Constructing cell data for diagram algebras

We show how the treatment of cellularity in families of algebras arising from diagram calculi, such as Jones’ Temperley-Lieb wreaths, variants on Brauer’s centralizer algebras, and the contour algebras of Cox et al. (of which many algebras are special cases), may be unified using the theory of tabular algebras. This improves an earlier result of the first author (whose hypotheses covered only the Brauer algebra from among these families).     

Cohomological stratification of diagram algebras

The class of cellularly stratified algebras is defined and shown to include large classes of diagram algebras. While the definition is in combinatorial terms, by adding extra structure to Graham and Lehrer’s definition of cellular algebras, various structural properties are established in terms of exact functors and stratifications of derived categories. The stratifications relate ‘large’ algebras such as Brauer algebras to ‘smaller’ ones such as group algebras of symmetric groups. Among the applications are relative equivalences of categories extending those found by Hemmer and Nakano and by Hartmann and Paget, as well as identities between decomposition numbers and cohomology groups of ‘large’ and ‘small’ algebras.     

The diagram for endomorphism algebras of two-term tilting complexes over self-injective algebras

We construct a diagram between the endomorphism algebra of a two-term projective module complex over a self-injective algebra and the endomorphism algebras of its homology groups and apply the diagram to the tilting theory of self-injective algebras. We give an example of recovering the endomorphism algebra of a two-term tilting complex over a self-injective algebra from the endomorphism algebras of its homology groups with some additional structures which appear in the diagram.     

On the isomorphism problem for finite dimensional diagram algebras

somorphism problems for finite dimensional algebras can be computationally hard. When the algebras are monomial, it is shown, refining work of Shirayanagi, that there is a simple definitive combinatorial method. However, examples show that no such criterion is possible if the class of algebras is expanded to that of diagram algebras (in the sense of Fuller). The presentation of a diagram algebra is field independent but the existence of an isomorphism between two such is not. (Subject classes: 16G30, 16P10, 20M25).     

Small models for chain algebras

Supported by the Deutsche Forschungsgemeinschaft under a Heisenberg grant     

Recollements of self-injective algebras,and classification of self-injective diagram algebras

Mathematische Zeitschrift - Diagram algebras, in particular Brauer algebras, Birman–Murakami-Wenzl algebras and partition algebras, are used in representation theory and invariant theory of...     

The Shilov boundary and the Gelfand spectrum of algebras of generalized analytic functions

Let S be a discrete abelian cancellation semigroup with identity. We consider an algebra of generalized analytic functions defined on the semigroup $ \hat S $ of semicharacters of S that is wider than the Arens-Singer algebra. We show that the strong boundary and the Shilov boundary of this algebra are unions of maximal subgroups of $ \hat S $ . If S contains no nontrivial simple ideals, then both boundaries coincide with the character group of S. In this case we calculate the Gelfand spectrum of the algebra under consideration.     

Geometric models for the spectra of certain Gelfand pairs associated with Heisenberg groups

Let K be a compact Lie group acting on a finite dimensional Hermitian vector space V via some unitary representation. Now K acts by automorphisms on the associated Heisenberg group ${H_V=V \times \mathbb{R}}$ , and we say that (K, H _V ) is a Gelfand pair when the algebra ${L^1_K(H_V)}$ of integrable K-invariant functions on H _V commutes under convolution. In this situation an application of the Orbit Method yields a injective mapping ${\Psi}$ from the space Δ(K, H _V ) of bounded K-spherical functions on H _V to the space ${\mathfrak{h}_V^{*}/K}$ of K-orbits in the dual of the Lie algebra for H _V . We prove that ${\Psi}$ is a homeomorphism onto its image provided that the action of K on V is “well-behaved” in a sense made precise in this work. Our result encompasses a widely studied class of examples arising in connection with Hermitian symmetric spaces.     

Oriented tree diagram Lie algebras and their abelian ideals

We introduce oriented tree diagram Lie algebras which are generalized from Xu＇s both upward and downward tree diagram Lie algebras, and study certain numerical invariants of these algebras related to abelian ideals.     

Reduced models of Albert algebras

Summary. We prove existence and uniqueness of reduced models for arbitrary Albert algebras and relate them to the Tits process. This relationship yields explicit noncohomological realizations of the invariants mod 2 due to Serre and Rost. We also construct nontrivial examples of Albert division algebras with nonvanishing invariants mod 2. Received 29 August 1994 / in final form 22 February 1995     

Reduced models of Albert algebras

Summary We prove existence and uniqueness of reduced models for arbitrary Albert algebras and relate them to the Tits process. This relationship yields explicit noncohomological realizations of the invariants mod 2 due to Serre and Rost. We also construct nontrivial examples of Albert division algebras with nonvanishing invariants mod 2.     

Inductions and restrictions on towers of cellularly stratified diagram algebras

Hartmann et al. defined the concept of cellularly stratified algebras that combine the features of both cellular algebras and stratified algebras. Many important diagram algebras in mathematics and physics, such as some Brauer, partition and BMW algebras, are cellularly stratified algebras, and each of these forms a tower of algebras. This article gives the concept of towers of cellularly stratified algebras in an axiomatic manner, and studies it in terms of induction and restriction functors. In particular, for certain towers of cellularly stratified algebras, we provide a criterion for semi-simplicity by using the cohomology groups of cell modules.     

A Gelfand model for wreath products

A Gelafand model for wreath products ℤ _r ≀ S _n is constructed. The proof relies on a combinatorial interpretation of the characters of the model, extending a classical result of Frobenius and Schur.