Hopf代数上的扭曲模和(斜)余配对Hopf模

本文第一部分主要把扭曲的方法运用到模上,从而得到扭曲模．作为特例,我们构造了H  M的Smaush模和量子模．当K是有限维Hopf代数,证明K*  M是一个右D(K)-Hopf模,因此得到了一个基本同构定理．第二部分主要把斜余配对双代数进行推广,得到了斜余配对Hopf模,并且给出判断斜余配对Hopf模的一个充要条件．     

On <Emphasis Type="Italic">osp</Emphasis>(<Emphasis Type="Italic">M</Emphasis>∣2<Emphasis Type="Italic">n</Emphasis>) Integrable Open Spin Chains

We consider open spin chains based on osp(M2n) Yangians and solve the reflection equations for some classes of reflection matrices, including the diagonal ones. Having then integrable open spin chains, we write the analytical Bethe Ansatz equations. More details and references can be found in D. Arnaudon et al.: Nucl. Phys B 668 (2003) 469 and 687 (2004) 257.     

Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

The mod 2 Steenrod algebra and Dyer-Lashof algebra have both striking similarities and differences arising from their common origins in ``lower-indexed' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra , whose module actions are equivalent to, but quite different from, those of and . The exact relationships emerge as ``sheared algebra bijections', which also illuminate the role of the cohomology of . As a bialgebra, has a particularly attractive and potentially useful structure, providing a bridge between those of and , and suggesting possible applications to the Miller spectral sequence and the structure of Dickson algebras.

     

Compatible associative bialgebras

We introduce a non-symmetric operad 𝒩, whose dimension in degree n is given by the Catalan number c_n?1. It arises naturally in the study of coalgebra structures defined on compatible associative algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a 𝒩-algebra. The data (As,As²,𝒩) is a good triple of operads, in J.-L. Loday’s sense. Our construction induces another triple of operads (As,As₂,As), where As₂ is the operad of matching dialgebras. Motivated by A. Goncharov’s Hopf algebra of paths P(S), we introduce the notion of bi-matching dialgebras and show that the Hopf algebra P(S) is a bi-matching dialgebras.     

The deformed twisted Heisenberg–Virasoro type Lie bialgebra

Abstract

In this article, we investigate Lie bialgebra structures on the deformed twisted Heisenberg–Virasoro Lie algebra. Sufficient and necessary conditions for this type Lie bialgebra structures to be triangular coboundary are given.

Communicated by K. C. Misra     

Quadratic Lie Algebras

In this article, the notion of universal enveloping algebra introduced in Ardizzoni [4 Ardizzoni , A. On primitively generated braided bialgebras . Algebr. Represent. Theory , to appear. (doi:10.1007/s10468-010-9257-z)  [Google Scholar]] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting, a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nichols algebra, which is a quadratic algebra.     

Aspects of free analysis

We survey the analysis around the free difference quotient derivation, which is the natural derivation for variables with the highest degree of noncommutativity. The analogue of the Fourier transform is then bialgebra duality for the bialgebra with derivation-comultiplication to which the free difference quotient gives rise and which involves fully matricial analytic functions. Some of the motivation from free probability, especially free entropy and random matrices are also discussed. Dan-Virgil Voiculescu; Research supported in part by NSF Grant DMS 0501178.     

A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

     

Orbits of Lagrangian Subalgebras of the Double sl(2, R)

We describe the variety of Lagrangian subalgebras of the Drinfeld double for an arbitrary bialgebra structure on sl(2, R). We determine the irreducible components and the orbit structure under the natural action of the group SL(2, R)