The Role of Solvable Groups in Quantization of Lie Algebras

The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras H with spectrum Q(H) in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups P (H). This provides utilities for a new algorithm of constructing quantum algebras especially useful for nonsemisimple ones. The quantization procedure can be carried out over an arbitrary field. The properties of the algorithm are demonstrated on examples. 1 Supported by Russian Foundation for Fundamental Research, Grant N 94-01-01157-a. 2 E-mail address: LYAKHOVSKY @ NIIF.SPB.SU Algebraic approach to quantization of Lie groups [1, 2] implies the following sequence of constructions G =⇒ Fun(G) =⇒ U(A) =⇒ Uq(A) =⇒ (Uq(A)) ∗ =⇒ Gq (1) Here U(A) is the universal enveloping algebra for Lie algebra A of group G, (Uq(A)) ∗ – the Hopf algebra dual to Uq(A). The quantum group Gq is understood as the spectrum of the Hopf algebra (Uq(A)) , i.e. it corresponds to (Uq(A)) ∗ just as the group G to the algebra of functions Fun(G). In the sequence (1) the main procedure is the construction of the quantum algebra U(A) =⇒ Uq(A). It was stressed in [3] that the main object in the quantization procedure is the quantum algebra of functions Funq(G) ≈ (Uq(A)) , moreover this interpretation is also valid for Uq(A) (but with respect to the dual group). At the same time in general situation the spectrum of a noncommutative Hopf algebra does not exist and the pair Funq(G) =⇒ Gq must be understood as unique entity. The problem of equivalence between the categories Uq(A) and (Uq(A)) ∗ was first mentioned in [1], where the enveloping algebra Uq(A) was incorporated in the family of Hopf algebras having the commutative classical limit. The quantum duality principle [4] allows to elucidate the nature of this equivalence. If one consideres the Lie bialgebra (A,A) as the starting object [1] then the quantization is the simultaneous deformation of Fun(G) and Fun(G), where G , G are the Lie groups with algebras A and A. So Funq(G ) (as an algebra dual to Funq(G) ≈ (Uq(A)) ∗ ) is equivalent to the Hopf algebra Uq(A) thus being not only a quantum algebra but also a quantum group. In this paper the scheme is proposed for the explicit realization of Uq(A) as a quantum group. It is shown that the wide class of quantum universal enveloping algebras are factor algebras of such noncommutative Hopf algebras H that the spectrum Q(H) does exist. The ’classical limit’ of Q(H) is a simply connected solvable group P (H) – the factor group of G. The established correspondence between Uq(A) and solvable Lie groups P (H) provides new possibilities to construct quantum Lie algebras. In sect.1 the properties of the selected class of Hopf algebras Uq(A) are formulated. For each Uq(A) the simply connected solvable Lie group P (H) strictly corresponds. In sect.2 the inverse problem is solved – for a given group P the Hopf algebra H is obtained that can be factorized to the quantum universal enveloping algebra Uq(A). In sect.3 the exposed scheme is demonstrated on some known constructions and an example of application of the new quantization method is given. For the elements of tensor products of Hopf algebras the abbreviated notation will be used: a ≡ a⊗ 1; a ≡ 1 ⊗ a. 1 1. Consider the quantum universal enveloping algebra Uq(A) of an algebra A over an arbitrary field K with generators {xl}, l = 1, . . . , n and a set of quantization parameters q. Set the following conditions u.1) for generators {xl} the tensor multipliers ∆ ′ j and ∆ ′′ j in the coproduct ∆(xl) = Σj∆ ′ j(xl)∆ ′′ j (xl) are either linear functions of generators, or convergent power series in {hi}. The elements of {hi}, i = 1, . . . , m < n , commute. u.2) in the coproduct ∆(hi) = Σj∆ ′ j(hi)∆ ′′ j (hi) ∆ and ∆ transfer hi to the subalgebra U (h) q generated by {hi, 1}. u.3) ε(xk) = 0. u.4) the relations (·)(S ⊗ id)∆ = (·)(id ⊗ S)∆ = η ◦ ε, applied to generators {xl}, can be solved (when subalgebra U (h) q is commutative) to define all the S(xl), whatever the other multiplications in Uq(A) are. u.5) limq→0 ∆(xl) = x ′ l + x ′′ l . It is known [5] that for each pair (X,R) of Hopf algebra X and commutative unital algebra R the coproduct ∆X , counit εX and antipode SX induce the group structure on the set of algebraic morphisms Hom(X,R) with the multiplication (χ1 ∗ χ2)(x) = (·)R(χ1 ⊗ χ2)∆χ(x); x ∈ X; χ1, χ2 ∈ Hom(X,R). (2) When R is noncommutative the previous statement fails, the map χ◦SX having no more the property of inverse element. The fact is that SX being an antihomomorphism forms the composition χ ◦ SX that does not belong to Hom(X,R) . We shall demonstrate that for algebras Uq(A) (with the properties u.1 u.5) this obstacle can be overcome. Let us consider Uq(A) together with such an associative algebra H (with the same set of generators) that h.1) the operators ∆H , SH , εH and ηH on the generators coinside with the corresponding defining compositions in Uq(A), h.2) the subalgebra H, generated by {hi, 1}, is equivalent to U (h) q , h.3) the algebra H is free modulo the relations of commutativity of H , h.4) the operators ∆ and ε are extended to H homomorphically and the antipode S – antihomomorphically. 2 These properties guarantee that H is a Hopf algebra. Let V and V (h) be the subspaces of the vector space of H – the corresponding lineals of {xl} and {hi} . Consider a free associative algebra L and the space of moprphisms Mor(V, L). In Mor(V, L) define the subset Mor such that its elements send the space (V ) to the fixed commutative subalgebra in L. The set Mor is obviously a vector space. Each ζ ∈ Mor is fixed by n coordinates ζ(xl). Let ζ↑H be the homomorphic extension of ζ to H . Such extensions always exist but do not constitute the vector space anymore. The multiplication on Mor will be introduced similarly to (2): ζ1 ∗ ζ2 = (·)L(ζ1↑H ⊗ ζ2↑H)∆. (3) For each ζ ∈ Mor the inverse will be given by ζ ≡ ζ↑H ◦ S. (4) Note that according to the definition of ζ the antipode S in ζ acts only on the linear combinations of the generators. From u.2 and u.4 it follows that ζ ∈ Mor for any ζ ∈ Mor. The map ζ(0) ≡ ηL ◦ εH (5) is the zero vector in the space Mor (see the property u.3). Let G be the Lie group with the algebra A. Denote by Q(H) the space Mor with the multiplication (3), the inversion (4) and the marked element (5). Proposition 1. Q(H) is a group. The 1-dimensional representation d of L transforms the group Q(H) into the vector solvable Lie group P (H) on the n-dimensional vector space. Groups P (H) and G are equivalent if and only if dimG = n. Proof. Consider the product ζ ∗ ζ = (·)L(ζ↑H ⊗ ζ↑H ◦ S↑H)∆ = = (·)L(ζ↑H ⊗ ζ↑H)(id⊗ S↑H)∆ = ζ↑H(·)H(id⊗ S↑H)∆. Note that S↑H , used here according to the definition (3), is not the antipode of H . This operator coinsides with SH on generators and is homomorphically extended to H . Nevertheless the properties u.1 and u.2 guarantee that the operator S↑H in the multiplication of Q(H) acts either on V , or on power series in U . In these situations S↑H coinsides with S and the last equality can be continued: ζ ∗ ζ = ζ↑H(·)H(id⊗ S)∆ = ζ↑HηHεH = ηLεH = ζ(0) (6)