Birman–Wenzl algebra

For each natural number n, the BMW algebra $\backslash mathrm\{C\}\_n(\backslash ell,m)$ is generated by $G\_1^\{\backslash pm1\},G\_2^\{\backslash pm1\},\backslash dots,\; G\_\{n-1\}^\{\backslash pm1\},E\_1,E\_2,\backslash dots,E\_\{n-1\}$ and relations:

:$G\_iG\_j\; =\; G\_jG\_i,\; \backslash mathrm\{if\}\; \backslash left\backslash vert\; i-j\; \backslash right\backslash vert\; \backslash geq\; 2,$

:$G\_i\; G\_\{i+1\}\; G\_i\; =\; G\_\{i+1\}\; G\_i\; G\_\{i+1\},$         $E\_i\; E\_\{i\backslash pm1\}\; E\_i\; =\; E\_i,$

:$G\_i\; +\; \{G\_i\}^\{-1\}\; =\; m(1+E\_i),$

:$G\_\{i\backslash pm1\}\; G\_i\; E\_\{i\backslash pm1\}\; =\; E\_i\; G\_\{i\backslash pm1\}\; G\_i\; =\; E\_i\; E\_\{i\backslash pm1\},$      $G\_\{i\backslash pm1\}\; E\_i\; G\_\{i\backslash pm1\}\; =\{G\_i\}^\{-1\}\; E\_\{i\backslash pm1\}\; \{G\_i\}^\{-1\},$

:$G\_\{i\backslash pm1\}\; E\_i\; E\_\{i\backslash pm1\}\; =\; \{G\_i\}^\{-1\}\; E\_\{i\backslash pm1\},$      $E\_\{i\backslash pm1\}\; E\_i\; G\_\{i\backslash pm1\}\; =E\_\{i\backslash pm1\}\; \{G\_i\}^\{-1\},$

:$G\_i\; E\_i=\; E\_i\; G\_i\; =\; l^\{-1\}\; E\_i,$     $E\_i\; G\_\{i\backslash pm1\}\; E\_i\; =\; l\; E\_i.$

These relations imply the further relations:

:$E\_i\; E\_j=E\_j\; E\_i,\; \backslash mathrm\{if\}\; \backslash left\backslash vert\; i-j\; \backslash right\backslash vert\; \backslash geq\; 2,$

:$(E\_i)^2\; =\; (m^\{-1\}(l+l^\{-1\})-1)\; E\_i,$

:$\{G\_i\}^2\; =\; m(G\_i+l^\{-1\}E\_i)-1.$

This is the original definition given by Birman 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)
2. :$G\_i\; -\; \{G\_i\}^\{-1\}=m(1-E\_i),$

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

2. (Idempotent relation)
3. :$(E\_i)^2\; =\; (m^\{-1\}(l-l^\{-1\})+1)\; E\_i,$
4. (Braid relations)
5. :$G\_iG\_j=G\_jG\_i,\; \backslash text\{if\; \}\; \backslash left\backslash vert\; i-j\; \backslash right\backslash vert\; \backslash geqslant\; 2,\; \backslash text\{\; and\; \}\; G\_i\; G\_\{i+1\}\; G\_i=G\_\{i+1\}\; G\_i\; G\_\{i+1\},$
6. (Tangle relations)
7. :$E\_i\; E\_\{i\backslash pm1\}\; E\_i=E\_i\; \backslash text\{\; and\; \}\; G\_i\; G\_\{i\backslash pm1\}\; E\_i\; =\; E\_\{i\backslash pm1\}\; E\_i,$
8. (Delooping relations)
9. :$G\_i\; E\_i=\; E\_i\; G\_i\; =\; l^\{-1\}\; E\_i\; \backslash text\{\; and\; \}\; E\_i\; G\_\{i\backslash pm1\}\; E\_i\; =l\; E\_i.$

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

It is proved by {{harvtxt|Morton|Wassermann|1989}} that the BMW algebra $\backslash mathrm\{C\}\_n(\backslash ell,m)$ is isomorphic to the Kauffman's tangle algebra $\backslash mathrm\{KT\}\_n$. The isomorphism $\backslash phi\; \backslash colon\; \backslash mathrm\{C\}\_n\; \backslash to\; \backslash mathrm\{KT\}\_n$ is defined by

Image:KauffmannTangleAlg 2.PNG and Image:KauffmannTangleAlg 3.PNG

Define the face operator as

:$U\_i(u)\; =\; 1-\; \backslash frac\{i\backslash sin\; u\}\{\backslash sin\; \backslash lambda\; \backslash sin\; \backslash mu\}(e^\{i(u-\backslash lambda)\}\; G\_i\; -e^\{-i(u-\backslash lambda)\}\{G\_i\}^\{-1\})$,

where $\backslash lambda$ and $\backslash mu$ are determined by

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

and

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

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 $E\_i=U\_i(\backslash lambda)$ with

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

In the limits $u\; \backslash to\; \backslash pm\; i\; \backslash infty$, the braids $\{G\_j\}^\{\backslash pm\}$ 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. {{harvtxt|Murakami|1987}} showed that the Kauffman polynomial can also be interpreted as a function $F$ on a certain associative algebra. In 1989, {{harvtxt|Birman|Wenzl|1989}} constructed a two-parameter family of algebras $\backslash mathrm\{C\}\_n(\backslash ell,m)$ with the Kauffman polynomial $K\_n(\backslash ell,m)$ as trace after appropriate renormalization.

• {{Citation | doi=10.1090/S0002-9947-1989-0992598-X | last1=Birman | first1=Joan S. | author1-link=Joan Birman | last2=Wenzl | first2=Hans | title=Braids, link polynomials and a new algebra | jstor= 2001074 | mr=992598 | year=1989 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=313 | issue=1 | pages=249–273 | publisher=American Mathematical Society| doi-access=free }}
• {{Citation | last1=Murakami | first1=Jun | title=The Kauffman polynomial of links and representation theory | url=http://projecteuclid.org/euclid.ojm/1200780357 | mr=927059 | year=1987 | journal=Osaka Journal of Mathematics | issn=0030-6126 | volume=24 | issue=4 | pages=745–758}}
• {{cite arXiv | last1=Morton | first1=Hugh R. | last2=Wassermann | first2=Antony J.|authorlink2=Antony Wassermann| title=A basis for the Birman–Wenzl algebra | eprint=1012.3116 | year=1989| class=math.QA }}

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