Birman–Wenzl algebra

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 ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ of dimension ${\displaystyle 1\cdot 3\cdot 5\cdots (2n-1)}$ 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 ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ is generated by ${\displaystyle G_{1}^{\pm 1},G_{2}^{\pm 1},\dots ,G_{n-1}^{\pm 1},E_{1},E_{2},\dots ,E_{n-1}}$ and relations:

${\displaystyle G_{i}G_{j}=G_{j}G_{i},\mathrm {if} \left\vert i-j\right\vert \geq 2,}$

${\displaystyle G_{i}G_{i+1}G_{i}=G_{i+1}G_{i}G_{i+1},}$         ${\displaystyle E_{i}E_{i\pm 1}E_{i}=E_{i},}$

${\displaystyle G_{i}+{G_{i}}^{-1}=m(1+E_{i}),}$

${\displaystyle G_{i\pm 1}G_{i}E_{i\pm 1}=E_{i}G_{i\pm 1}G_{i}=E_{i}E_{i\pm 1},}$      ${\displaystyle G_{i\pm 1}E_{i}G_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1}{G_{i}}^{-1},}$

${\displaystyle G_{i\pm 1}E_{i}E_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1},}$      ${\displaystyle E_{i\pm 1}E_{i}G_{i\pm 1}=E_{i\pm 1}{G_{i}}^{-1},}$

${\displaystyle G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i},}$     ${\displaystyle E_{i}G_{i\pm 1}E_{i}=lE_{i}.}$

These relations imply the further relations:

${\displaystyle E_{i}E_{j}=E_{j}E_{i},\mathrm {if} \left\vert i-j\right\vert \geq 2,}$

${\displaystyle (E_{i})^{2}=(m^{-1}(l+l^{-1})-1)E_{i},}$

${\displaystyle {G_{i}}^{2}=m(G_{i}+l^{-1}E_{i})-1.}$

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

1. (Kauffman skein relation)

   ${\displaystyle G_{i}-{G_{i}}^{-1}=m(1-E_{i}),}$

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

2. (Idempotent relation)

   ${\displaystyle (E_{i})^{2}=(m^{-1}(l-l^{-1})+1)E_{i},}$

3. (Braid relations)

   ${\displaystyle G_{i}G_{j}=G_{j}G_{i},{\text{if }}\left\vert i-j\right\vert \geqslant 2,{\text{ and }}G_{i}G_{i+1}G_{i}=G_{i+1}G_{i}G_{i+1},}$

4. (Tangle relations)

   ${\displaystyle E_{i}E_{i\pm 1}E_{i}=E_{i}{\text{ and }}G_{i}G_{i\pm 1}E_{i}=E_{i\pm 1}E_{i},}$

5. (Delooping relations)

   ${\displaystyle G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i}{\text{ and }}E_{i}G_{i\pm 1}E_{i}=lE_{i}.}$

Properties

• The dimension of ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ is ${\displaystyle (2n)!/(2^{n}n!)}$.
• The Iwahori–Hecke algebra associated with the symmetric group ${\displaystyle S_{n}}$ is a quotient of the Birman–Murakami–Wenzl algebra ${\displaystyle \mathrm {C} _{n}}$.
• The Artin braid group embeds in the BMW algebra: ${\displaystyle B_{n}\hookrightarrow \mathrm {C} _{n}}$.

Isomorphism between the BMW algebras and Kauffman's tangle algebras

It is proved by Morton & Wassermann (1989) that the BMW algebra ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ is isomorphic to the Kauffman's tangle algebra ${\displaystyle \mathrm {KT} _{n}}$. The isomorphism ${\displaystyle \phi \colon \mathrm {C} _{n}\to \mathrm {KT} _{n}}$ is defined by
and

Baxterisation of Birman–Murakami–Wenzl algebra

Define the face operator as

${\displaystyle U_{i}(u)=1-{\frac {i\sin u}{\sin \lambda \sin \mu }}(e^{i(u-\lambda )}G_{i}-e^{-i(u-\lambda )}{G_{i}}^{-1})}$,

where ${\displaystyle \lambda }$ and ${\displaystyle \mu }$ are determined by

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

and

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

Then the face operator satisfies the Yang–Baxter equation.

${\displaystyle U_{i+1}(v)U_{i}(u+v)U_{i+1}(u)=U_{i}(u)U_{i+1}(u+v)U_{i}(v)}$

Now ${\displaystyle E_{i}=U_{i}(\lambda )}$ with

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

In the limits ${\displaystyle u\to \pm i\infty }$, the braids ${\displaystyle {G_{j}}^{\pm }}$ 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. Murakami (1987) showed that the Kauffman polynomial can also be interpreted as a function ${\displaystyle F}$ on a certain associative algebra. In 1989, Birman & Wenzl (1989) constructed a two-parameter family of algebras ${\displaystyle \mathrm {C} _{n}(\ell ,m)}$ with the Kauffman polynomial ${\displaystyle K_{n}(\ell ,m)}$ as trace after appropriate renormalization.

References

• Birman, Joan S.; Wenzl, Hans (1989), "Braids, link polynomials and a new algebra", Transactions of the American Mathematical Society, 313 (1), American Mathematical Society: 249–273, doi:10.1090/S0002-9947-1989-0992598-X, ISSN 0002-9947, JSTOR 2001074, MR 0992598
• Murakami, Jun (1987), "The Kauffman polynomial of links and representation theory", Osaka Journal of Mathematics, 24 (4): 745–758, ISSN 0030-6126, MR 0927059
• Morton, Hugh R.; Wassermann, Antony J. (1989). "A basis for the Birman–Wenzl algebra". arXiv:1012.3116 [math.QA].