In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Birman–Murakami–Wenzl (BMW) algebra, introduced by and , is a two-parameter family of
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
of dimension
having the
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators, which are named after Erich Hecke.
Properties
The algebra is a commutative ring.
In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ' ...
of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
as a
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
. It is related to the
Kauffman polynomial of a
link. It is a deformation of the
Brauer algebra
In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the g ...
in much the same way that Hecke algebras are deformations of the
group algebra of the symmetric group.
Definition
For each
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
''n'', the BMW algebra
is generated by
and relations:
:
:
:
:
:
:
These relations imply the further relations:
:
:
:
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)
#:
Given invertibility of ''m'', the rest of the relations in Birman & Wenzl's original version can be reduced to
#
(Idempotent relation)
#:
# (Braid relations)
#:
# (Tangle relations)
#:
# (Delooping relations)
#:
Properties
*The dimension of is .
*The Iwahori–Hecke algebra
In mathematics, the Iwahori–Hecke algebra, or Hecke algebra, named for Erich Hecke and Nagayoshi Iwahori, is a deformation of the group algebra of a Coxeter group. The Hecke algebra can also be viewed as a ''q''-analog of the group algebra ...
associated with the symmetric group is a quotient of the Birman–Murakami–Wenzl algebra .
*The Artin braid group
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...
embeds in the BMW algebra: .
Isomorphism between the BMW algebras and Kauffman's tangle algebras
It is proved by that the BMW algebra is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the Kauffman's tangle algebra . The isomorphism is defined by
and
Baxterisation of Birman–Murakami–Wenzl algebra
Define the face operator as
:,
where and are determined by
:
and
:.
Then the face operator satisfies the Yang–Baxter equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their ...
.
:
Now with
:.
In the limits , the braid
A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strand ...
s can be recovered up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
a scale factor
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions ( isotropically). The result of uniform sc ...
.
History
In 1984, Vaughan Jones
Sir Vaughan Frederick Randal Jones (31 December 19526 September 2020) was a New Zealand mathematician known for his work on von Neumann algebras and knot polynomials. He was awarded a Fields Medal in 1990.
Early life
Jones was born in Gisbo ...
introduced a new polynomial invariant of link isotopy types which is called the Jones polynomial
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...
. The invariants are related to the traces of irreducible representations of Hecke algebras associated with the symmetric groups. showed that the Kauffman polynomial can also be interpreted as a function on a certain associative algebra. In 1989, constructed a two-parameter family of algebras with the Kauffman polynomial as trace after appropriate renormalization.
References
*
*
*
{{DEFAULTSORT:Birman-Wenzl algebra
Representation theory
Knot theory
Diagram algebras