Quantization of Lie bialgebras, II

This paper is a continuation of [EK]. We show that the quantization procedure of [EK] is given by universal acyclic formulas and defines a functor from the category of Lie bialgebras to the category of quantized universal enveloping algebras. We also show that this functor defines an equivalence between the category of Lie bialgebras over k [[h]] and the category of quantized universal enveloping (QUE) algebras.     

Quantization of Lie Bialgebras, Part VI: Quantization of Generalized Kac–Moody Algebras

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 the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid for the Lie algebra $\mathfrak{g}(A)$ corresponding to any symmetrizable matrix A (not necessarily with integer entries), which answers another question of Drinfeld. We also prove the Drinfeld-Kohno theorem for the algebra $\mathfrak{g}(A)$ (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).     

A New Approach to Leibniz Bialgebras

A study of Leibniz bialgebras arising naturally through the double of Leibniz algebras analogue to the classical Drinfeld’s double is presented. A key ingredient of our work is the fact that the underline vector space of a Leibniz algebra becomes a Lie algebra and also a commutative associative algebra, when provided with appropriate new products. A special class of them, the coboundary Leibniz bialgebras, gives us the natural framework for studying the Yang-Baxter equation (YBE) in our context, inspired in the classical Yang-Baxter equation as well as in the associative Yang-Baxter equation. Results of the existence of coboundary Leibniz bialgebra on a symmetric Leibniz algebra under certain conditions are obtained. Some interesting examples of coboundary Leibniz bialgebras are also included. The final part of the paper is dedicated to coboundary Leibniz bialgebra structures on quadratic Leibniz algebras.     

Continuous Hecke algebras

The theory of PBW properties of quadratic algebras, to which this paper aims to be a modest contribution, originates from the pioneering work of Drinfeld (see [Dr1]). In particular, as we learned after publication of [EG] (to the embarrassment of two of us!), symplectic reflection algebras, as well as PBW theorems for them, were discovered by Drinfeld in the classical paper [Dr2] 15 years before [EG] (namely, they are a special case of degenerate affine Hecke algebras for a finite group G introduced in [Dr2, Section 4]).     

Compatible Lie Brackets and the Yang-Baxter Equation

We show that any pair of compatible Lie brackets with a common invariant form produces a nonconstant solution of the classical Yang-Baxter equation. We describe the corresponding Poisson brackets, Manin triples, and Lie bialgebras. It turns out that all bialgebras associated with the solutions found by Belavin and Drinfeld are isomorphic to some bialgebras generated by our solutions. For any compatible pair, we construct a double with a common invariant form and find the corresponding solution of the quantum Yang-Baxter equation for this double. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 195–207, February, 2006.     

Quantization of Γ-Lie bialgebras

We introduce the notion of Γ-Lie bialgebras, where Γ is a group. These objects give rise to cocommutative co-Poisson bialgebras, for which we construct quantization functors. This enlarges the class of co-Poisson algebras for which a quantization is known. Our result relies on our earlier work, where we showed that twists of Lie bialgebras can be quantized; we complement this work by studying the behavior of this quantization under compositions of twists.     

On Quantization Functors of Lie Bialgebras

We show that the set of quantization functors of Lie bialgrebras has the structure of a torsor. Then we show that the Etingof–Kazhdan map is a morphism of torsors. We compute the infinitesimal of this map. As a corollary, we show that the quantization functors of finite-dimensional Lie bialgebras are independent of the choice of an associator.     

Krein duality,positive 2-algebras,and dilation of comultiplications

The Krein-Tannaka duality for compact groups was a generalization of the Pontryagin-van Kampen duality for locally compact Abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found applications in algebraic combinatorics (“Krein algebras”). Later, this duality was substantially extended: in [29], the notion of involutive algebras in positive vector duality was introduced. In this paper, we reformulate the notions of this theory using the language of bialgebras (and Hopf algebras) and introduce the class of involutive bialgebras and positive 2-algebras. The main goal of the paper is to give a precise statement of a new problem, which we consider as one of the main problems in this field, concerning the existence of dilations (embeddings) of positive 2-algebras in involutive bialgebras, or, in other words, the problem of describing subobjects of involutive bialgebras; we define two types of subobjects of bialgebras, strict and nonstrict ones. The dilation problem is illustrated by the example of the Hecke algebra, which is viewed as a positive involutive 2-algebra. We consider in detail only the simplest situation and classify two-dimensional Hecke algebras for various values of the parameter q, demonstrating the difference between the two types of dilations. We also prove that the class of finite-dimensional involutive semisimple bialgebras coincides with the class of semigroup algebras of finite inverse semigroups.     

Quantization of Lie bialgebras revisited

We describe a new method of quantization of Lie bialgebras, based on a construction of Hopf algebras out of a cocommutative coalgebra and a braided comonoidal functor.     

Lie bialgebras of generalized Witt type

In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.     

Gröbner–Shirshov bases for some Lie algebras

Siberian Mathematical Journal - We give Gröbner–Shirshov bases for the Drinfeld–Kohno Lie algebra L n in [1] and the Kukin Lie algebra A P in [2], where P is a semigroup. By way of...     

Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof–Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are “compatible with duality”, i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof–Kazhdan quantizations of some affine Lie superalgebras coincide with their Drinfeld–Jimbo-type quantizations. To the memory of Paulette Libermann (1919–2007)     

Lie comodules and the constructions of Lie bialgebras

In this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.     

Quantization of lie Bialgebras via the Formality of the operad of Little Disks

We give a proof of the Etingof–Kazhdan theorem on quantization of Lie bialgebras based on the formality of the chain operad of little disks and show that the Grothendieck–Teichmüller group acts non-trivially on the corresponding quantization functors. Partially supported by an NSF Grant and A. Sloan Research Fellowship Received: November 2004 Revision: April 2006 Accepted: May 2006     

Lie Bialgebras of Generalized Virasoro-like Type

In this paper, Lie bialgebra structures on generalized Virasoro-like algebras are studied. It is proved that all such Lie bialgebras are triangular coboundary.     

Quasidiagonal Solutions of the Yang–Baxter Equation, Quantum Groups and Quantum Super Groups

This paper answers a few questions about algebraic aspects of bialgebras, associated with the family of solutions of the quantum Yang–Baxter equation in Acta Appl. Math. 41 (1995), pp. 57–98. We describe the relations of the bialgebras associated with these solutions and the standard deformations of GL_n and of the supergroup GL(m|n). We also show how the existence of zero divisors in some of these algebras are related to the combinatorics of their related matrix, providing a necessary and sufficient condition for the bialgebras to be a domain. We consider their Poincaré series, and we provide a Hopf algebra structure to quotients of these bialgebras in an explicit way. We discuss the problems involved with the lift of the Hopf algebra structure, working only by localization.     

A Koszul duality for props

The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.

     

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.     

Lie Bialgebras of Generalized WITT Type,II

In an article by Michaelis, a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H¹(𝒲, 𝒲 ? 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.     

Simplicity of quadratic Lie conformal algebras

In this paper, simplicity of quadratic Lie conformal algebras is investigated. From the view point of the corresponding Gel’fand–Dorfman bialgebras, some su?cient conditions and necessary conditions to ensure simplicity of quadratic Lie conformal algebras are presented. By these observations, we present several new classes of infinite simple Lie conformal algebras. These results will be useful for classification purposes.