Cyclotomic Nazarov-Wenzl algebras

TL;DR

This paper introduces and analyzes cyclotomic quotients of Nazarov-Wenzl algebras, constructing their irreducible representations, proving their freeness and cellularity, and classifying their simple modules over arbitrary fields.

Contribution

It provides the first construction of irreducible representations and a cellular structure for cyclotomic Nazarov-Wenzl algebras, extending understanding of their module theory.

Findings

Algebras are free of rank r^n(2n-1)!! when dmissible.

Algebras are shown to be cellular.

Classification of simple modules over arbitrary fields.

Abstract

Nazarov \cite{Nazarov:brauer} introduced an infinite dimensional algebra, which he called the \textit{affine Wenzl algebra}, in his study of the Brauer algebras. In this paper we study certain ``cyclotomic quotients'' of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank $r^n(2n-1)!!$ (when $\Omega$ is $\bu$--admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov--Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Weyl algebra (when $\Omega$ is admissible).