en>Jarekadam: /* Asymptotic behavior */ reference update

2011-10-03T12:24:18Z

Asymptotic behavior: reference update

New page

In mathematics, the Birman-Murakami-Wenzl (BMW) algebra, introduced by {{harvtxt|Birman|Wenzl|1989}} and {{harvtxt|Murakami|1986}}, is a two-parameter family of [[algebra (mathematics)|algebra]]s C''n''(''ℓ'', ''m'') of dimension 1·3·5 ··· (2''n'' − 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 C''n''(''ℓ'', ''m'') is generated by G''1'',G''2'',...,G''n-1'',E''1'',E''2'',...,E''n-1'' and relations:

:<math> G_iG_j=G_jG_i, \mathrm{if} \left\vert i-j \right\vert \geqslant 2,</math>

:<math>G_i G_{i+1} G_i=G_{i+1} G_i G_{i+1},</math>         <math> E_i E_{i\pm1} E_i=E_i, </math>

:<math> G_i + {G_i}^{-1}=m(1+E_i), </math>

:<math> G_{i\pm1} G_i E_{i\pm1} = E_i G_{i\pm1} G_i = E_i E_{i\pm1},</math>      <math> G_{i\pm1} E_i G_{i\pm1} ={G_i}^{-1} E_{i\pm1} {G_i}^{-1}, </math>

:<math> G_{i\pm1} E_i E_{i\pm1}={G_i}^{-1} E_{i\pm1}, </math>      <math> E_{i\pm1} E_i G_{i\pm1} =E_{i\pm1} {G_i}^{-1}, </math>

:<math> G_i E_i= E_i G_i = l^{-1} E_i,</math>     <math> E_i G_{i\pm1} E_i =l E_i. </math>

These relations imply the further relations: 
:<math> E_i E_j=E_j E_i, \mathrm{if} \left\vert i-j \right\vert \geqslant 2,</math> 
:<math> (E_i)^2 = (m^{-1}(l+l^{-1})-1) E_i, \,\! </math> 
:<math> {G_i}^2 = m(G_i+l^{-1}E_i)-1. </math>

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)
::<math> G_i - {G_i}^{-1}=m(1-E_i), </math>

:Given invertibility of ''m'', the rest of the relations in Birman & Wenzl's original version can be reduced to 
(2) (Idempotent relation)
::<math> (E_i)^2 = (m^{-1}(l-l^{-1})+1) E_i, \,\! </math> 
(3) (Braid relations)
::<math> 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}, \,\!</math> 
(4) (Tangle relations)
::<math> E_i E_{i\pm1} E_i=E_i \text{ and } G_i G_{i\pm1} E_i = E_{i\pm1} E_i,</math> 
(5) (Delooping relations)
::<math> G_i E_i= E_i G_i = l^{-1} E_i \text{ and } E_i G_{i\pm1} E_i =l E_i. </math>

==Properties==
*The dimension of C''n''(''ℓ'', ''m'') is <math> (2n)!/(2^nn! )</math>.
*[[Iwahori-Hecke algebra]] associated with the [[symmetric group]] <math>\mathfrak{S}_n,</math> is a quotient of the Birman-Murakami-Wenzl algebra C''n''.
*The [[Braid group]] embeds in the BMW algebra <math> {\mathcal{B}}_n \hookrightarrow {\mathcal{C}}_n </math>.

==Isomorphism between the BMW algebras and Kauffman's tangle algebras==
It is proved by {{harvtxt|Morton|Wassermann|1989}} that the BMW algebra C''n''(''ℓ'', ''m'') is isomorphic to the Kauffman's tangle algebra KT''n'', the [[isomorphism]] <math>\phi : C_n \to KT_n </math> is defined by 
[[Image:KauffmannTangleAlg 2.PNG]] and [[Image:KauffmannTangleAlg 3.PNG]]

==Baxterisation of Birman-Murakami-Wenzl algebra==
Define the face operator as
:<math> 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})</math>
where <math>\lambda</math> and <math>\mu</math> are determined by
:<math> 2\cos \lambda=1+(l-l^{-1})/m</math>
and
:<math> 2\cos \lambda = 1+(l-l^{-1})/(\lambda \sin \mu)</math>.

Then the face operator satisfies the [[Yang-Baxter equation]].
:<math> U_{i+1}(v) U_i(u+v) U_{i+1}(u) = U_i(u) U_{i+1}(u+v) U_i(v)\,\!</math>
Now <math> E_i=U_i(\lambda) </math> with
:<math> \rho(u)=\frac{\sin (\lambda-u) \sin (\mu+u)}{\sin \lambda \sin \mu} </math>.
In the [[limit (mathematics)|limits]] <math> u \to \pm i \infty </math>, the [[braid theory|braid]]s <math> {G_j}^{\pm} </math> 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 group]]s. In 1986, {{harvtxt|Murakami|1986}} showed that the [[Kauffman polynomial]] can also be interpreted as a function <math>F</math> on a certain associative algebra. In 1989, {{harvtxt|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.

==References==
*{{Citation | doi=10.1090/S0002-9947-1989-0992598-X | last1=Birman | first1=Joan S. | 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}}
*{{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=A.J.| title=A basis for the Birman-Wenzl algebra | eprint=1012.3116 | year=1989}}

{{DEFAULTSORT:Birman-Wenzl algebra}}
[[Category:Representation theory]]
[[Category:Knot theory]]

Weingarten function - Revision history

en>Jarekadam: /* Asymptotic behavior */ reference update