In mathematics, the Birman–Murakami–Wenzl (BMW) algebra, introduced by Joan Birman and Hans Wenzl (1989) and Jun Murakami (1987), is a two-parameter family of algebras
having the Hecke algebra of the symmetric group as a quotient.
It is related to the Kauffman polynomial of a link.
It is a deformation of the Brauer algebra in much the same way that Hecke algebras are deformations of the group algebra of the symmetric group.
For each natural number n, the BMW algebra
and relations: These relations imply the further relations: This is the original definition given by Birman and Wenzl.
However a slight change by the introduction of some minus signs is sometimes made, in accordance with Kauffman's 'Dubrovnik' version of his link invariant.
In that way, the fourth relation in Birman & Wenzl's original version is changed to Given invertibility of m, the rest of the relations in Birman & Wenzl's original version can be reduced to It is proved by Morton & Wassermann (1989) that the BMW algebra
is isomorphic to the Kauffman's tangle algebra
are determined by and Then the face operator satisfies the Yang–Baxter equation.
can be recovered up to a scale factor.
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.
Murakami (1987) showed that the Kauffman polynomial can also be interpreted as a function
on a certain associative algebra.
In 1989, Birman & Wenzl (1989) constructed a two-parameter family of algebras