Birman–Wenzl algebra

In mathematics, the Birman-Murakami-Wenzl (BMW) algebra, introduced by Birman & Wenzl (1989) and Murakami (1986), is a two-parameter family of algebras Cn(ℓ, m) of dimension 1·3·5 ··· (2n − 1) 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.

Definition

For each natural number n, the BMW algebra Cn(ℓ, m) is generated by G1,G2,...,Gn-1,E1,E2,...,En-1 and relations:

G_iG_j=G_jG_i, \mathrm{if} \left\vert i-j \right\vert \geqslant 2,
G_i G_{i+1} G_i=G_{i+1} G_i G_{i+1},         E_i E_{i\pm1} E_i=E_i,
G_i + {G_i}^{-1}=m(1+E_i),
G_{i\pm1} G_i E_{i\pm1} = E_i G_{i\pm1} G_i = E_i E_{i\pm1},      G_{i\pm1} E_i G_{i\pm1} ={G_i}^{-1} E_{i\pm1} {G_i}^{-1},
G_{i\pm1} E_i E_{i\pm1}={G_i}^{-1} E_{i\pm1},      E_{i\pm1} E_i G_{i\pm1} =E_{i\pm1} {G_i}^{-1},
G_i E_i= E_i G_i = l^{-1} E_i,      E_i G_{i\pm1} E_i =l E_i.

These relations imply the further relations:

E_i E_j=E_j E_i, \mathrm{if} \left\vert i-j \right\vert \geqslant 2,
(E_i)^2 = (m^{-1}(l+l^{-1})-1) E_i, \,\!
{G_i}^2 = m(G_i+l^{-1}E_i)-1.

This is the original definition given by Birman & 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
(1) (Kauffman skein relation)

G_i - {G_i}^{-1}=m(1-E_i),
Given invertibility of m, the rest of the relations in Birman & Wenzl's original version can be reduced to

(2) (Idempotent relation)

(E_i)^2 = (m^{-1}(l-l^{-1})+1) E_i, \,\!

(3) (Braid relations)

G_iG_j=G_jG_i, \text{if } \left\vert i-j \right\vert \geqslant 2, \text{ and } G_i G_{i+1} G_i=G_{i+1} G_i G_{i+1}, \,\!

(4) (Tangle relations)

E_i E_{i\pm1} E_i=E_i \text{ and } G_i G_{i\pm1} E_i = E_{i\pm1} E_i,

(5) (Delooping relations)

G_i E_i= E_i G_i = l^{-1} E_i \text{ and } E_i G_{i\pm1} E_i =l E_i.

Properties

Isomorphism between the BMW algebras and Kauffman's tangle algebras

It is proved by Morton & Wassermann (1989) that the BMW algebra Cn(ℓ, m) is isomorphic to the Kauffman's tangle algebra KTn, the isomorphism \phi : C_n \to KT_n is defined by
and

Baxterisation of Birman-Murakami-Wenzl algebra

Define the face operator as

U_i(u)=1- \frac{i\sin u}{\sin \lambda \sin \mu}(e^{i(u-\lambda)} G_i -e^{-i(u-\lambda)}{G_i}^{-1})

where \lambda and \mu are determined by

2\cos \lambda=1+(l-l^{-1})/m

and

2\cos \lambda = 1+(l-l^{-1})/(\lambda \sin \mu).

Then the face operator satisfies the Yang-Baxter equation.

U_{i+1}(v) U_i(u+v) U_{i+1}(u) = U_i(u) U_{i+1}(u+v) U_i(v)\,\!

Now E_i=U_i(\lambda) with

\rho(u)=\frac{\sin (\lambda-u) \sin (\mu+u)}{\sin \lambda \sin \mu} .

In the limits u \to \pm i \infty , the braids {G_j}^{\pm} can be recovered up to a scale factor.

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 Cn(ℓ, m) with the Kauffman polynomial Kn(ℓ, m) as trace after appropriate renormalization.

References