2-parameter family of algebras with the Hecke algebra of the symmetric group as a quotient
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
C
n
(
ℓ
,
m
)
{\displaystyle \mathrm {C} _{n}(\ell ,m)}
of dimension
1
⋅
3
⋅
5
⋯
(
2
n
−
1
)
{\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
C
n
(
ℓ
,
m
)
{\displaystyle \mathrm {C} _{n}(\ell ,m)}
is generated by
G
1
±
1
,
G
2
±
1
,
…
,
G
n
−
1
±
1
,
E
1
,
E
2
,
…
,
E
n
−
1
{\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:
G
i
G
j
=
G
j
G
i
,
i
f
|
i
−
j
|
⩾
2
,
{\displaystyle G_{i}G_{j}=G_{j}G_{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
,
{\displaystyle G_{i}G_{i+1}G_{i}=G_{i+1}G_{i}G_{i+1},}
E
i
E
i
±
1
E
i
=
E
i
,
{\displaystyle E_{i}E_{i\pm 1}E_{i}=E_{i},}
G
i
+
G
i
−
1
=
m
(
1
+
E
i
)
,
{\displaystyle 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
,
{\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},}
G
i
±
1
E
i
G
i
±
1
=
G
i
−
1
E
i
±
1
G
i
−
1
,
{\displaystyle G_{i\pm 1}E_{i}G_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1}{G_{i}}^{-1},}
G
i
±
1
E
i
E
i
±
1
=
G
i
−
1
E
i
±
1
,
{\displaystyle G_{i\pm 1}E_{i}E_{i\pm 1}={G_{i}}^{-1}E_{i\pm 1},}
E
i
±
1
E
i
G
i
±
1
=
E
i
±
1
G
i
−
1
,
{\displaystyle E_{i\pm 1}E_{i}G_{i\pm 1}=E_{i\pm 1}{G_{i}}^{-1},}
G
i
E
i
=
E
i
G
i
=
l
−
1
E
i
,
{\displaystyle G_{i}E_{i}=E_{i}G_{i}=l^{-1}E_{i},}
E
i
G
i
±
1
E
i
=
l
E
i
.
{\displaystyle E_{i}G_{i\pm 1}E_{i}=lE_{i}.}
These relations imply the further relations:
E
i
E
j
=
E
j
E
i
,
i
f
|
i
−
j
|
⩾
2
,
{\displaystyle 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
,
{\displaystyle (E_{i})^{2}=(m^{-1}(l+l^{-1})-1)E_{i},}
G
i
2
=
m
(
G
i
+
l
−
1
E
i
)
−
1.
{\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
(Kauffman skein relation)
G
i
−
G
i
−
1
=
m
(
1
−
E
i
)
,
{\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
(Idempotent relation)
(
E
i
)
2
=
(
m
−
1
(
l
−
l
−
1
)
+
1
)
E
i
,
{\displaystyle (E_{i})^{2}=(m^{-1}(l-l^{-1})+1)E_{i},}
(Braid relations)
G
i
G
j
=
G
j
G
i
,
if
|
i
−
j
|
⩾
2
,
and
G
i
G
i
+
1
G
i
=
G
i
+
1
G
i
G
i
+
1
,
{\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},}
(Tangle relations)
E
i
E
i
±
1
E
i
=
E
i
and
G
i
G
i
±
1
E
i
=
E
i
±
1
E
i
,
{\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},}
(Delooping relations)
G
i
E
i
=
E
i
G
i
=
l
−
1
E
i
and
E
i
G
i
±
1
E
i
=
l
E
i
.
{\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
C
n
(
ℓ
,
m
)
{\displaystyle \mathrm {C} _{n}(\ell ,m)}
is
(
2
n
)
!
/
(
2
n
n
!
)
{\displaystyle (2n)!/(2^{n}n!)}
.
The Iwahori–Hecke algebra associated with the symmetric group
S
n
{\displaystyle S_{n}}
is a quotient of the Birman–Murakami–Wenzl algebra
C
n
{\displaystyle \mathrm {C} _{n}}
.
The Artin braid group embeds in the BMW algebra,
B
n
↪
C
n
{\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
C
n
(
ℓ
,
m
)
{\displaystyle \mathrm {C} _{n}(\ell ,m)}
is isomorphic to the Kauffman's tangle algebra
K
T
n
{\displaystyle \mathrm {KT} _{n}}
, the isomorphism
ϕ
:
C
n
→
K
T
n
{\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
U
i
(
u
)
=
1
−
i
sin
u
sin
λ
sin
μ
(
e
i
(
u
−
λ
)
G
i
−
e
−
i
(
u
−
λ
)
G
i
−
1
)
{\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
λ
{\displaystyle \lambda }
and
μ
{\displaystyle \mu }
are determined by
2
cos
λ
=
1
+
(
l
−
l
−
1
)
/
m
{\displaystyle 2\cos \lambda =1+(l-l^{-1})/m}
and
2
cos
λ
=
1
+
(
l
−
l
−
1
)
/
(
λ
sin
μ
)
{\displaystyle 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
)
{\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
E
i
=
U
i
(
λ
)
{\displaystyle E_{i}=U_{i}(\lambda )}
with
ρ
(
u
)
=
sin
(
λ
−
u
)
sin
(
μ
+
u
)
sin
λ
sin
μ
{\displaystyle \rho (u)={\frac {\sin(\lambda -u)\sin(\mu +u)}{\sin \lambda \sin \mu }}}
.
In the limits
u
→
±
i
∞
{\displaystyle u\to \pm i\infty }
, the braids
G
j
±
{\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
F
{\displaystyle F}
on a certain associative algebra. In 1989, Birman & Wenzl (1989) constructed a two-parameter family of algebras
C
n
(
ℓ
,
m
)
{\displaystyle \mathrm {C} _{n}(\ell ,m)}
with the Kauffman polynomial
K
n
(
ℓ
,
m
)
{\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 , American Mathematical Society, 313 (1): 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 ].