A cohomological construction of quantization functors of Lie bialgebras

We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_Φ associated to an associator Φ using 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 gives a construction of quantization functors, equivalent to the Etingof-Kazhdan construction.     

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.     

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.     

Quantizations of the W-Algebra W（2, 2）

We quantize the W-algebra W(2, 2), whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.     

Lie Bialgebra Structures on the q-Analog Virasoro-Like Algebras

We prove that all the bialgebra structures on a q-analog Virasoro-like algebra are triangular coboundary.     

The Long Dimodules Category and Nonlinear Equations

Let H be a bialgebra and _H L^H be the category of Long H-dimodules defined, for a commutative and co-commutative H, by F. W. Long and studied in connection with the Brauer group of a so-called H-dimodule algebra. For a commutative and co-commutative H, _H L^H =_H YD^H (the category of Yetter–Drinfel'd modules), but for an arbitrary H, the categories _H L^H and _H YD^H are basically different. Keeping in mind that the category _H YD^H is deeply involved in solving the quantum Yang–Baxter equation, we study the category _H L^H of H-dimodules in connection with what we have called the D-equation: R¹² R²³ = R²³ R¹², where R End_k(M M) for a vector space M over a field k. The main result is a FRT-type theorem: if M is finite-dimensional, then any solution R of the D-equation has the form R = R_{(M, , )}, where (M, , ) is a Long D(R)-dimodule over a bialgebra D(R) and R_{(M, , )} is the special map R_{(M, , )}(m n) : = n₁ m n₀. In the last section, if C is a coalgebra and I is a coideal of C, we introduce the notion of D-map on C, that is a k-bilinear map : C C / I k satisfying a condition which ensures on the one hand that, for any right C-comodule, the special map R is a solution of the D-equation and, on the other, that, in the finite case, any solution of the D-equation has this form.     

General twisting of algebras

We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A,μ,u) in a monoidal category, as a morphism satisfying a list of axioms ensuring that (A,μ○T,u) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevich's braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms.     

对偶双代数

在文献[1]中,作者引入了扭曲积（twistingproduct）概念,并指出量子偶（Drinfel’ddouble）D（H）为张量积代数的扭曲积．本文把扭曲积加以推广为扭曲模,并给出它的基本结构定理．同时,我们引进对偶Hopf模,它是[2]中对偶Hopf模的发展．     

Lie Bialgebra Structures on Lie Algebras of Generalized Weyl Type

Lie bialgebra structures on Lie algebras of generalized Weyl type are studied. They are shown to be triangular coboundary.