Birman–Wenzl algebra

Definition

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

G i G j = G j G i , i f | i − j | ⩾ 2 , G i G i + 1 G i = G i + 1 G i G i + 1 ,         E i E i ± 1 E i = E i , G i + G i − 1 = m ( 1 + E i ) , G i ± 1 G i E i ± 1 = E i G i ± 1 G i = E i E i ± 1 ,      G i ± 1 E i G i ± 1 = G i − 1 E i ± 1 G i − 1 , G i ± 1 E i E i ± 1 = G i − 1 E i ± 1 ,      E i ± 1 E i G i ± 1 = E i ± 1 G i − 1 , G i E i = E i G i = l − 1 E i ,      E i G i ± 1 E i = l E i .

These relations imply the further relations:

E i E j = E j E i , i f | i − j | ⩾ 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)

Given invertibility of m, the rest of the relations in Birman & Wenzl's original version can be reduced to

(2) (Idempotent relation)

(3) (Braid relations)

(4) (Tangle relations)

(5) (Delooping relations)

Properties

The dimension of C_n(ℓ, m) is ( 2 n ) ! / ( 2 n n ! ) .

Iwahori-Hecke algebra associated with the symmetric group S n , is a quotient of the Birman-Murakami-Wenzl algebra C_n.

The Braid group embeds in the BMW algebra B n ↪ C n .

Isomorphism between the BMW algebras and Kauffman's tangle algebras

It is proved by Morton & Wassermann (1989) that the BMW algebra C_n(ℓ, m) is isomorphic to the Kauffman's tangle algebra KT_n, the isomorphism ϕ : C n → K T n is defined by
and

Baxterisation of Birman-Murakami-Wenzl algebra

Define the face operator as

U i ( u ) = 1 − i sin ⁡ u sin ⁡ λ sin ⁡ μ ( e i ( u − λ ) G i − e − i ( u − λ ) G i − 1 )

where λ and μ are determined by

2 cos ⁡ λ = 1 + ( l − l − 1 ) / m

and

2 cos ⁡ λ = 1 + ( l − l − 1 ) / ( λ sin ⁡ μ ) .

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 ( λ ) with

ρ ( u ) = sin ⁡ ( λ − u ) sin ⁡ ( μ + u ) sin ⁡ λ sin ⁡ μ .

In the limits u → ± i ∞ , the braids G j ± 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 C_n(ℓ, m) with the Kauffman polynomial K_n(ℓ, m) as trace after appropriate renormalization.