Any quantization of a quasitriangular Lie bialgebra g gives rise to a braiding of the dual Poisson-Lie formal group G^*. We show that this braiding always coincides with the Weinstein-Xu braiding. We also define the lifts of the classical r-matrix r as certain functions on G^* x G^*, prove their existence and uniqueness using co-Hochschild cohomology arguments and show that the lift can be expressed in terms of r by universal formulas.
Let G be a connected, simply connected Poisson-Lie group with quasitriangular Lie bialgebra g. An explicit description of the double D(g) is given, together with the embeddings of g and g^*. This description is then used to provide a construction of the double D(G). The aim of this work is to describe D(G) in sufficient detail to be able to apply the procedures of Semenov-Tian-Shansky and Drinfeld for the classification of symplectic leaves and Poisson homogeneous spaces for Poisson-Lie groups.
We introduce and study the triple of a quasitriangular Lie bialgebra as a natural extension of the Drinfeld double. The triple is itself a quasitriangular Lie bialgebra. We prove several results about the algebraic structure of the triple, analogous to known results for the double. Among them, we prove that in the factorisable case the triple is isomorphic to a twisting of $g \oplus g \oplus g$ by a certain cocycle. We also consider real forms of the triple and the triangular case.
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized as a ``cocycle perturbation'' of the linear Poisson bracket. We construct a certain twisted group C*-algebra A, which is shown to be a strict deformation quantization of G. Motivated by the data at the Poisson (classical) level, we then construct on A its locally compact quantum group structures: comultiplication, counit, antipode and Haar weight, as well as …
We introduce a new 2-parameter family of sigma models exhibiting Poisson-Lie T-duality on a quasitriangular Poisson-Lie group $G$. The models contain previously known models as well as a new 1-parameter line of models having the novel feature that the Lagrangian takes the simple form $L=E(u^{-1}u_+,u^{-1}u_-)$ where the generalised metric $E$ is constant (not dependent on the field $u$ as in previous models). We characterise these models in terms of a global conserved $G$-invariance. The models on $G=SU_2$ and its dual $G^\star$ are computed explicitly. The general theory of Poisson-Lie T-dual…
We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_Φassociated to an associator Φusing cohomological techniques. We then introduce a criterion ensuring that the ``left Hopf algebra'' of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This provides a construction of quantization functors, equivalent to the Etingof-Kazhdan construction.
Let g be a quasitriangular Lie bialgebra over a field k of characteristic zero, and let g^* be its dual Lie bialgebra. We prove that the formal Poisson group F[[g^*]] is a braided Hopf algebra. More generally, we prove that if (U_h,R) is any quasitriangular QUEA, then (U_h', Ad(R)|_{U_h' \otimes U_h'}) --- where U_h' is defined by Drinfeld --- is a braided QFSHA. The first result is then just a consequence of the existence of a quasitriangular quantization (U_h,R) of U(g) and of the fact that U_h' is a quantization of F[[g^*]]. ----- Soit g une bigèbre de Lie quasitriangulaire sur un corps k d…
We solve a functional version of the problem of twist quantization of a coboundary Lie bialgebra (g,r,Z). We derive from this the following results: (a) the formal Poisson manifolds g^* and G^* are isomorphic; (b) we construct a subalgebra of U(g^*), isomorphic to S(g^*)^g. When g can be quantized, we construct a deformation of the morphism S(g^*)^g subset U(g^*). When g is quasitriangular and nondegenerate, we compare our construction with Semenov-Tian-Shansky's construction of a commutative subalgebra of U(g^*). We also show that the canonical derivation of the function ring of G^* is Hamilt…
This paper is a continuation of the series of papers "Quantization of Lie bialgebras (QLB) I-V". We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with th…
For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra L, let T(L) be the vector space of tensors over L equipped with the Ito Hopf algebra structure derived from the associative multiplication in L. We show a necessary and sufficient condition that the double product integral satisfy the quantum Yang-Baxter equation over T(L). We construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on L in the unital associative subalgebra of T(L)[[h]] consisting of formal power series whose zero order coefficient lies in the space S(L) of s…
Let $ \mathfrak{g} $ be a quasitriangular Lie bialgebra over a field $ K $ of characteristic zero, and let $ \mathfrak{g}^* $ be its dual Lie bialgebra. We prove that the formal Poisson group $ K\big[\big[\mathfrak{g}^*\big]\big] $ is a braided Hopf algebra, thus generalizing a result due to Reshetikhin (in the case $ \, \mathfrak{g} = \mathfrak{sl}(2,K) \, $). The proof is via quantum groups, using the existence of a quasitriangular quantization of $ \mathfrak{g}^* $, as well as the fact that this one provides also a quantization of $ K\big[\big[\mathfrak{g}^*\big]\big] \, $.
Let $\g$ be a finite dimensional complex Lie algebra and $ł\subset \g$ a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group $\Exp ł$ with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra $\Dł$. Let $\Nc_ł(0)\subset ł$ be an open domain parameterizing a neighborhood of the identity in $\Exp ł$ by the exponential map. We present dynamical $r$-matrices with values in $\g\wedge \g$ over the Poisson Lie base manifold $\Nc_ł(0)$.
We introduce the categories of infinitesimal Hopf modules and bimodules over an infinitesimal bialgebra. We show that they correspond to modules and bimodules over the infinitesimal version of the double. We show that there is a natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary infinitesimal bialgebra and a dendriform algebra from a quasitriangular infinitesimal bialgebra. As consequences, we obtain a pre-Lie structure on the space of paths on an arbitrary quiver, and a striking dendriform structure on the space of endomorphisms of an arbitrary infinitesimal bialgeb…
Certain quantization problems are equivalent to the construction of morphisms from "quantum" to "classical" props. Once such a morphism is constructed, Hensel's lemma shows that it is in fact an isomorphism. This gives a new, simple proof that any Etingof-Kazhdan quantization functor is an equivalence of categories between quantized universal enveloping (QUE) algebras and Lie bialgebras over a formal series ring (dequantization). We apply the same argument to construct dequantizations of formal solutions of the quantum Yang-Baxter equation and of quasitriangular QUE algebras. We also give stru…
To any field K of characteristic 0, we associate a set Sha(K). Elements of Sha(K) are equivalence classes of families of Lie polynomials subject to associativity relations. We construct an injection and a retraction between Sha(K) and the set of quantization functors of Lie bialgebras over K. This construction involves the following steps. 1) To each element \varpi of Sha(K), we associate a functor g\mapsto Sh(g) from the category of Lie algebras to that of Hopf algebras; Sh(g) contains Ug. 2) When g and h are Lie algebras, and r_{gh} \in g\otimes h, we construct an element R(r_{gh}) of Sh(g)\…
In the paper "On some unsolved problems in quantum group theory", V.Drinfeld formulated the problem of the existence of a universal quantization for Lie bialgebras. When the paper "Tensor structures arising from affine Lie algebras, III", by Kazhdan and Lusztig, appeared, Drinfeld asked whether its methods could be useful for the problem of universal quantization of Lie bialgebras. In this paper we use these methods to construct the universal quantization, which gives a positive answer to Drinfeld's question. We also show the existence of universal quantization of classical r-matrices, unitary…
Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra of g. In particular we investigate such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of g and V. We classify these structure in the important special case, when g is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce more a general notion, co-Poisson module algebras and their…
