Partition Algebras are Cellular

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.     

Cyclotomic Extensions of Diagram Algebras

A uniform approach to cyclotomic extensions of diagram algebras is given, focussing on cellular structures. Cyclotomic Temperley–Lieb algebras, cyclotomic Brauer algebras and cyclotomic walled Brauer algebras are discussed as examples.     

Automorphism groups of pointed Hopf algebras

The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.      

On Some Brauer Subgroups Arising From Twists of Matrix Algebras

We study some relations between groups of continuous characters and subgroups of Brauer groups. Explicit determination of some subgroups of Brauer groups arising from this connection are done for some cases in the local fields via local class field theory.     

The Brauer-Clifford group for locally finite (S,H)-Azumaya algebras

In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.     

Schur algebras of Brauer algebras I

Schur algebras of Brauer algebras are defined as endomorphism algebras of certain direct sums of ‘permutation modules’ over Brauer algebras. Explicit combinatorial bases of these new Schur algebras are given; in particular, these Schur algebras are defined integrally. The new Schur algebras are related to the Brauer algebra by Schur–Weyl dualities on the above sums of permutation modules. Moreover, they are shown to be quasi-hereditary. Over fields of characteristic different from two or three, the new Schur algebras are quasi-hereditary 1-covers of Brauer algebras, and hence the unique ‘canonical’ Schur algebras of Brauer algebras.     

Homogeneous Deformations of Brauer Tree Algebras

Abstract

Based on tilting theory, we demonstrate the existence of homogeneous deformations for the Brauer tree algebras, which are derived naturally from a tilting complex.     

On the algebra structure of some bismash products

We study several families of semisimple Hopf algebras, arising as bismash products, which are constructed from finite groups with a certain specified factorization. First we associate a bismash product H_q of dimension q(q−1)(q+1) to each of the finite groups PGL₂(q) and show that these H_q do not have the structure (as algebras) of group algebras (except when q=2,3). As a corollary, all Hopf algebras constructed from them by a comultiplication twist also have this property and are thus non-trivial. We also show that bismash products constructed from Frobenius groups do have the structure (as algebras) of group algebras.     

Jordan Algebras Over Algebraic Varieties

General results on the module structure of Jordan algebras over locally ringed spaces are obtained. Albert algebras over a Brauer–Severi variety with associated central simple algebra of degree 3 are constructed using generalizations of the Tits process and the first Tits construction.     

Finite dimensional Nichols algebras over certain p-groups

We investigate pointed Hopf algebras over finite nilpotent groups of odd order, with nilpotency class 2. For such a group G, we show that if its commutator subgroup coincides with its center, then there exists no non-trivial finite-dimensional pointed Hopf algebra with kG as its coradical. We apply these results to non-abelian groups of order p³, p⁴ and p⁵, and list all the pointed Hopf algebras of order p⁶, whose group of grouplikes is non-abelian.     

Group algebras of Kleinian type and groups of units

The algebras of Kleinian type are finite-dimensional semisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such that the group of units of RG is virtually a direct product of free-by-free groups.     

一类Heisenberg n-李代数的自同构群(英文)

本文主要研究Heisenberg n-李代数的结构.给出了一类(3m+1)-维Heisenberg3-李代数及(nm+1)-维Heisenberg n-李代数的自同构群.且给出了自同构的具体表达式.     

Conjugation in Brauer algebras and applications to character theory

The Brauer algebra has a basis of diagrams and these generate a monoid H consisting of scalar multiples of diagrams. Following a recent paper by Kudryavtseva and Mazorchuk, we define and completely determine three types of conjugation in H. We are thus able to define Brauer characters for Brauer algebras which share many of the properties of Brauer characters defined for finite groups over a field of prime characteristic. Furthermore, we reformulate and extend the theory of characters for Brauer algebras as introduced by Ram to the case when the Brauer algebra is not semisimple.     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

Non-denseness of factorable matrix functions

It is proved that for certain algebras of continuous functions on compact abelian groups, the set of factorable matrix functions with entries in the algebra is not dense in the group of invertible matrix functions with entries in the algebra, assuming that the dual abelian group contains a subgroup isomorphic to Z³. These algebras include the algebra of all continuous functions and the Wiener algebra. More precisely, it is shown that infinitely many connected components of the group of invertible matrix functions do not contain any factorable matrix functions, again under the same assumption. Moreover, these components actually are disjoint with the subgroup generated by the triangularizable matrix functions.     

Quasitilted algebras admit a preprojective component

Quasitilted algebras are generalizations of tilted algebras. As a main result we show here that the Auslander-Reiten quiver of such an algebra has a preprojective component

     

Brauer groups for commutative S-algebras

We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a $K\left(n\right)$-local variant of the notion of Brauer groups.     

A new characterization of semisimple Lie algebras

Using Casimir elements, we characterize the semisimple Lie algebras among the quadratic Lie algebras. This characterization gives, in particular, a generalization of a consequence of Cartan's second criterion.

     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.