Birman–Murakami-Wenzl algebra

introduction

Birman–Murakami-Wenzl algebra, a deformation of the Brauer algebra.
has the Hecke algebra of type A as a quotien
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.

Ariki, Susumu. 2006. “Algebras Arising from Schur-Weyl Type Dualities.” In Proceedings of the 38th Symposium on Ring Theory and Representation Theory, 1–10. Symp. Ring Theory Represent. Theory Organ. Comm., Yamanashi. http://www.ams.org/mathscinet-getitem?mr=2264119.
Benkart, Georgia. 1996. “Commuting Actions—a Tale of Two Groups.” In Lie Algebras and Their Representations (Seoul, 1995), 194:1–46. Contemp. Math. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=1395593.
Schüler, Axel. 1993. “The Brauer Algebra and the Birman-Wenzl-Murakami Algebra.” Seminar Sophus Lie 3 (1): 3–11. http://www.heldermann-verlag.de/jlt/jlt03/SCHUELAT.PDF

Morton, H. R. “A Basis for the Birman-Wenzl Algebra.” arXiv:1012.3116 [math], December 14, 2010. http://arxiv.org/abs/1012.3116.