Birman–Murakami-Wenzl algebra

introduction

Birman–Murakami-Wenzl algebra, a deformation of the Brauer algebra.
has the Hecke algebra of type A as a quotient
its specializations play a role in types B,C,D akin to that of the symmetric group in Schur-Weyl duality

In 1984, Vaughan Jones introduced a new polynomial invariant of link isotopy types which is called the Jones polynomial.
The invariants are related to the traces of irreducible representations of Hecke algebras associated with the symmetric groups.
In 1986, Murakami (1986) showed that the Kauffman polynomial can also be interpreted as a function F on a certain associative algebra.
In 1989, Birman & Wenzl (1989) constructed a two-parameter family of algebras \(C_n(\ell, m)\) with the Kauffman polynomial \(K_n(\ell, m)\) as trace after appropriate renormalization.

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Rui, Hebing, and Linliang Song. “Mixed Schur-Weyl Duality between General Linear Lie Algebras and Cyclotomic Walled Brauer Algebras.” arXiv:1509.05855 [math], September 19, 2015. http://arxiv.org/abs/1509.05855.
Rui, Hebing, and Linliang Song. ‘Decomposition Matrices of Birman-Murakami-Wenzl Algebras’. arXiv:1411.3067 [math], 11 November 2014. http://arxiv.org/abs/1411.3067.
Li, Ge. “A KLR Grading of the Brauer Algebras.” arXiv:1409.1195 [math], September 3, 2014. http://arxiv.org/abs/1409.1195.
Morton, H. R. “A Basis for the Birman-Wenzl Algebra.” arXiv:1012.3116 [math], December 14, 2010. http://arxiv.org/abs/1012.3116.