum symmetrizers. OISSON ORDERS ON LARGE QUANTUM GROUPS 15 ◦ Let K / F be a faithfully flat extension of commutative rings. Let U be a F -algebra withgenerators ( y j ) j ∈ J and U K = U ⊗ F K which is also generated by ( y j ) j ∈ J . Let ( r t ) t ∈ T be a set of elements in the tensor algebra over F of the free module F ( J ) . Then theseare defining relations of U if and only if they are defining relations of U K .The discussions in §3.5 and §3.6 apply to the matrix q . Let q : Z I × Z I → ( C [ ν ± ]) × as in §3.5; we also have the notation q αβ for α, β ∈ Z I as in (3.8). We denote by W q thecorresponding Weyl groupoid, by ρ i ( q ) the related braiding matrices, etc. As in Remark3.4, there are ς q i ∈ Aut alg ( B q ) and (id , ς q i ) -derivations ∂ q i : B q → B q , for every i ∈ I . Remark 4.1.
Crucially, β is a Cartan root of q if and only if ord q ββ = ∞ .4.2. The quantum group U q . Here we work over C ( ν ) . Let W C ( ν ) the C ( ν ) -vectorspace with basis ( y i ) i ∈ I . The group Γ acts on V C ( ν ) ⊕ W C ( ν ) by K i · x j = q ij x j , K i · y j = q − ij y j , L i · x j = q ji x j , L i · y j = q − ji y j , (4.2) i, j ∈ I . The vector space V C ( ν ) ⊕ W C ( ν ) is Γ -graded by deg x i = K i , deg y i = L i , i ∈ I . (4.3)Thus V C ( ν ) ⊕ W C ( ν ) ∈ C ( ν )Γ C ( ν )Γ YD with coaction given by the grading. In particular, W C ( ν ) is a braided vector space with braiding matrix q ′ where q ′ ij = q − ji , i, j ∈ I .We define U q as the quotient Hopf algebra of the bosonization T ( V C ( ν ) ⊕ W C ( ν ) ) C ( ν )Γ modulo the ideal generated by J ( V C ( ν ) ) , J ( W C ( ν ) ) , x i y j − q − ij y j x i − δ ij ( K i L i − , i, j ∈ I . The images of x i , y i , K i and L i in U q will again be denoted by the same symbols. Let E i := x i , F i := y i L − i in U q , i ∈ I . Then for all i, j ∈ I we have K i E j = q ij E j K i , L i E j = q ji E j L i , (4.4) K i F j = q − ij F j K i , L i F j = q − ji F j L i , (4.5) E i F j − F j E i = δ ij ( K i − L − i ) , (4.6) ∆( E i ) = K i ⊗ E i + E i ⊗ , ∆( F i ) = 1 ⊗ F i + F i ⊗ L − i . (4.7)We consider the following subalgebras of U q : U +0 q = C ( ν )[ K ± i : i ∈ I ] , U − q = C ( ν )[ L ± i : i ∈ I ] , U q = C ( ν )[ K ± i , L ± i : i ∈ I ] ,U + q = C ( ν ) h E i : i ∈ I i , U − q = C ( ν ) h F i : i ∈ I i ,U > q = C ( ν ) h E i , K ± i : i ∈ I i , U q = C ( ν ) h F i , L ± i : i ∈ I i . The multiplication map induces linear isomorphisms U q ≃ U + q ⊗ C ( ν ) U q ⊗ C ( ν ) U − q ≃ U > q ⊗ C ( ν ) U q . (4.8)We have canonical isomorphisms of Hopf algebras U +0 q ≃ C ( ν )Γ + , U − q ≃ C ( ν )Γ − , U q ≃ C ( ν )Γ . The algebra U + q has a canonical structure of a Hopf algebra in C ( ν )Γ + C ( ν )Γ + YD and there areisomorphisms of (braided) Hopf algebras U + q ≃ B q , U > q ≃ U + q U +0 q , see e.g. [ARS] for details. Define the module V ∗ C ( ν ) ∈ YD C ( ν )Γ − C ( ν )Γ − with basis { x ∗ i : i ∈ I } by x ∗ j · L i = q ji x ∗ j , deg x ∗ i = L − i , i, j ∈ I . Let π − : U q → U − q be the canonical Hopf algebra morphism; then co π − U q = U − q ,cf. [ARS, Corollary 3.9 (2)]. Hence U − q has a canonical structure of a Hopf algebra in YD C ( ν )Γ − C ( ν )Γ − . By Remark 3.2, we have isomorphisms of (braided) Hopf algebras U − q ≃ B ( V ∗ C ( ν ) ) ≃ B q ( − , U q ≃ U − q U − q . Here q ( − means the matrix obtained by inverting every entry of q .Now there is a unique Hopf skew-pairing h· , ·i : U q × U > q → C ( ν ) determined by h L i , K j i = q − ji , h F i , E j i = δ ij , h L i , E j i = h F i , K j i = 0 , i, j ∈ I , see [ARS, Theorem 3.7]. By [ARS, Theorem 3.11 (1)], we have h x − g − , x + g + i = h x − , x + ih g − , g + i , x ± ∈ U ± q , g ± ∈ Γ ± . The restriction h· , ·i : U − q × U + q → C ( ν ) is non-degenerate by [ARS, Theorem 3.11 (3)]and is a Hopf skew-pairing of braided Hopf algebras by Lemma 3.3.4.3. The large quantum group U q . Recall that q ∈ (cid:0) C × (cid:1) I × I belongs to a one param-eter family given by a matrix q , cf. §4.1. Definition 4.2.
The large quantum group U q is the Drinfeld double of the bosoniza-tion of the distinguished pre-Nichols algebra e B q .The complex Hopf algebra U q was defined in [An3] for arbitrary q with dim B q < ∞ .Explicitly, let W the C -vector space with basis ( y i ) i ∈ I . The group Γ acts on V ⊕ W by K i · x j = q ij x j , K i · y j = q − ij y j , L i · x j = q ji x j , L i · y j = q − ji y j , i, j ∈ I . Now V ⊕ W is Γ -graded by (4.3), so W is a braided vector space with braiding matrix q ′ with entries q ′ ij = q − ji for i, j ∈ I . Recall the defining ideal I ( V ) of e B q . Then U q isthe bosonization T ( V ⊕ W ) C Γ modulo the ideal generated by I ( V ) , I ( W ) , x i y j − q − ij y j x i − δ ij ( K i L i − , i, j ∈ I . The images of x i , y i , K i and L i in U q will again be denoted by the same symbols. Let e i = x i , f i = y i L − i in U q , i ∈ I . Then for all i, j ∈ I we have K i e j = q ij e j K i , L i e j = q − ji e j L i , (4.9) K i f j = q − ij f j K i , L i f j = q ji f j L i , (4.10) e i f j − f j e i = δ ij ( K i − L − i ) , (4.11) ∆( e i ) = K i ⊗ e i + e i ⊗ , ∆( f i ) = 1 ⊗ f i + f i ⊗ L − i . (4.12)We consider the following subalgebras of U q : U +0 q = C [ K ± i : i ∈ I ] , U − q = C [ L ± i : i ∈ I ] , U q = C [ K ± i , L ± i : i ∈ I ] ,U + q = C h e i : i ∈ I i , U − q = C h f i : i ∈ I i ,U > q = C h e i , K ± i : i ∈ I i , U q = C h f i , L ± i : i ∈ I i . Definition 4.3.
The algebras U > q and U q will be called large quantum Borel alge-bras and the algebras U ± q large quantum unipotent algebras . OISSON ORDERS ON LARGE QUANTUM GROUPS 17
The multiplication map induces the linear isomorphisms U q ≃ U + q ⊗ C U q ⊗ C U ( − ) q ≃ U > q ⊗ C U q . (4.13)We have canonical isomorphisms of Hopf algebras U +0 q ≃ C Γ + , U − q ≃ C Γ − , U q ≃ C Γ . The algebra U + q has a canonical structure of a Hopf algebra in C Γ + C Γ + YD . We have isomor-phisms of (braided) Hopf algebras: U + q ≃ e B q , U > q ≃ U + q U +0 q , see [An3]. Define the module V ∗ ∈ YD C Γ − C Γ − with basis { x ∗ i : i ∈ I } by x ∗ j · L i = q ji x ∗ j , deg x ∗ i = L − i , i, j ∈ I . Let π − : U q → U − q be the canonical Hopf algebra projection; then co π − U q = U − q as in [ARS, Corollary 3.9 (2)]. Hence U − q is a Hopf algebra in YD C Γ − C Γ − and because ofthe defining relations of U − q , it is isomorphic to the distinguished pre-Nichols algebra of V ∗ ∈ YD C Γ − C Γ − . Combining the above, we get isomorphisms of (braided) Hopf algebras: U − q ≃ e B q ( − , U q ≃ U − q U − q . (4.14)Here, again, q ( − denotes the matrix obtained by inverting every entry of q .4.4. Lusztig isomorphisms and root vectors.
As in [H3, §3] we consider λ q ij = ( q − c q ij ii q ij q ji ) c q ij ( − c q ij ) ! q ii Y ≤ s< − c q ij ( q sii q ij q ji − ∈ C [ ν ± ] × , i = j ∈ I . (4.15)By [H3, Proposition 6.8], there exist algebra isomorphisms T q i : U ρ i ( q ) → U q such that T q i ( K j ) = K j K − c q ij i ; T q i ( E i ) = ( F i L i , j = i, (ad c E i ) − c q ij E j , j = i,T q i ( L j ) = L j L − c q ij i ; T q i ( F i ) = ( K − i E i , j = i, ( λ q ij ) − (ad c F i ) − c q ij F j , j = i, (4.16)where the underlined letters denote the generators of U ρ i ( q ) .Let ω q be the element of W q of maximal lenght ending at q and ω q = σ q i σ i · · · σ i ℓ be a reduced expression. By [H3, Theorem 6.20], E β k := T q i . . . T i k − ( E i k ) ∈ U + q , F β k := T q i . . . T i k − ( F i k ) ∈ U − q , k ∈ I ℓ . (4.17)By [HY2, Theorem 4.5] the sets n E n β E n β . . . E n ℓ β ℓ : 0 ≤ n j < e N β j , j ∈ I ℓ o , n F m β F m β . . . F m ℓ β ℓ : 0 ≤ m j < e N β j , j ∈ I ℓ o (4.18)are bases of U + q and U − q , respectively. Indeed, this follows from Property (c) in theAppendix A and Remark 4.1. Thus the following set is a basis of U q : { E m β . . . E m ℓ β ℓ K a . . . K a θ θ L b . . . L b θ θ F n β . . . F n ℓ β ℓ : 0 ≤ m j , n j < e N β , a i , b i ∈ Z } . (4.19) We now turn to the algebras U q . Let λ q ij is defined as (4.15) with q in place of q . By[An3, Proposition 10], there exist algebra isomorphisms T q i : U ρ i ( q ) → U q such that T q i ( K j ) = K j K − c q ij i ; T q i ( e i ) = ( f i L i , j = i, (ad c e i ) − c q ij e j , j = i,T q i ( L j ) = L j L − c q ij i ; T q i ( f i ) = ( K − i e i , j = i, ( λ q ij ) − (ad c f i ) − c q ij f j , j = i, (4.20)The underlined letters denote the generators of U ρ i ( q ) .Analogously, e β k = T q i . . . T i k − ( e i k ) and f β k = T q i . . . T i k − ( f i k ) belong to U + q and U − q ,respectively and by [An3, Theorem 11] the sets (cid:8) e n β e n β . . . e n ℓ β ℓ : 0 ≤ n i < e N β i (cid:9) and (cid:8) f m β f m β . . . f m ℓ β ℓ : 0 ≤ m j < e N β j (cid:9) (4.21)are bases of U + q and U − q , respectively. Thus the following set is a basis of U q : (cid:8) e m β . . . e m ℓ β ℓ K a . . . K a θ θ L b . . . L b θ θ f n β . . . f n ℓ β ℓ : 0 ≤ m j , n j < e N β j , a i , b i ∈ Z (cid:9) . (4.22)4.5. The central subalgebras Z q , Z ± q , Z > q , Z q . In this subsection and the next q does not need to be in a family, just dim B q < ∞ is assumed. To start with, we considerthe subalgebra Z q of U q introduced right after (3.17); as shown in [An3, p. 18], Z q isgenerated by e N β β , f N β β , K ± N β β , L ± N β β , β ∈ O q ; this is a normal Q q -graded Hopf subalgebra of U q [An3, Proposition 21, Theorem 33].For Z q to be central in U q we need the following condition that we assume from now on: q N β αβ = 1 , α ∈ ∆ q , β ∈ O q . (4.23) Remark 4.4. (a) If (4.23) holds, then q N β βα = 1 [An3, Lemma 24].(b) Condition (4.23) is equivalent to the following one: q N β α i β = 1 , for all i ∈ I , β ∈ Π q . (4.24)The reduction to simple roots is clear. Since q N β αβ = q αβ and Π q is a basis of the rootsystem O q , the reduction from O q + to Π q holds.(c) Let i ∈ I . Condition (4.23) holds for q if and only if it holds for ρ i ( q ) .Indeed, ρ i ( q ) αβ = q s q i ( α ) s q i ( β ) for all α, β ∈ Z θ by (3.11), and by [AAR3, Lemma 2.3]we have s q i ( O q ) = O ρ i ( q ) , N ρ i ( q ) s q i ( β ) = N q β for all β .When q is symmetric, we can quotient the large quantum by a central group subalgebrato remove the extra Cartan generators as in quantum groups. However the condition of q being symmetric is not always compatible with (4.23) as we see next. Example 4.5.
Assume that q has Dynkin diagram − ◦ ξ − ◦ , ξ ∈ G ′ N , N > : it isof super type A (1 | . In this case, ∆ q + = { α , α + α , α } , O q + = { α + α } . OISSON ORDERS ON LARGE QUANTUM GROUPS 19
Condition (4.24) becomes q q ) N = ( − q ) N , q q ) N = ( − q ) N ⇐⇒ q N = ( − N = q N . We have two possibilities: if N is even, then q = ξ k for some k ∈ I N , so q = ξ − k ,and q is not symmetric. If N is odd, then q = − ξ k for some k ∈ I N , so q = − ξ − k .In this case q is symmetric only when k = N +12 .We consider also the Hopf subalgebras Z +0 q = C h K ± N β β : β ∈ O q + i , Z − q = C h L ± N β β : β ∈ O q + i , Z q = Z +0 q Z − q ,Z + q = C h e N β β : β ∈ O q + i , Z − q = C h f N β β : β ∈ O q + i , Z > q = Z + q Z +0 q , Z q = Z − q Z − q . Remark 4.6.
The following properties hold:(a) [An3, Th. 23]. Z ± q is a polynomial ring in variables e N β β , respectively f N β β , β ∈ O q + .(b) The multiplication gives linear isomorphisms Z + q ⊗ Z +0 q ⊗ Z − q ⊗ Z − q ≃ Z q ≃ Z > q ⊗ Z q .(c) Recall the skew-derivations ∂ q i , ∂ q ( − i of U ± q , cf. (4.14). By [An3, Theorem 31], Z + q = \ i ∈ I ker ∂ q i , Z − q = \ i ∈ I ker ∂ q ( − i . (4.25)(d) The algebras U q , U > q , U q and U ± q are module finite over their central subalgebras Z q , Z > q , Z q and Z ± q ; just consider the PBW-bases in §4.4.4.6. Action of the Weyl groupoid on Z q . Next we prove invariance of the centralHopf subalgebras Z q under the Lusztig isomorphisms T q i : U ρ i ( q ) → U q , cf. §4.4. Theorem 4.7.
Let i ∈ I . Then T q i restricts to an algebra isomorphism T q i : Z ρ i ( q ) → Z q .Proof. By (4.25), Z q does not depend on the expression of ω q ; in particular we maychoose ω q = σ q i . . . σ i ℓ such that i = i . For simplicity we set p = ρ i ( q ) . As σ p i . . . σ i ℓ isreduced, we may extend it to a reduced expression of ω p [HY1, Corollary 3]: ω p = σ p i . . . σ i ℓ σ j for some j ∈ I . We set β ′ k = σ i ( β k ) = σ p i . . . σ i k − ( α i k ) , k ∈ I ,ℓ . Hence { β ′ k : k ∈ I ,ℓ } = s p i (cid:0) ∆ q + − { α i } (cid:1) = ∆ p + − { α i } . As σ p i . . . σ i ℓ ( α j ) ∈ ∆ p + , σ p i . . . σ i ℓ ( α j ) = β ′ k for k ∈ I ,ℓ , we have that σ p i . . . σ i ℓ ( α j ) = α i .Let β ∈ O ρ i ( q ) . If β = β ′ k for some k ∈ I ,ℓ , then s q i ( β ′ k ) = β k and N β ′ k = N β k , hence T q i (cid:0) K ± N β ′ k β ′ k (cid:1) = K ± N β ′ k s q i ( β ′ k ) = K ± N βk β k ∈ Z q , T q i (cid:0) e N β ′ k β ′ k (cid:1) = T q i T i . . . T i k − ( e N βk i k ) = e N βk β k ∈ Z q . Otherwise β = α i , so i is a Cartan vertex and T q i ( K ± N β β ) = K ∓ N αi i ∈ Z q , T q i ( e N β β ) = T q i ( e N αi i ) = ( f i L i ) N αi = q ( Nαi ) ii f N αi i L N αi i ∈ Z q . Analogously, T q i ( L ± N β β ) , T q i ( f N β β ) ∈ Z q for all β ∈ O ρ i ( q ) , so T q i ( Z ρ i ( q ) ) ⊆ Z q . Applying T p i we get the opposite inclusion. (cid:3) The specialization setting for large quantum groups
In this section we construct the non-restricted integral form of U q and prove that thelarge quantum group U q is a specialization of it. We also construct restricted integralforms of the subalgebras U ± q and establish pairing results for the corresponding special-izations. The latter integral forms will play a key role in our treatment of Poisson orderstructures on the large quantum groups U q and their Borel and unipotent subalgebras.5.1. Integral forms.
In order to implement the ideas of Section 2, we need to considerforms over suitable rings, generalizing [DP]. For simplicity, we set A := C [ ν ± , ( q sii q ij q ji − − : i = j ∈ I , ≤ s < − c q ij ] ⊂ C ( ν ) . (5.1)We now define the (non-restricted) integral forms as the A -subalgebras U + q , A = A h E i : i ∈ I i ⊂ U + q , U q , A = A [ K ± i , L ± i : i ∈ I ] ⊂ U q ,U − q , A = A h F i : i ∈ I i ⊂ U − q , U q , A = A h K ± i , L ± i , E i , F i : i ∈ I i ⊂ U q ,U > q , A = U + q , A ⊗ A A [ K ± i : i ∈ I ] , U q , A = U − q , A ⊗ A A [ L ± i : i ∈ I ] . These are crucial for our purposes. We have again a triangular decomposition U + q , A ⊗ A U q , A ⊗ A U − q , A ≃ U q , A . (5.2)The surjectivity of this multiplication map follows from the cross relations (4.4), (4.5)and (4.6), while the injectivity follows from (4.8). Recall (4.15) for the next result. Lemma 5.1.
For all i = j , ( λ q ij ) − ∈ A .Proof. If q c q ij ii q ij q ji = 1 , then using that q ii ∈ C [ ν ± ] × we have ( λ q ij ) − = ( − c q ij q c q ij ( c q ij − ii ( q ii − − c q ij Y ≤ s< − c q ij ( q sii q ij q ji − − ∈ A Otherwise q ii is a root of unity of order − c q ij , so because ( − c q ij ) ! q ii ∈ C × , we have ( λ q ij ) − = ( q − ii q ij q ji ) − c q ij ( − c q ij ) ! q ii Y ≤ s< − c q ij ( q sii q ij q ji − − ∈ A . (cid:3) Example 5.2.
Let q be of modular type br (2) , respectively wk (4) , see §A.3. Then A = C [ ν ± , ( ν − − , ( ν − ζ ) − ] , respectively A = C [ ν ± , ( ν − − , ( ν + 1) − ] .We now define restricted integral forms that also play a central role in this paper.Recall the Hopf skew-pairing from §4.2. The A -submodules U res − q , A := { y ∈ U − q |h y, U + q , A i ⊂ A } , U res + q , A := { x ∈ U + q |h U − q , A , x i ⊂ A } . (5.3)are A -subalgebras of U − q and U + q , respectively. This follows from the fact that U ± q , A arebraided Hopf subalgebras of U ± q over A and the properties of Hopf skew-pairings. OISSON ORDERS ON LARGE QUANTUM GROUPS 21
PBW-bases of integral forms.
Recall the Lusztig isomorphisms T q i from §4.4. Lemma 5.3. (a) T q i restricts to an A -algebra isomorphism T q i : U ρ i ( q ) , A → U q , A , i ∈ I . (b) Let β ∈ ∆ + . Then E β , F β ∈ U q , A .Proof. (a) follows from (4.20) and Lemma 5.1, while (b) from (a) and (4.17). (cid:3) Proposition 5.4.
The sets (4.18) and (4.19) are A -bases of U ± q , A and U q , A , respectively.Proof. We consider the case of U + q , A , the other being analogous. Let Y be the set ofPBW monomials of U + q from (4.18). By Lemma 5.3, Y ⊂ U + q , A . The defining relationsof U + q involve products of E i with coefficients in A , hence we may prove recursively that,for j > k , E β j E β k ∈ A Y , the A -module generated by Y , where each monomial in theexpansion has letters E β t , j > t > k ; see the proof of [HY2, Theorem 4.8]. Thus A Y is a left ideal containing , so A Y = U + q , A . This fact and the direct sum decomposition U + q = ⊕ y ∈ Y C ( ν ) y imply that U + q , A = ⊕ y ∈ Y A y . (cid:3) Recall the notation e N β in §3.7. Next we consider the quantum divided powers F ( n ) β j = F nβ j ( n ) ! q βjβj , E ( n ) β j = E nβ j ( n ) ! q βjβj , ≤ n < e N β j . Proposition 5.5.
For j ∈ I ℓ , let n j , m j be such that ≤ n j , m j < e N β j . Then h F ( n ) β . . . F ( n ℓ ) β ℓ , E m β . . . E m ℓ β ℓ i = δ n m . . . δ m ℓ n ℓ . Proof.
Let η j = h F β j , E β j i , j ∈ I ℓ . The same proof as [AnY, Proposition 4.6] shows that h F ( n ) β . . . F ( n ℓ ) β ℓ , E m β . . . E m ℓ β ℓ i = δ n m . . . δ m ℓ n ℓ η n · · · η n ℓ ℓ , As in [AnY, 4.7], we see that η j = 1 : here h F i , E i i = 1 , there h F i , E i i = − for i ∈ I . (cid:3) Propositions 5.4 and 5.5 imply the following:
Corollary 5.6.
The following sets are A -basis of U res − q , A and U res + q , A , respectively: n F ( n ) β . . . F ( n ℓ ) β ℓ : 0 ≤ n j < e N β j o and n E ( m ) β . . . E ( m ℓ ) β ℓ : 0 ≤ m j < e N β j o . (5.4)5.3. The specialization of U q , A . We consider the setting in Section 2 assuming R = A , h = ν − ξ and the R -algebra A being either U q , A or its subalgebras U ± q , A . We claim thatthe map C [ ν ± ] → C , q ξ extends to an isomorphism A / ( ν − ξ ) ≃ C . For, if q sii q ij q ji − q sii q ij q ji − for some i = j, ≤ s < − c q ij , then ≤ − c q ij ≤ s < − c q ij , which contradicts Property (c) in the Appendix A. Here andbelow we will use the bar notation x x for specializations. Theorem 5.7.
There are Hopf algebra (respectively, braided Hopf algebra) isomorphisms Ξ q : U q → U q , A / ( ν − ξ ) and Ξ q | U ± q : U ± q → U ± q , A / ( ν − ξ ) given by e i E i , f i F i , K ± i K ± i , L ± i L ± i for all i ∈ I . For each i ∈ I , thefollowing diagram is commutative: U ρ i ( q ) Ξ ρi ( q ) / / T q i (cid:15) (cid:15) U ρ i ( q ) f, A / ( ν − ξ ) T q i (cid:15) (cid:15) U q Ξ q / / U q , A / ( ν − ξ ) . (5.5) Proof.
The defining relations of U q hold in U q , A / ( ν − ξ ) by the definition of U q in [An3]and the presentation of U q in [An2]. Therefore, the map Ξ q as above is well-defined.Moreover Ξ q is surjective, since E i , F i , K ± i , L ± i generate U q , A / ( ν − ξ ) as C -algebra.Now we check that (5.5) is a commutative diagram. Indeed, since Property (c) in theAppendix A holds and q q under the evaluation map, we have that Ξ q ◦ T q i ( e j ) = (ad c E i ) − c q ij E j = T q i ◦ Ξ ρ i ( q ) ( e j ) , Ξ q ◦ T q i ( f j ) = (ad c F i ) − c q ij F j = T q i ◦ Ξ ρ i ( q ) ( f j ) for j = i . Since Ξ q ◦ T q i ( X ) = T q i ◦ Ξ ρ i ( q ) ( X ) for X ∈ { e i , f i , K ± j , L ± j } , the claim follows.By (5.5), Ξ q ( E β ) = E β and Ξ q ( F β ) = F β for all β ∈ ∆ + . Hence Ξ q sends the PBWbasis of U q to that of U q , A / ( ν − ξ ) , so Ξ q , and its restrictions to U ± q , are isomorphisms.Clearly Ξ q (and its restrictions) are isomorphisms of (braided) Hopf algebras. (cid:3) The specialization of U res ± q , A . Recall the Lusztig algebra L q §3.9 and the identifi-cation of e B q with U + q as in §4.3. For β ∈ O q , n ∈ N , define η ( n ) β ∈ L q such that ( η ( n ) β j , e m β . . . e m ℓ β ℓ ) = ( , m j = n, m k = 0 for k = j, , otherwise , (5.6)By [AAR1, Proposition 4.6], the set { η ( n ) β · · · η ( n ℓ ) β ℓ : 0 ≤ n j < e N β j } is a basis of L q and the algebra L q is generated by { η α i : i ∈ I } ∪ { η ( N β ) β : β ∈ Π q } . The Lie algebra n − q from (3.15) has a C -basis { ι ∗ ( η ( N β ) β ) : β ∈ O q } and set of simpleroot vectors { ι ∗ ( η ( N β ) β ) : β ∈ Π q } . Similar results hold for the Lusztig algebra L q ( − associated to e B q ( − ) ≃ U − q . The corresponding elements of L q ( − , defined as in (5.6) using f mβ instead of e mβ , will be denoted by θ ( n ) β , where β ∈ O q and n ∈ N . Remark 5.8.
The Lie algebras associated to L q and L q ( − as in Remark 3.7 are iso-morphic to each other, see the list in the Appendix A. Hence we have a Lie algebraisomorphism n − q ( − ≃ n + q (5.7)where ι ∗ ( θ ( N β ) β ) ∈ n + q ( − , β ∈ Π q are mapped to the simple root vectors of n − q .For a braided Hopf algebra B denote the braided opposite algebra B op with product µ op := µc − where µ : B × B → B is the product in B . OISSON ORDERS ON LARGE QUANTUM GROUPS 23
Proposition 5.9.
There are C -algebra anti-isomorphisms φ − : U res − q , A / ( ν − ξ ) → L q ,φ + : (cid:16) U res + q , A / ( ν − ξ ) (cid:17) op → L q ( − , given by F ( n ) β η ( n ) β ,E ( n ) β θ ( n ) β , β ∈ O q , n ∈ N . Proof.
We prove the statement in the minus case, the plus case is analogous. By Propo-sition 5.5 the Hopf skew-pairing h , i : U − q × U + q → C ( ν ) restricts to a perfect pairing h , i : U res − q , A × U + q , A → A . Since U + q , A / ( ν − ξ ) ≃ U + q as braided Hopf algebras, the latter pairing induces a non-degenerate pairing h , i : (cid:0) U res − q , A / ( ν − ξ ) (cid:1) × U + q → C such that h yy ′ , x i = h y ⊗ y ′ , ∆( x ) i , y, y ′ ∈ U res − q , A / ( ν − ξ ) , x ∈ U + q (5.8)and we have the commutative diagram U res − q , A × U + q , A / / (cid:15) (cid:15) A (cid:15) (cid:15) (cid:0) U res − q , A / ( ν − ξ ) (cid:1) × U + q / / C By the definition of L q , we have a canonical vector space isomorphism φ − : U res − q , A / ( ν − ξ ) → L q such that h Y, x i = ( φ − ( Y ) , x ) for all Y ∈ U res − q , A / ( ν − ξ ) , x ∈ U + q . Comparing (3.17) and (5.8), we see that φ − is analgebra anti-isomorphism. Using again Proposition 5.5 and the definition (5.6) of η ( n ) β ,we get that φ − is given by F ( n ) β η ( n ) β for β ∈ O q , n ∈ N . (cid:3) Poisson orders on large quantum groups
By Theorem 5.7, the large quantum group U q fits in the context of Section 2 andconsequently the pair ( U q , Z ( U q )) inherits a structure of Poisson order from deformationtheory. However the Poisson algebra Z ( U q ) is often singular. We prove that the centralHopf subalgebra Z q introduced in §4.5 (which is of course regular) is a Poisson subalgebraof Z ( U q ) of the same dimension. Thus ( U q , Z q ) has a structure of Poisson order thatrestricts to the corresponding large quantum Borel and unipotent algebras.6.1. Poisson structure on Z q . We show that Z q , Z > q , Z q , Z + q and Z − q are Poissonsubalgebras of Z ( U q ) , respectively Z ( U > q ) , Z ( U q ) , Z ( U + q ) and Z ( U − q ) .We need first to introduce the matrix P q ∈ C Π q × Π q . Let β, γ ∈ O q + . As q βγ = q βγ ( ξ ) ,(4.23) implies that there exists ℘ q βγ ( ν ) ∈ A such that − q N β N γ βγ = ( ν − ξ ) ℘ q βγ ( ν ) . (6.1)Recall the notation β from (3.14) and the set O q from Theorem 3.5. Then we define P q = ( ℘ q βγ ( ξ )) β,γ ∈ Π q . (6.2) Lemma 6.1.
Let i ∈ I . Then P ρ i ( q ) = P q .Proof. First, Π ρ i ( q ) = s q i ( Π q ) . Thus ℘ ρ i ( q ) s q i ( β ) s q i ( γ ) ( ν ) = ℘ q βγ ( ν ) for β, γ ∈ Π q by (3.11). (cid:3) The next theorem is the main result of this section.
Theorem 6.2.
There are structures of Poisson order on the pairs ( U q , Z q ) , ( U > q , Z > q ) , ( U q , Z q ) , ( U + q , Z + q ) and ( U − q , Z − q ) (6.3) arising by restriction from the Poisson order on the corresponding algebra and its centerwith Poisson bracket (2.2) . The central algebras Z q , Z > q and Z q are Poisson-Hopf while Z ± q are coideal Poisson subalgebras over the former. Because of Theorem 5.7 and Proposition 2.4 we are reduced to prove:
Proposition 6.3.
The subalgebras Z ± q , Z > q , Z q and Z q are Poisson subalgebras of Z ( U ± q ) , Z ( U > q ) , Z ( U q ) and Z ( U q ) , respectively, under the Poisson bracket (2.2) . Observe that Z ± q , Z > q and Z q are Poisson subalgebras of Z q . Proof.
We apply Theorem 2.3 to the algebra U + q , A , the automorphisms ς q i and the (id , ς q i ) -derivations ∂ q i , i ∈ I to conclude that Z ′ defined as in (2.4) is a Poisson subalgebra of Z ( U + q ) . Now we have that ς q i ⋆ = ς q i , ∂ q i ∗ = ∂ q i , Z + q ⊂ ∩ i ∈ I ker( ς q i − id) . The equality ⋆ holds since q = q ( ξ ) , while ∗ follows by a direct computation on thegenerators of U + q . The inclusion holds since Z + q ⊂ Z ( U q ) : indeed ς q i ( x ) = K i xK − i = x for all x ∈ Z + q . From this inclusion and (4.25), Z ′ = Z + q . The proof for Z − q is analogous.The restriction of the Poisson structure to Z ± q vanishes by the definition (2.2).Next we prove the statement for Z > q . Let β, γ ∈ O q + . We compute { e N β β , K N γ γ } = [ E N β β , K N γ γ ] ν − ξ = 1 − q N β N γ βγ ν − ξ E N β β K N γ γ (6.1) = ℘ q βγ ( ξ ) e N β β K N γ γ ∈ Z > q . This proves the claim since it suffices to check the bracket between generators. Similarly, { e N β β , L N γ γ } ∈ A e N β β L N γ γ , { f N β β , K N γ γ } ∈ A e N β β K N γ γ , { f N β β , L N γ γ } ∈ Z q . This finishes the proof for Z q and reduces that of Z q to prove that { e N β β , f N γ γ } ∈ Z q . Forthis we use the enumeration of the positive roots using the longest element of the Weylgroupoid. First we assume that β = β j , γ = β k for ≤ j < k ≤ ℓ . Let p = ρ i j . . . ρ i ( q ) , γ ′ = s p i j . . . s i ( γ ) , so N γ ′ = N γ . We have that { e N β β , f N γ γ } = [ E N β β , F N γ γ ] ν − ξ = T q i . . . T i j ([ K − N ij i j F N ij i j , F N γ ′ γ ′ ]) ν − ξ = T q i . . . T i j [ K − N ij i j F N ij i j , F N γ ′ γ ′ ] ν − ξ = T q i . . . T i j (cid:16)n K − N ij i j f N ij i j , f N γ ′ γ ′ o(cid:17) . By the statements already proved, n K − N ij i j f N ij i j , f N γ ′ γ ′ o ∈ Z p . Hence { e N β β , f N γ γ } ∈ T q i . . . T i j ( Z p ) Theorem 4.7 = Z q . OISSON ORDERS ON LARGE QUANTUM GROUPS 25
The case j > k is proved analogously. Now assume that β = γ . We start with the case β = α i for some i ∈ I (a simple Cartan root). Using (4.6) we prove recursively that [ E Ni , F Ni ] = N X t =1 ( t ) ! q ii (cid:18) Nt (cid:19) q ii F N − ti t − Y s =0 (cid:16) K i q t − N − sii − L − i (cid:17) E N − ti , N ∈ N . (6.4)Let t ∈ I N i − . As q ii is a primitive N i -th root of unity and q ii = q ii , ℘ q α i α i ( ξ ) = 1 − q N i ii ν − ξ = (1 − q N i ii )(1 + q N i ii + · · · + q N i ( N i − ii ) ν − ξ = N i − q N i ii ν − ξ . Hence, ( N i ) ! q ii ν − ξ = 1 − q N i ii ν − ξ · (1 − q ii ) . . . (1 − q N i − ii )(1 − q ii ) N i = ℘ q α i α i ( ξ )(1 − q ii ) N i · (6.5)From this we obtain, { e N i i , f N i i } = [ E N i i , F N i i ] ν − ξ = ( N i ) ! q ii ν − ξ N i − Y s =0 ( K i q − sii − L i ) = − ℘ q α i α i ( ξ )( q ii − N i ( K N i i − L − N i i ) ∈ Z q . Next, if β is not simple, say β = β j for some j ∈ I ℓ , then using Theorem 4.7 again { e N β β , f N β β } = T q i . . . T i j − (cid:16) [ E N ij i j , F N ij i j ] ν − ξ (cid:17) = − ℘ q ββ ( ξ )( q ββ − N β ( K N β β − L N β β ) ∈ Z q . (cid:3) (6.6) 7. The associated Poisson algebraic groups
In this section we describe the Poisson algebraic groups that correspond to the Poisson-Hopf algebras Z q , Z > q and Z q . We prove that, as algebraic groups, they are isomorphicto Borel subgroups of connected semisimple algebraic groups but of adjoint type (andnot of simply connected type as in previous works) and direct products of such Borelsubgroups. The dual Lie bialgebras of the three tangent Lie algebras are proved toconstitute a Manin triple, the ample Lie algebra in which is reductive. It is shown thatthe resulting Lie bialgebra structures are the ones from the Belavin–Drinfeld classification[BD] for the standard BD-triple (containing the empty subsets of the Dynkin graph)and arbitrary choice of the continuous parameters The results completely determine thePoisson structures on the three kinds of algebraic groups in question.7.1. The positive and negative parts of the dual tangent Lie bialgebra of M q . Let M q , M ± q , M ± q , M > q and M q be the complex algebraic groups which are equalto the maximal spectra of the commutative Hopf algebras Z q , Z ± q , Z ± q , Z > q and Z q ,respectively. Here the Hopf algebra structures on Z ± q are the restrictions of the braidedHopf algebra structures on U ± q to Z ± q [An3].Since Z q is a finitely generated Poisson-Hopf algebra which is an integral domain, M q is a connected Poisson algebraic group (see §B.2 for background). Analogously, M > q , M q and M ± q are connected Poisson algebraic groups, and M ± q are connected unipotentalgebraic groups. The latter are not Poisson algebraic groups; they are isomorphic tocertain Poisson homogeneous spaces for M > q and M q (see §8.3). The tensor productdecompositions Z q ≃ Z > q ⊗ Z q from §4.5 give rise to the decomposition of algebraicgroups(7.1) M q ≃ M > q × M q . This is not a direct product decomposition of Poisson algebraic groups (because Z q ≃ Z > q ⊗ Z q is a tensor product decomposition of commutative but not Poisson algebras).However, the canonical projections M q ։ M > q and M q ։ M q are homomorphisms ofPoisson algebraic groups because Z > q and Z q are Poisson-Hopf subalgebras of Z q .Denote by m q , m > q and m q the tangent Lie bialgebras of M q , M > q and M q (see §B.1for background and notations). Eq. (7.1) gives rise to the direct sum decomposition ofLie algebras m q ≃ m q ⊕ m > q . The Lie coalgebra structure on m q , fully described below, has cross terms. The dual ofthe tangent Lie bialgebra m ∗ q = T ∗ M q is computed as the linearization at the identityelement of M q of its Poisson structure by using (B.1). The maximal ideal M of C [ M q ] ≃ Z q of functions vanishing at 1 coincides with the augmentation ideal of Z q . Inthe proofs below we will use the identification T ∗ M ≃ M / M where the differential d ( g ) of a function g ∈ C [ M q ] at ∈ M q is sent to the class of g − g (1) in M / M for g ∈ C [ M q ] . The Lie algebra m ∗ q has the C -basis: (cid:8) d ( e N β β ) , d ( f N β β ) , d ( K N γ γ ) , d ( L N γ γ ) : β ∈ O q + , γ ∈ Π q (cid:9) . (7.2)By Proposition 6.3, the subspaces ( m + q ) ∗ := ⊕ β ∈ O q + C d ( e N β β ) and ( m − q ) ∗ := ⊕ β ∈ O q + C d ( f N β β ) are Lie subalgebras of m ∗ q . The dual Lie bialgebras ( m > q ) ∗ := ( m > q ) ∗ and ( m q ) ∗ := ( m q ) ∗ are canonically identified with the Lie sub-bialgebras of m ∗ q ( m + q ) ∗ ⊕ (cid:0) ⊕ γ ∈ Π q d ( K N γ γ ) (cid:1) and ( m − q ) ∗ ⊕ (cid:0) ⊕ γ ∈ Π q d ( L N γ γ ) (cid:1) . Recall the notation from §5.4.It follows from the triangular decomposition (3.15) of the semisimple Lie algebra g q associated to the large quantum group U q that the set of simple root of g q can be identifiedwith Π q . Denote the entries of the Cartan matrix of g q by c β γ , β, γ ∈ Π q . Throughout the section we will assume the identification n − q ( − ≃ n + q from (5.7), so g q = n + q ⊕ h q ⊕ n − q will be identified with n − q ( − ⊕ h q ⊕ n − q . By the definitions of n − q ( − and n − q , g q has a set of Chevalley generators { x β , y β , h β : β ∈ Π q } such that x β ∈ C × ι ∗ ( θ β ) and y β ∈ C × ι ∗ ( η β ) , respectively. In this way the root lattice of g q is identified with Q q by setting deg x β = − deg y β = N β β , deg h β = 0 for β ∈ Π q . Proposition 7.1.
We have a Q q -graded Lie algebra isomorphism ( m ± q ) ∗ ≃ n ± q given by d ( e N β β ) s β ι ∗ ( θ ( N β ) β ) , respectively d ( f N β β )
7→ − s β ι ∗ ( η ( N β ) β ) for all β ∈ O q + , where s β := ℘ q ββ ( ξ )(1 − q ββ ) N β . In the plus case we use the identification (5.7) . OISSON ORDERS ON LARGE QUANTUM GROUPS 27
Proof.
First we prove the minus case. Let β, γ ∈ O q + . Since F N β β = f N β β , F N γ γ = f N γ γ ∈ Z q and the subalgebra Z q is closed under the Poisson bracket {· , ·} by Proposition 6.3, usingProposition 5.4 we obtain [ F N β β , F N γ γ ] ≡ X δ ∈ O q + ( ν − ξ ) a δβγ ( ν ) F N δ δ + ( ν − ξ ) g βγ mod ( ν − ξ ) U − q , A , (7.3)where a δβγ ( ν ) ∈ A and g βγ is a non-commutative polynomial in { F N δ δ : δ ∈ O q + } involvingmonomials of degree ≥ . Since U q is Z I -graded, the sum in the right-hand side has atmost one non-zero term, when N β β + N γ γ = N δ δ for some δ ∈ O q + . Therefore [ d ( f N β β ) , d ( f N γ γ )] = d (cid:0) { f N β β , f N γ γ } (cid:1) = d [ F N β β , F N γ γ ] ν − ξ ! (7.4) = X δ ∈ O q + a δβγ ( ξ ) d ( f N δ δ ) + d ( g βγ ) = X δ ∈ O q + a δβγ ( ξ ) d ( f N δ δ ) , because g βγ ∈ M . From (7.3) and since U − q , A is N -graded connected, we see that [ F ( N β ) β , F ( N γ ) γ ] ≡ X δ ∈ O q + a δβγ ( ν ) ( ν − ξ )( N δ ) ! q δδ ( N β ) ! q ββ ( N γ ) ! q γγ F ( N δ ) δ mod ( ν − ξ ) U res − q , A . It follows from (6.5) that ( N β ) ! q ββ ν − ξ = ℘ q ββ ( ξ )(1 − q ββ ) N β = s β . Hence in U res − q , A / ( ν − ξ ) we have [ s β F ( N β ) β , s γ F ( N γ ) γ ] = X δ ∈ O q + a δβγ ( ξ ) s δ F ( N δ ) δ . (7.5)The statement of the lemma follows from this identity, (7.4) and Proposition 5.9. Theplus case is proved analogously, using Remark 5.8 and that q βδ = 1 for all β, γ ∈ O q + . (cid:3) The last part of the proof gives the following fact about the structure of Lusztigalgebras which is of independent interest:
Corollary 7.2.
The braided Hopf algebra projection ι ∗ : L q ։ U ( n − q ) (recall (3.18) ) hasan algebra section U ( n − q ) → L q given by ι ∗ ( η ( N β ) β ) η ( N β ) β , β ∈ O q + . The dual tangent Lie bialgebra of M q .Lemma 7.3. The following hold in the Lie algebra m ∗ q : [ d ( e N β β ) , d ( f N γ γ )] = − δ βγ ℘ q ββ ( ξ )( q ββ − N β ( d ( K N β β ) + d ( L N β β )) , β, γ ∈ Π q , and [ d ( K N β β ) , d ( e N γ γ )] = − ℘ q βγ ( ξ ) d ( e N γ γ ) , [ d ( K N β β ) , d ( f N γ γ )] = ℘ q βγ ( ξ ) d ( f N γ γ ) , [ d ( L N β β ) , d ( e N γ γ )] = − ℘ q γβ ( ξ ) d ( e N γ γ ) , [ d ( L N β β ) , d ( f N γ γ )] = ℘ q γβ ( ξ ) d ( f N γ γ ) for all β, γ ∈ O q + .Proof. The case of β = γ ∈ Π q of the first identity follows from the fact that (7.2) is abasis of the Lie algebra m ∗ q and that the latter is Q q -graded. The case β = γ ∈ Π q is aconsequence of (6.6) since d ( L N β β ) = − d ( L − N β β ) , which in turn follows since the value of L N β β at the identity of M q equals . The other four identities follow from (4.4)–(4.5). (cid:3) Since the polynomials ν n − a are separable over C for a = 0 , we infer from (6.1) that ℘ q ββ ( ξ ) = 0 for all β ∈ O q + . Theorem 7.4. (a)
The Cartan matrix of the semisimple Lie algebra g q is given by c β γ = ℘ q βγ ( ξ ) + ℘ q γβ ( ξ ) ℘ q ββ ( ξ ) , β, γ ∈ Π q . (b) There is a ( Q q -graded) Lie algebra isomorphism g q ⊕ h q ≃ m ∗ q such that x β d ( f N β β ) , y β ( q β − N β ℘ q ββ ( ξ ) d ( e N β β ) , h β ℘ q ββ ( ξ ) ( d ( K N β β ) + d ( L N β β )) for β ∈ Π q and h q maps to the subspace (cid:8) X β ∈ Π q a β d ( K N β β ) + b β d ( L N β β ) : X β ∈ Π q ℘ q βγ ( ξ ) a β + ℘ q γβ ( ξ ) b β = 0 , ∀ γ ∈ Π q (cid:9) of the abelian Lie algebra ⊕ β ∈ Π q ( C d ( K N β β ) + C d ( L N β β )) .Proof. (a) For β ∈ Π q , define the following elements of m ∗ q : e x β := d ( f N γ γ ) , e y β := ( q β − N β ℘ q ββ ( ξ ) d ( e N β β ) , e h β := 1 ℘ q ββ ( ξ ) ( d ( K N β β ) + d ( L N β β )) and the Lie subalgebra g q ( β ) := C e x β ⊕ C e h β ⊕ C e y β . Lemma 7.3 implies that [ e h β , e x β ] = 2 e x β , [ e h β , e y β ] = − e y β , [ e x β , e y β ] = e h β , so g q ( β ) ≃ sl .Now take β = γ ∈ Π q and consider g q as a g q ( β ) -module under the adjoint action. Itfollows from Lemma 7.3 that [ e x β , e y γ ] = 0 and [ e h β , e y γ ] = − ℘ q βγ ( ξ ) + ℘ q γβ ( ξ ) ℘ q ββ ( ξ ) e x γ , so e y γ is a highest weight vector for g q ( β ) ≃ sl of weight − ℘ q βγ ( ξ )+ ℘ q γβ ( ξ ) ℘ q ββ ( ξ ) ω where ω denotes the fundamental weight of sl . The isomorphism of Proposition 7.1 and theSerre relations in n − q imply that ad − c βγ +1 e y β ( e y γ ) = 0 and ad j e y β ( e y γ ) = 0 for j − c βγ . Hence, ad − c βγ e y β ( e y γ ) is the lowest weight vector of the (irreducible) g q ( β ) -module generatedby e y γ , which forces c β γ = ℘ q βγ ( ξ ) + ℘ q γβ ( ξ ) ℘ q ββ ( ξ ) · This proves part (a). It also proves that the assignment x β e x β , y β e y β , h β e h β for β ∈ Π q defines a Q q -graded Lie algebra homomorphism η : g q → m ∗ q which is an OISSON ORDERS ON LARGE QUANTUM GROUPS 29 embedding by Proposition 7.1 and the linear independence of { d ( K N β β ) , d ( L N β β ) : β ∈ O q + } . Here we use the canonical isomorphism n ± q → n ∓ q obtained by restricting theChevalley involution of g ± q . Denote ( m q ) ∗ := ⊕ β ∈ Π q (cid:16) C d ( K N β β ) ⊕ C d ( L N β β ) (cid:17) . Let ( m q ) ′ be the intersection of the kernels of the functionals { l β : β ∈ Π q } on ( m q ) ∗ given by l β ( d ( K N γ γ )) := ℘ q γβ ( ξ ) , l β ( d ( L N γ γ )) := ℘ q βγ ( ξ ) . Proposition 7.1 and Lemma 7.3 imply that [( m q ) ′ , Im η ] = 0 . Since the number of theabove functionals equals | O q + | = dim h q , we have dim( m q ) ′ > dim h q . It follows from part(a) that dim( m q ) ′ dim h q , Hence, dim h q = dim( m q ) ′ and taking a linear isomorphism h q ≃ ( m q ) ′ extends η to the needed Lie algebra isomorphism for part (b). (cid:3) Let ( · , · ) be the invariant symmetric bilinear form on g q for which the induced formon the dual of the Cartan subalgebra of g q satisfies ( β, β ) = 2 for short roots β . As it iscommon, we will identify h q with h ∗ q via this form. The scalar κ β := 2 ℘ q ββ ( ξ )( β, β ) − only depends on the simple factor of g q of which β is a root, because by Theorem 7.4(a),(7.6) c βγ = ℘ q βγ ( ξ ) + ℘ q γβ ( ξ ) ℘ q ββ ( ξ ) = 2( β, γ )( β, β ) · Proposition A.3 (i) tells us that each large quantum group U q is realized as a specil-iazation of an integral form of a one-parameter quantum group U q in infinitely manydifferent ways parametrized by integers t ij ∈ Z for i < j ∈ I . Furthermore, by part (ii)of that proposition, for a generic choice of the parameters t ij ∈ Z , i < j ∈ I , the matrixwith entries ℘ q βγ ( ξ ) for β, γ ∈ Π q is non-degenerate. In the remaining part of the paperwe will assume the following: Non-degeneracy Assumption 7.5.
The specialization parameters t ij ∈ Z , i < j ∈ I inProposition A.3 are chosen in such a way that the matrix P q in (6.2) is non-degenerate. Remark 7.6.
In what follows we will identify the Lie algebras(7.7) m ∗ q ≃ g q ⊕ h q via the isomorphism from Theorem 7.4. In particular, x β , y β , h β for β ∈ Π q will beviewed as elements of m ∗ q . We also fix the identification of abelian Lie algebras(7.8) (cid:8) X β ∈ Π q a β d ( K N β β ) + b β d ( L N β β ) : X β ∈ Π q ℘ q βγ ( ξ ) a β + ℘ q γβ ( ξ ) b β = 0 , ∀ γ ∈ Π q (cid:9) ≃ h q for Theorem 7.4(b) by sending P β ∈ Π q a β d ( K N β β ) + b β d ( L N β β ) P β ∈ Π q b β κ β β , usingthe identification of h q with h ∗ q via the form ( ., . ) . Since both Lie algebras in (7.8) havethe same dimensions, we only need to show that this map is injective. An element inthe kernel has b β = 0 for β ∈ Π q and thus, P ℘ q βγ ( ξ ) a β = 0 . The Non-degeneracyAssumption 7.5 implies that a β = 0 for β ∈ Π q . Let b ± q be the Borel subalgebras of g q with respect to these Chevalley generators. Then(7.9) ( m > q ) ∗ ⊂ b − q ⊕ h q and ( m q ) ∗ ⊂ b + q ⊕ h q . Using the Non-degeneracy Assumption 7.5 one more time, we obtain that the projectioninto the first component m ∗ q ≃ g q ⊕ h q → g q restricts to the Lie algebra isomorphisms(7.10) ( m > q ) ∗ ≃ b − q and ( m q ) ∗ ≃ b + q . We next describe the embeddings (7.9). Denote the linear maps P , P T ∈ End( h q ) : P ( β ) = X γ ∈ Π q ℘ q βγ ( ξ ) γ, P T ( β ) = X γ ∈ Π q ℘ q γβ ( ξ ) γ. (7.11)Because of the Non-degeneracy Assumption 7.5 both endomorphisms are invertible.Denote by (( · , · )) the invariant symmetric bilinear form on g q which is a rescaling of ( · , · ) on each simple factor of g q by κ − β . It satisfies (( d ( K N β β ) + d ( L N β β ) , d ( K N γ γ ) + d ( L N γ γ ))) = ℘ q ββ ( ξ ) ℘ q γγ ( ξ )(( h β , h γ ))= ℘ q ββ ( ξ ) ℘ q γγ ( ξ ) κ − γ c βγ ( γ, γ ) = ℘ q βγ ( ξ ) + ℘ q γβ ( ξ ) . · This implies that the form (( · , · )) has a unique extension to an invariant symmetric bilinearform on m ∗ q such that (( d ( K N β β ) , d ( L N γ γ ))) = ℘ q βγ ( ξ ) , (7.12) (( d ( K N β β ) , d ( L N β β ))) = (( d ( K N γ γ ) , d ( L N γ γ ))) = 0 (7.13)for β, γ ∈ Π q . The Non-degeneracy Assumption 7.5 implies that the bilinear form (( · , · )) on m ∗ q is non-degenerate.One easily verifies that the orthogonal complement in m ∗ q of g q equals h q . Proposition 7.7.
For all large quantum groups U q satisfying the Non-degeneracy As-sumption 7.5, the subalgebras ( m > q ) ∗ ⊂ b − q ⊕ h q and ( m q ) ∗ ⊂ b + q ⊕ h q are given by ( m > q ) ∗ = { ( y + h, − h ) : y ∈ n − q , h ∈ h q } , ( m q ) ∗ = { ( x + h, P − P T ( h )) : x ∈ n + q , h ∈ h q } . Proof.
Denote the first (abelian) Lie algebra in (7.8) by h (2) q . Fix h := X β ∈ Π q c β ( d ( K N β β ) + d ( L N β β )) , h := X β ∈ Π q a β d ( K N β β ) , h := X β ∈ Π q b β d ( L N β β ) . If h + h ∈ h (2) q , then l γ ( h ) = − l γ ( h ) for all γ ∈ Π q , which is equivalent to(7.14) P (cid:0) X β ∈ Π q a β β (cid:1) = − P T (cid:0) X β ∈ Π q b β β (cid:1) . By Theorem 7.4(b), in the identification (7.7), d ( K N β β )+ d ( L N β β ) corresponds to κ β β .Hence, the first statement of the proposition is equivalent to proving that for all h , h , h as above, if h + h ∈ h (2) q and h + h + h ∈ ( m > q ) ∗ , then c β = − b β for β ∈ Π q . Fromthe condition h + h ∈ h (2) q we obtain (( d ( L N γ γ ) , h + h )) = 0 for all γ ∈ Π q . Thus X β ∈ Π q ℘ q βγ ( ξ )( c β + b β ) = 0 , ∀ γ ∈ Π q . OISSON ORDERS ON LARGE QUANTUM GROUPS 31
Now the first statement follows from the Assumption 7.5. The second follows from thefirst by interchanging the roles of ( m > q ) ∗ and ( m q ) ∗ and applying (7.14). (cid:3) We next describe the Lie coalgebra structure on m ∗ q and the corresponding Manintriple; see §B.1 for background. Theorem 7.8.
For every choice of the specialization parameters t ij ∈ Z satisfying theNon-degeneracy Assumption 7.5 the following hold: (a) The Lie coalgebra structure of the Lie bialgebra m ∗ q is given by δ ( x β ) = d ( L N β β ) ∧ x β , δ ( y β ) = d ( K N β β ) ∧ y β , δ ( d ( L N β β )) = δ ( d ( L N β β )) = 0 for all β ∈ Π q . (b) With respect to the bilinear form (( · , · )) , ( m ∗ q , ( m q ) ∗ , ( m > q ) ∗ ) is a Manin triple. (c) The Lie coalgebra structures of ( m > q ) ∗ and ( m q ) ∗ satisfy (( δ ( y ) , x ⊗ x )) = − (( y, [ x , x ])) , (( δ ( x ) , y ⊗ y )) = (( x, [ y , y ])) (7.15) for all x, x , x ∈ ( m q ) ∗ and y, y , y ∈ ( m > q ) ∗ . Remark 7.9. (a) Part (a) of the theorem uniquely determines the Lie coalgebra struc-tures of m ∗ q , ( m q ) ∗ and ( m > q ) ∗ , since the set { x β , y β , d ( L N β β ) , d ( L N β β ) : β ∈ Π q } and its appropriate subsets generate m ∗ q , ( m q ) ∗ and ( m > q ) ∗ .(b) By part (c) of the theorem, the Lie coalgebra structures of m ∗ q , ( m q ) ∗ and ( m > q ) ∗ ,are precisely the ones that are associated to a Manin triple as in Remark B.1(c). Inparticular, we have the isomorphism of Lie bialgebras m ∗ q ≃ D (( m q ) ∗ ) , ( m > q ) ∗ ≃ ((( m q ) ∗ ) ∗ ) op ≃ ( m q ) op . (7.16)(c) The Lie bialgebra structures on the reductive Lie algebras m ∗ q ≃ g q ⊕ h q from part(a) of the theorem correspond to standard Belavin–Drinfeld triples (containing theempty subsets of the Dynkin graph) and arbitrary choice of the continuous param-eters in their classification [BD]. Proof of Theorem 7.8.
Part (a) follows from Lemma B.2 and the identities ∆( e N β β ) = K N β β ⊗ e N β β + e N β β ⊗ , ∆( f N β β ) = 1 ⊗ f N β β + f N β β ⊗ L − N β β . (7.17)for β ∈ Π q and the fact that K N β β and L N β β are group-like elements.(b) The subalgebras ( m > q ) ∗ and ( m q ) ∗ are orthogonal to their nilradicals because of theembeddings (7.9). This, combined with (7.13), implies that they are isotropic subalgebrasof m ∗ q with respect to the form (( · , · )) . The direct sum decomposition m q ≃ m q ⊕ m > q yields the desired result.(c) Part (a) of the theorem and the isomorphism in Theorem 7.4(b) imply at once thevalidity of the identities (7.15) for y = d ( e N β β ) , y = d ( K N β β ) , x = d ( f N β β ) , x = d ( L N β β ) , β ∈ Π q and all possible choices of x , x , y , y . The general case follows by inductionon root height when x , y are chosen to be root vectors by using the invariance of thebilinear form (( · , · )) . (cid:3) The Poisson algebraic groups M > q and M q . Combining the isomorphisms(7.10) and (7.16), we get the Lie algebra isomorphisms(7.18) m > q ≃ (( m q ) ∗ ) op ≃ ( b + q ) op ≃ b + q and m q ≃ ( m > q ) ∗ ≃ b − q , where ( . ) op stands for the opposite Lie algebra structure and ( b + q ) op ≃ b + q is the stan-dard Lie algebra isomorphism x
7→ − x . The proof of Proposition 7.7 shows that thecorresponding pull back maps on the level of duals send β
7→ − d ( K N β β ) , β d ( L N β β ) , ∀ β ∈ Π q . (7.19)The scalars κ β do not appear here because the form (( · , · )) is a rescaling of the form ( · , · ) on each simple factor of g q by κ − β .Denote by G q the adjoint semisimple algebraic group with Lie algebra g q and by B ± q its Borel subgroups corresponding to b ± q . Let T q := B + q ∩ B − q be the correspondingmaximal torus of G q . Denote by N ± q the unipotent radicals of B ± q .The groups of group-like elements of Z > q and Z q are the free abelian groups on K ± N β β , β ∈ Π q and L ± N β β , β ∈ Π q , respectively. Theorem 7.10.
For every choice of the specialization parameters t ij ∈ Z satisfyingthe Non-degeneracy Assumption 7.5, the Lie algebra isomorphisms (7.18) integrate toisomorphisms of algebraic groups. τ + : M > q ≃ −→ B + q and τ − : M q ≃ −→ B − q . Theorem 7.10 describes explicitly the algebraic groups M > q and M q . As an algebraicgroup, M q ≃ B + q × B − q . The Poisson structures on M > q , M q and M q are the uniquePoisson algebraic group structures that integrate the Lie bialgebras m > q , m q and m q ,whose dual Lie bialgebras are described in Theorem 7.8. Proof.
We prove the first statement, the second being analogous. Since G q is of adjointtype, the Borel subgroup B + q is canonically identified with the identity component of Aut( b + q ) . The adjoint action of M > q on m > q ≃ b + q induces a surjective homomorphism τ + : M > q ։ B + q . The latter restricts to an isomorphism τ + : N ( M > q ) ≃ −→ N + q , where N ( M > q ) is the unipotent radical of M > q . The homomorphism τ + also restricts to asurjective homomorphism(7.20) τ + : T ( M > q ) ։ T q , where T ( M > q ) is a maximal torus of M > q . The tori T ( M > q ) and T q are connected because M > q and B + q are connected algebraic groups. Invoking the Levi decompositions of M > q and B + q , to show that τ + is an isomorphism, it is sufficient to show that the restriction(7.20) is an isomorphism. However, C [ T ( M > q )] ≃ C [ M > q /N ( M > q )] ≃ C [ G ( C [ M > q ])] , C [ T q ] ≃ C [ B + q /N + q ] ≃ C [ G ( C [ B + q ])] , where G ( H ) denotes the group of group-like elements of a Hopf algebra H .The group of group-like elements of C [ M > q ] ≃ Z > q is the free abelian group withgenerators K N β β , β ∈ Π q and the group of group-like elements of B + q is canonicallyidentified with the roots lattice Z Π q of G q . The differentials at the identity element ofthe two generating sets are respectively d ( K N β β ) and β , where β ∈ Π q . Eq. (7.19) implies OISSON ORDERS ON LARGE QUANTUM GROUPS 33 that τ ∗ + : G ( C [ B + q ]) → G ( C [ M > q ]) is an isomorphism. Hence, τ ∗ + : C [ T q ] → C [ T ( M > q ))] isan isomorphism and same holds for (7.20). This completes the proof of the theorem. (cid:3) Example 7.11.
Let q be of type wk (4) and fix N = ord q , M = ord( − q ) , see §A.3. Let γ = α + 2 α + 3 α + α . Then N α = N α = N , N α = N γ = M , O q + = { α , α , α + α , α , γ, α + γ } . As shown previously, ∆( e N β β ) = e N β β ⊗ K N β β ⊗ e N β β for β ∈ Π q = { α , α , α , γ } . Wecan check that e α + α = [ e , e ] c , e α + γ = [ e γ , e ] c and ∆( e Nα + α ) = e Nα + α ⊗ q − N e Nα K Nα ⊗ e Nα + K Nα K Nα ⊗ e Nα + α , ∆( e Mα + γ ) = e Mα + γ ⊗ q + 1) M e Mγ K Mα ⊗ e Mα + K Mα K Mγ ⊗ e Mα + γ . We now construct an explicit isomorphism between Z > q and the algebra of functionsover the Borel subgroup of PSL ( C ) × PSL ( C ) . Consider the Levi decomposition e B ≃ N ⋊ e T of the Borel subgroup of SL ( C ) , where e T = { diag( a , a , a ) : a i ∈ C × , a a a = 1 } , N = n t t t : t ij ∈ C o . The coproducts of these coordinate functions are given by ∆( a i ) = a i ⊗ a i and ∆( x ) = x ⊗ a a − ⊗ x , ∆( x ) = x ⊗ a a − ⊗ x , ∆( x ) = x ⊗ x a a − ⊗ x + a a − ⊗ x . Denote Z = h ( ζ, ζ, ζ ) i , where ζ is a primitive 3rd root of unity. The Borel subgroup B of SL ( C ) has Levi decomposition B ≃ N ⋊ T where T = e T / Z , so C [ B ] = C [ N ] ⊗ C [ e T ] Z = C [ x , x , x , a ± , a ± ] , where a := a a − and a = a a − . The coproducts of the coordinate functions on B are given by ∆( a ii +1 ) = a ii +1 ⊗ a ii +1 and ∆( x ) = x ⊗ a ⊗ x , ∆( x ) = x ⊗ a ⊗ x , ∆( x ) = x ⊗ x a ⊗ x + a a ⊗ x . The Borel subgroup of
PSL ( C ) × PSL ( C ) is isomorphic to B × B . We denote thecoordinate functions a ii +1 and x ij on the first and second copy of B by superscripts 1and 2. Now, clearly the map τ + : Z > q → C [ B × B ] given by K Nα a , K Nα a , K Nα a , K Mγ a and e Nα x , e Nα x , e Nα + α ( q − N x ,e Mα x , e Mγ x , e Mα + γ ( q + 1) M x is a Hopf algebra isomorphism. Poisson geometry and representations
In this section we describe the symplectic foliations and the torus orbits of symplecticleaves of the Poisson algebraic groups M q , M > q and M q , and the Poisson homogeneousspaces M + q and M − q . Previous work in this direction dealt with the so called standardPoisson structures on simple algebraic groups (and their Borel subgroups) [HL], thedual Poisson algebraic groups [DKP] and the related flag varieties [GY]. The Poissonstructures in Remark 7.9 are not of standard type in general and the results in thissection can not be deduced from [HL, DKP, GY]. For z ∈ M q , respectively M > q , M q , M + q , M − q , let H z , respectively H > z , H z , H + z , H − z be the algebra defined in Theorem A (c),respectively (1.2). The Poisson geometric results described above provide informationon the irreducible representations of the large quantum groups U q by reduction to thesheaf of algebras H z , z ∈ M q . Analogous results hold for U > q , U q , and U ± q .8.1. Representations of the large quantum groups and symplectic foliations.
The Manin triple described in Theorem 7.8 and the identification m ∗ q ≃ g q ⊕ h q equip g q ⊕ h q with a quasitriangular Lie bialgebra structure, which turns G q × T q into a Poissonalgebraic group. The Poisson structure on G q × T q equals L g ( r ) − R q ( r ) for g ∈ G q × T q ,where r ∈ ∧ ( g q ⊕ h q ) is the r -matrix for the Lie bialgebra structure on g q ⊕ h q , and L g ( − ) and R g ( − ) refer to the left and right-invariant bivector fields on G q × T q .Let f M > q and f M q be the connected Lie subgroups of G q × T q with Lie algebras ( m q ) ∗ and ( m > q ) ∗ . Proposition 7.7 implies that f M q is an algebraic subgroup, while f M > q is notnecessarily a closed Lie subgroup. The projection onto the first component π : G q × T q ։ G q gives the surjective Lie group homomorphisms π + : f M > q ։ B + q , π − : f M q ։ B − q . Since G q is of adjoint type, the kernel of the exponential map exp : h q ։ T q equals πiP ∨ q , where P ∨ q denotes the coweight lattice of g q . Denote the subgroup C q := exp (cid:0) πi P − P T ( P ∨ q ) (cid:1) ⊂ T q , (8.1)cf. (7.11). Proposition 7.7 and the solvability of f M > q and f M q give that f M > q = ( N + q × { } ) × { exp( h, P − P T ( h )) : h ∈ h q } , f M q = ( N − q × { } ) × { ( t, t − ) : t ∈ T q } , from which one obtains that Ker π + = { } , Ker π − = { } × C q . Composing π ± with the isomorphisms from Theorem 7.10 leads to the isomorphisms τ − π + : f M > q ≃ −→ M > q , τ − − π − : f M q / Ker π − ≃ −→ M q . Their inverses give the canonical embeddings j + : M > q ֒ → G q × (cid:0) T q /C q (cid:1) , j − : M q ֒ → G q × (cid:0) T q /C q (cid:1) . (8.2)Here we use that G q × (cid:0) T q /C q (cid:1) ≃ (cid:0) G q × T q (cid:1) / Ker π − and f M q ∩ Ker π − = { } . Remark 8.1.
If the matrix q is symmetric, then so is the matrix P q . This implies that P = P T and that the group C q is trivial. Then the continuous parameter accompanyingthe BD triple is trivial and the Poisson structure is the standard one. OISSON ORDERS ON LARGE QUANTUM GROUPS 35
Theorem 8.2.
Let U q be a large quantum group. For every choice of the specializationparameters t ij ∈ Z satisfying the Non-degeneracy Assumption 7.5 the following hold: (a) The symplectic leaves of the Poisson algebraic group M q ≃ M > q × M q are the inverseimages j − ( O × t ) under the map j : M q → G q × (cid:0) T q /C q (cid:1) , j ( m + , m − ) := j + ( m + ) − j − ( m − ) , m + ∈ M > q , m − ∈ M q , where O is a conjugacy class of G q and t ∈ T q /C q . The dimension of the symplecticleaf j − ( O × { t } ) equals dim O . (b) If j ( z ) and j ( z ′ ) are in the same conjugacy class of G q × (cid:0) T q /C q (cid:1) , then there is analgebra isomorphism H z ≃ H z ′ . Note that, since T q /C q is abelian, each conjugacy class of G q × (cid:0) T q /C q (cid:1) has the form O × { t } , where O is a conjugacy class of G q and t ∈ (cid:0) T q /C q (cid:1) . Proof. (a) By [RSTS], since the Poisson algebraic group G q × T q is quasitriangular, itsdouble Poisson algebraic group is canonically isomorphic to D ( G q × T q ) ≃ (cid:0) G q × T q (cid:1) × (cid:0) G q × T q (cid:1) . Theorem 7.8(b) implies that the dual Poisson Lie group of G q × T q is f M > q × f M q ֒ → (cid:0) G q × T q (cid:1) × (cid:0) G q × T q (cid:1) with the opposite Poisson structure to the restriction of the one of the double. Both f M > q × f M q and M q ≃ M > q × M q have the same tangent Lie bialgebra, hence the map τ := ( τ − π + , τ − − π − ) : f M > q × f M q ։ M > q × M q ≃ M q is a Poisson covering map. By the Semenov-Tian-Shansky dressing method [STS], we getthat the symplectic leaves of f M > q × f M q are the connected components of the intersections f M q ∩ (cid:0) diag( G q × T q ) · g · diag( G q × T q ) (cid:1) , where diag (cid:0) G q × T q (cid:1) denotes the diagonal of ( G q × T q ) × and g ∈ ( G q × T q ) × . Nowwe apply [Y, Theorem 1.10] to obtain that each such intersection is a dense, open andconnected subset of diag (cid:0) G q × T q (cid:1) · g · diag (cid:0) G q × T q (cid:1) . Consider the map e j : f M > q × f M q → G q × T q , e j ( m + , m − ) := m − m − , m + ∈ f M > q , m − ∈ f M q . By a direct argument we conclude that each symplectic leaf of f M > q × f M q is of the form S O ′ := (cid:0) f M > q × f M q (cid:1) ∩ O ′ , where O ′ is a conjugacy class of G q × T q , and that dim S O ′ = dim O ′ . Since τ : f M > q × f M q ։ M q is a covering of Poisson Lie groups, each symplectic leaf of M q is of the form τ ( S O ′ ) . One easily verifies that the diagram f M > q × f M q e j / / τ (cid:15) (cid:15) G q × T q ψ (cid:15) (cid:15) M q j / / G q × (cid:0) T q /C q (cid:1) commutes, where ψ : G q × T q ։ G q × (cid:0) T q /C q (cid:1) is the canonical projection. Clearly, ψ ( O ′ ) = O × { t } , where O is a conjugacy class of G q and t ∈ T q /C q . Thereforeall symplectic leaves of M q are of the form τ ( S O ′ ) = j − ψ ( O ′ ) = j − ( O × { t } ) and dim j − ( O × { t } ) = dim O ′ = dim O .Part (b) follows from part (a), Theorem 6.2, and the theorem for isomorphisms ofcentral quotients across symplectic leaves [BG, Theorem 4.1]. (cid:3) In regard to the irreducible representations of U q we wonder whether the De Concini–Kac–Procesi conjecture could be extended to the setting of Theorem 8.2, see [DKP]. Question 8.3.
Let O be a conjugacy class of G q , t ∈ T q /C q and z ∈ j − ( O × { t } ) . Does ℓ dim O / divide the dimension of any irreducible representation of H z ? The torus orbits of symplectic leaves and the representations of the largequantum Borel algebras.
The algebras U q , U > q , U q and U ± q are Z I -graded with grad-ing deg e i = − deg f i = α i , deg K i = deg L i = 0 for i ∈ I . This leads to a canonicalaction of the torus ( C × ) I on these algebras by algebra automorphisms, which preservesthe central subalgebras Z q , Z > q , Z q and Z ± q .By a direct comparison, one obtains that the ( C × ) I -action on Z > q corresponds to theleft action of τ − ( T q ) on M > q in the sense that every automorphism from the first onecorresponds to an automorphism from the second and vice versa. Similarly, the ( C × ) I -action on Z q corresponds to the left action of τ − − ( T q ) on M q . Theorem 8.2(a) impliesthat the induced action of ( C × ) I on M q preserves the symplectic leaves of M q . So, inregard to irreps of U q , the ( C × ) I -automorphisms of U q do not provide any additionalinformation to that in Theorem 8.2(a).But for U > q and U q , we do obtain additional representation theoretic information fromthe ( C × ) I -action, as stated in next theorem. Let W q be the Weyl group of G q . Theorem 8.4.
For every choice of the specialization parameters t ij ∈ Z satisfying theNon-degeneracy Assumption 7.5 the following hold: (a) The Poisson structure on M > q is invariant under the left and right actions of τ − ( T q ) .The τ − ( T q ) -orbits of symplectic leaves of M > q are the double Bruhat cells τ − ( B + q ∩ B − q wB − q ) , w ∈ W q . (b) If τ + ( z ) and τ + ( z ′ ) are in the same double Bruhat cell, then there is an algebraisomorphism H > z ≃ H > z ′ . Proof. (a) For a Lie subalgebra of g q ⊕ h q , denote by N ( − ) its normalizer in G q × T q .By [LY, Lemma 2.12], the left and right actions of f M > ∩ N (( m > q ) ∗ ) on the Lie group f M > preserve its Poisson structure. By the definition of τ + , these actions correspond to theleft and right actions of τ − ( T q ) on M > q , so the latter preserve the Poisson structure on M > q , because f M > q ։ M > q is a Poisson map.Applying [LY, Theorem 2.7 and Proposition 2.15] and the Bruhat decomposition of G q , we obtain that the f M > ∩ N (( m > q ) ∗ ) -orbits of symplectic leaves of f M > (with respectto either action) are the intersections f M > ∩ (cid:0) ( G q × T q ) w ( G q × T q ) (cid:1) OISSON ORDERS ON LARGE QUANTUM GROUPS 37 for w ∈ W q . Since f M > q ։ M > q is a Poisson covering map and τ + : M > q ≃ −→ B + q isan isomorphism (Theorem 7.10), the τ − ( T q ) -orbits of symplectic leaves of M > q (withrespect to either action) are the double Bruhat cells τ − ( B + q ∩ B − q wB − q ) for w ∈ W q .Part (b) follows from part (a), the constructed Poisson orders in Theorem 6.2, [BG,Theorem 4.1] and the fact that the left action of τ − ( T q ) on M > q comes from the ( C × ) I -action on U > q by algebra automorphisms. (cid:3) Example 8.5.
Let q be of type wk (4) . By Example 7.11, the corresponding algebraicgroup G q is isomorphic to PSL ( C ) × PSL ( C ) whose Weyl groups is S × S . Theorem 8.4implies that among the quotients U > q / M z U > q for z in the maximal spectrum of Z > q , thereare at most | S × S | = (3!) = 36 isomorphism classes of finite dimensional algebras.Analogously to Theorem 8.4 one proves the following: Proposition 8.6.
For every choice of the specialization parameters t ij ∈ Z satisfyingthe Non-degeneracy Assumption 7.5 the following hold: (a) The Poisson structure on M q is invariant under the left and right actions of τ − − ( T q ) .The τ − − ( T q ) -orbits of symplectic leaves of M q are the double Bruhat cells τ − − ( B − q ∩ B + q wB + q ) , w ∈ W q . (b) If τ − ( z ) and τ − ( z ′ ) are in the same double Bruhat cell, then H z ≃ H z ′ as algebras. Poisson homogeneous spaces and irreps of large quantum unipotent al-gebras.
Since Z + q is the algebra of coinvariants for the coaction of Z q on Z > q obtainedby restricting the coaction of U q on Z > q , and analogously for the negative part, we haveisomorphisms of Poisson algebras Z + q ≃ C [ M > q /τ − ( T q )] , Z − q ≃ C [ M q /τ − − ( T q )] . (8.3)As shown in the previous subsection, the left and right actions of τ − ( T q ) and τ − − ( T q ) onthe Poisson algebraic groups M > q and M q preserve their Poisson structures. The righthand sides of the isomorphisms (8.3) involve the coordinate rings of the resulting Poissonhomogeneous spaces M > q /τ − ( T q ) and M q /τ − − ( T q ) obtained by taking quotients withrespect to the right actions. The Poisson structures on M > q /τ − ( T q ) and M q /τ − − ( T q ) are invariant under the induced left actions of τ − ( T q ) and τ − − ( T q ) . By Theorem 7.10, τ + restricts to the isomorphism of homogeneous spaces τ + : M > q /j − ( T q ) ≃ −→ B + q /T q .Denote the canonical isomorphism υ : B + q /T q ≃ −→ B + q B − q /B − q ⊂ G q /B − q . Theorem 8.7.
For every choice of the specialization parameters t ij ∈ Z satisfying theNon-degeneracy Assumption 7.5 the following hold: (a) The τ − ( T q ) -orbits of symplectic leaves of M > q /τ − ( T q ) are the open Richardsonvarieties τ − υ − (cid:0) ( B + q B − q ∩ B − q wB − q ) /B − q (cid:1) , w ∈ W q . (b) If υτ + ( z ) and υτ + ( z ′ ) are in the same open Richardson variety, then there is anisomorphism of algebras H + z ≃ H + z ′ . Proof.
Part (a) is proved arguing as in the proof of Theorem 8.4(a). Then (b) is aconsequence of (a), Theorem 6.2 and [BG, Theorem 4.1]. (cid:3)
An analogous result holds for the large quantum unipotent algebra U − q and the torusorbits of symplectic leaves of the Poisson homogeneous space M q /τ − − ( T q ) . Appendix A. Families of finite-dimensional Nichols algebras
Let θ ∈ N , I = I θ . We fix a matrix q = ( q ij ) ∈ C I × I such that dim B q < ∞ . To insurecentrality of Z q we require (a). The matrix q satisfies (4.24) , i.e. q N β α i β = 1 , for all i ∈ I , β ∈ Π q . Remark A.1.
If the Dynkin diagram of q ′ is as in Tables 1, 2 and 3, then there is q with the same Dynkin diagram that satisfies (4.24); the proof is straightforward.If q satisfies (4.24), then any matrix in its Weyl-equivalence class also does. Let C [ ν ± ] be the algebra of Laurent polynomials; its group of units is C [ ν ± ] × = C × ν Z . Let q = ( q ij ) ∈ (cid:0) C [ ν ± ] × (cid:1) I × I (A.1)For x ∈ C × , we denote by q ( x ) the matrix obtained by the evaluation ev : C [ ν ± ] → C , ev( ν ) = x . We seek for matrices (A.1) with the following properties (b) and (d). (b). The Nichols algebra of the C ( ν ) -braided vector space of diagonal type with braidingmatrix (A.1) has the same arithmetic root system as q . By inspection of the list in [H2]–see also the exposition in [AA]–we conclude that theonly possible matrices (A.1) are those Weyl-equivalent to the ones with Dynkin diagramsas in Tables 1, 2 and 3 and that the following property holds. (c).
There exists an open subset ∅ 6 = O ⊆ C × such that for any x ∈ O , the root systemsand Weyl groupoids associated to q and q ( x ) are isomorphic. Also there exists ξ ∈ G ′∞ ∩ O with N := ord ξ ∈ [2 , ∞ ) such that q = q ( ξ ) . Remark A.2. (i). The Dynkin diagrams of the matrices q and q locally have theform q ii ◦ e q ij q jj ◦ , respectively q ii ◦ e q ij q jj ◦ , where e q ij = q ij q ji , e q ij = q ij q ji ; i.e. theDynkin diagram does not determine completely the braiding matrix. We deal with thisas follows. Let p = ( p ij ) ∈ C I × I with the same Dynkin diagram as q . Then there exists p ∈ (cid:0) C [ ν ± ] × (cid:1) I × I with the same Dynkin diagram as q such that p = p ( ξ ) . For, take p ii = q ii and p ij ∈ C [ ν ± ] × such that p ij = p ij ( ξ ) for i < j ; then p ji = e q ij p − ij .(ii). Assume that q satisfies (b). Let p be another matrix with the same Dynkindiagram as (A.1). Then q ij = p ij q h ij N , i < j for a unique family ( h ij ) i There exist matrices C = ( c ij ) ∈ Z I × I and ( p ij ) ∈ ( C × ) I × I such that C is symmetric and: (i) There are infinitely many matrices T = ( t ij ) ∈ Z I × I fulfilling t ii = c ii , t ij + t ji = c ij for all i = j ∈ I (A.2) OISSON ORDERS ON LARGE QUANTUM GROUPS 39 such that the matrix q = ( q ij ) defined by q ij = p ij ν t ij , for all i, j ∈ I (A.3) satisfies (b). (ii) Among those T in (i) , there infinitely many such that q satisfies (d).Proof. It suffices to fix one matrix for each Weyl-equivalence class, see Lemma 6.1. Wecheck below (i) by case-by-case considerations computing also T q and proving that it isinvertible for infinitely many T .A.1. Cartan type. Let q be in this class; then there is a Cartan matrix A = ( a ij ) i,j ∈ I such that q ij q ji = q a ij ii . We fix d i ∈ I such that d i a ij = d j a ji for all i, j ∈ I . The Liealgebra g q has the same type except when N is even and A is of type B θ or C θ , whenthey are interchanged. In this case Π q = { N i α i : i ∈ I } , so (4.24) becomes: q N j ij = 1 , for all i, j ∈ I . (A.4)The matrix q we are looking for should also satisfy q ij q ji = q a ij ii for all i = j . In all caseswe take ξ = q except for B θ , where ξ = q θθ ; see Table 1. Set t ii = d i and q ii = ν t ii .Thus q ii ( ξ ) = q ii for all i ∈ I . Recall that ( ν − ξ ) ℘ q α i α j ( ν ) = 1 − q N i N j ij . For instance ℘ q α i α i ( ν ) = − ν diN i ν − ξ hence ℘ q α i α i ( ξ ) = − ξ − d i N i = − ξ − t ii N i . (A.5)Let i < j . We see that there exists d j ∈ I such that N j = N/d j . By (A.4), q ij is a power of ξ d j ; choose t ij ∈ d j Z such that q ij = ν t ij satisfies q ij ( ξ ) = ξ t ij = q ij .Set t ji = d i a ij − t ij and q ji = ν t ji . We have defined T satisfying (A.2) and q turnsout to be given by (A.3) with p ij = 1 for all i, j , i.e. (i) holds. Also for all i = j , ( ν − ξ ) ℘ q α i α j ( ν ) = 1 − ν t ij N i N j and ℘ q α i α j ( ξ ) = − ξ − t ij N i N j . (A.6)Therefore T q = T . Observe that if t ij = 0 for i < j , then det T q = 0 . By a standardargument, (ii) holds.A.2. Super type. Assume that the braiding matrix q is of super type; see [AA] fordetails (see [AA] for details and below for D (2 , α ) ). Going over the list, we see thatthere exist ◦ ξ ∈ C × , a root of 1 of order N > ; ◦ a symmetric matrix B = ( b ij ) i,j ∈ I ∈ Z I × I with b ij = 1 for at least one pair ( i, j ) ; ◦ a parity vector p = ( p , . . . , p θ ) ∈ {± } I with p i = − when b ii = 0 ; such that q ij q ji = ξ b ij , i = j ; q ii = p i ξ b ii , i ∈ I . We describe in Table 2 matrices q of super type, one for each Weyl-equivalence class(here α ( ij ) := α i + · · · + α j for i < j ). Since the matrix q has an analogous shape, wemay assume that ◦ there exists k ∈ I such that { i ∈ I : p i = − } = { k } ; ◦ there exists h ∈ I , h = k , such that ξ = q hh .Therefore we have: Table 1. Cartan typeType q NA θ ν ◦ ν − ν ◦ ν − ν ◦ ν ◦ ν − ν ◦ B θ , θ ≥ ν ◦ ν − ν ◦ ν ◦ ν − ν ◦ > C θ , θ ≥ ν ◦ ν − ν ◦ ν − ν ◦ ν ◦ ν − ν ◦ > D θ , θ ≥ ν ◦ ν ◦ ν − ν ◦ ν ◦ ν − ν ◦ ν − ν − ν ◦ B θ , θ ∈ I , ν ◦ ν ◦ ν − ν ◦ ν ◦ ν − ν − ν ◦ ν − ν ◦ F ν ◦ ν − ν ◦ ν − ν ◦ ν − ν ◦ > G ν ◦ ν − ν ◦ > ◦ either Π q = { N i α i : i ∈ I , i = k } for type A ( k − | θ − k ) or else there exists a uniquepositive non-simple root β such that Π q = { N i α i : i ∈ I , i = k } ∪ { N β β } ; ◦ for i ∈ I , i = k , we may (and do) choose b ii ∈ {± , ± , ± } . Then N i = LCD ( b ii , N ) ;set d i = N/N i .We start defining the matrix q . First we take t ii = b ii and q ii = p i q b ii for all i ∈ I .Condition (4.24) says that q N j ij = 1 , for all i ∈ I , j ∈ I \{ k } . Let i < j with j = k ;choose t ij ∈ d j Z and set t ji = b ji − t ij . Then q ij = ν t ij and q ji = ν t ji satisfy q ij ( ξ ) = ξ t ij = q ij and q ij q ji = ν b ij .Similarly, q N i ki = 1 for k > i , so choose t ik ∈ d i Z and set t ki = b ki − t ik , q ki = ν b ki − t ik and q ik = ν t ik so that q ki ( ξ ) = ξ t ki = q ki . We have defined T satisfying (A.2) and q turns out to be given by (A.3) with p ii = p i and p ij = 1 for all i = j , i.e. (i) holds.It remains to compute the matrix T q . Arguing as in the Cartan case we see that ℘ q α i α i ( ξ ) = − ξ − p i b ii N i , ℘ q α i α j ( ξ ) = − ξ − t ij N i N j , i, j ∈ I \{ k } . Assume that there exists γ ∈ Π q \ I (a non-simple Cartan root). Then there exist p γ ∈ {± } and b γγ , b iγ ∈ Z such that q γγ = p γ q b γγ , q iγ q γi = ν b iγ . Extend ( t ij ) to a bilinear form t : Z I × Z I → Z . Then for k = i ∈ I , ℘ q γγ ( ξ ) = − ξ − p γ b γγ N γ , ℘ q α i γ ( ξ ) = − ξ − t iγ N i N γ , ( ν − ξ ) ℘ q γα i ( ν ) = − ξ − t γi N i N γ . OISSON ORDERS ON LARGE QUANTUM GROUPS 41 All in all, T q is of the form ( t αβ ) ∈ Z Π q × Π q , where t αα = p α b αα , t rαβ + t βα = b αβ for α = β . Arguing as in the Cartan case, we conclude that (ii) holds. Table 2. Super typeType q N Π q g q A ( k − | θ − k ) ,k ∈ I ⌊ θ +12 ⌋ ν − ◦ ν ν − ◦ − ◦ k ν ◦ ν − ν ◦ > { N α j | j = k } A k − × A θ − k B ( k | θ − k ) ,k ∈ I θ − ν − ◦ ν ν − ◦ − ◦ k ν ◦ ν − ν ◦ = 2 , (A.7) C k × B θ − k C k × C θ − k D ( k | θ − k ) , ν − ◦ ν ν − ◦ − ◦ k ν ◦ ν − ν ◦ > (A.8) D k × C θ − k k < θ D k × B θ − k D (2 , α ) ,d , d ∈ N ν d ◦ ν − d − ◦ ν − d ν d ◦ { , , } A × A × A F (4) ν ◦ ν − ν ◦ ν − ν ◦ − ◦ ν − > (A.9) A × B G (3) − ◦ ν − ν ◦ ν − ν ◦ , N > (A.10) A × G { N j α j | j = k } ∪ { N α ( kθ ) α ( kθ ) } , (A.7) { N j α j | j = k } ∪ { N α ( k − θ ) + α ( k θ − ( α ( k − θ ) + α ( k θ − ) } , (A.8) { N α , N α , N α , N α +2 α +3 α +2 α ( α + 2 α + 3 α + 2 α ) } , (A.9) { N α +2 α + α ( α + 2 α + α ) , N α , N α } . (A.10) Type D (2 , α ) . The diagrams of this type are Weyl equivalent to the following one r ◦ r − − ◦ s − s ◦ , with r, s, rs = 1 . The corresponding Nichols algebra has finite di-mension if and only if r, s ∈ G ′∞ , rs = 1 . Let q be a braiding matrix with this diagramsatisfying (4.24). Fix a generator ξ of the subgroup of G ∞ generated by r, s ; we choose d , d ∈ N minimal such that r = ξ d , s = ξ d . Then there exists a braiding matrix q asin Table 2 such that q = q ( ξ ) .A.3. Modular type. The Nichols algebras in this family could be thought of as quan-tizations in char 0 of the 34-dimensional Lie algebras in char 2 from [KaW], respectivelythe 10-dimensional Lie algebras in char 3 introduced in [Br]. The information on thistype is given in Table 2. The matrices T and T q are worked out as in the super case. (cid:3) Appendix B. Lie bialgebras and Poisson algebraic groups We gather minimal background material on Lie bialgebras and Poisson algebraicgroups for Sections 7 and 8. We refer to [ES, Section 2-7] for a full treatment. Table 3. Modular typeType q N Π q g q wk (4) q ◦ q − q ◦ q − − ◦ − q − q − ◦ > (A.11) A × A br (2) ζ ◦ ξ − ξ ◦ , ζ ∈ G ′ = 3 { M α + M α , N α } A × A { N α , N α , M α , M α + 2 M α + 3 M α + M α } . (A.11)B.1. Lie bialgebras. Recall that a Lie bialgebra is a Lie algebra g equipped with alinear map δ : g → ∧ g such that(i) the dual of the map δ defines a Lie algebra structure of g ∗ and(ii) δ is a 1-cocycle, i.e., δ ([ a, b ]) = ad a ( δ ( b )) − ad b ( δ ( a )) for all a, b ∈ g .The Lie bialgebras with opposite cobracket (same bracket) and opposite bracket (samecobracket) will be denoted by g op and g op , respectively. The dual Lie bialgebra g ∗ of g is the Lie bialgebra with Lie bracket and cobracket given by h [ f, g ] , a i = h f ⊗ g, δ ( a ) i , h δ ( f ) , a ⊗ b i = h f, [ a, b ] i , ∀ a, b ∈ g , f, g ∈ g ∗ . The Drinfeld double D ( g ) of the Lie bialgebra g is a Lie bialgebra which is isomorphicto g ⊕ g ∗ as a vector space and is uniquely defined by the conditions:(a) The canonical embeddings ι : g ֒ → D ( g ) and ι ∗ : ( g ∗ ) op ֒ → D ( g ) are embeddings ofLie bialgebras;(b) For a ∈ g ⊂ D ( g ) , f ∈ g ∗ ⊂ D ( g ) , [ x, f ] = ad ∗ x ( f ) − ad ∗ f ( x ) in terms of the coadjointactions of g and g ∗ .A quadratic Lie algebra is a Lie algebra g equipped with an non-degenerate invariantsymmetric bilinear form ( ., . ) . A Manin triple is a triple ( g , g + , g − ) consisting of aquadratic Lie algebra ( g , ( ., . )) and a pair of isotropic Lie subalgebras g ± ⊂ g . Remark B.1. The notions of Drinfeld double and Manin triple are equivalent in thecase of finite dimensional Lie algebras:(a) Each Drinfeld double D ( g ) is a quadratic Lie algebra with symmetric bilinear form ( a + f, b + g ) = h f, b i + h g, a i , a, b ∈ g , f, g ∈ g ∗ . With respect to this form, ( D ( g ) , g , g ∗ ) is a Manin triple.(b) For a Manin triple ( g , g + , g − ) , g ± have canonical Lie bialgebra structures given by ( δ ( a ) , f ⊗ g ) = ( a, [ f, g ]) , ( δ ( f ) , a ⊗ b ) = − ( f, [ a, b ]) , ∀ a, b ∈ g + , f, g ∈ g − . Then g , equipped with the Lie cobracket δ g + + δ g − , is isomorphic to the Drinfelddouble of g + , and g − ≃ ( g ∗ + ) op .B.2. Poisson algebraic groups. A (complex) Poisson algebraic group is an algebraicgroup G equipped with a bivector field π such that the product map ( G, π ) × ( G, π ) → ( G, π ) is Poisson. The coordinate ring C [ G ] has a canonical structure of commutative Poisson-Hopf algebra with Poisson bracket given by { f, g } := h df ⊗ dg, π i , f, g ∈ C [ G ] , OISSON ORDERS ON LARGE QUANTUM GROUPS 43 where df denotes the differential of f . Conversely, every finitely generated commutativePoisson-Hopf algebra H gives rise to the Poisson algebraic group MaxSpec H .The tangent Lie algebra g = T G of every Poisson algebraic group G has a canonicalLie bialgebra structure. The Poisson structure π automatically vanishes at the identityelement of G . The Lie cobracket on g , or equivalently the Lie bracket on g ∗ ≃ T ∗ G , isdefined as the linearization of π at : [ d ( f ) , d ( g )] := d (cid:0) { f, g } (cid:1) , f, g ∈ C [ G ] . (B.1)In Hopf algebra situations it is advantageous to describe the tangent Lie algebra g ofan algebraic group G by describing the corresponding Lie cobracket on g ∗ = T ∗ G . Lemma B.2. Let G be a complex algebraic group; as usual ∆( f ) = f (1) ⊗ f (2) for f ∈ C [ G ] . Then the canonical Lie coalgebra structure on T ∗ G ≃ g ∗ is given by δ ( d f ) = d f (1) ∧ d f (2) , f ∈ C [ G ] . References [A] N. Andruskiewitsch. An Introduction to Nichols Algebras . 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FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n,Ciudad Universitaria, 5000 Córdoba, República Argentina E-mail address : (andrus|angiono)@famaf.unc.edu.ar Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,U.S.A. E-mail address ::