Cyclotomic Birman--Wenzl--Murakami algebras, II: Admissibility Relations and Freeness

TL;DR

This paper investigates the structure and representation theory of cyclotomic BMW algebras, establishing conditions for their freeness, isomorphism to tangle algebras, and criteria for semisimplicity, advancing understanding of their algebraic properties.

Contribution

It introduces admissibility conditions ensuring freeness and isomorphism of cyclotomic BMW algebras, and characterizes their semisimplicity and trace weights.

Findings

Algebras are free modules over admissible ground rings.

Isomorphism to cyclotomic Kauffman tangle algebras is established.

Provides criteria for semisimplicity and recursive formulas for trace weights.

Abstract

The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We study admissibility conditions on the ground ring for these algebras, and show that the algebras defined over an admissible integral ground ring $S$ are free $S$--modules and isomorphic to cyclotomic Kauffman tangle algebras. We also determine the representation theory in the generic semisimple case, obtain a recursive formula for the weights of the Markov trace, and give a sufficient condition for semisimplicity.