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

history

• 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.

related items

• Schur-Weyl duality for general linear groups

expositions

• 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

articles

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

encyclopedia

• http://en.wikipedia.org/wiki/Brauer_algebra
• http://en.wikipedia.org/wiki/Birman–Wenzl_algebra