The Birman-Murakami-Algebras Algebras of Type Dn

TL;DR

This paper proves the semisimplicity and cellularity of the BMW algebra of type Dn, establishes its rank, and explores its relationship with Brauer and Temperley-Lieb algebras, advancing algebraic understanding of type Dn structures.

Contribution

It demonstrates the semisimplicity, freeness, and cellularity of the BMW algebra of type Dn, and links it to Brauer and Temperley-Lieb algebras, providing new structural insights.

Findings

BMW algebra of type Dn is semisimple and free of specified rank.

Brauer algebra of type Dn is semisimple and related to BMW algebra.

Temperley-Lieb algebra of type Dn is a subalgebra of BMW algebra.

Abstract

The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image af an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the polynomial ring Z with delta and its inverse adjoined. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type.