HOME

TheInfoList



OR:

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 ...
\mathrm_n(\ell,m) of dimension 1\cdot 3\cdot 5\cdots (2n-1) 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 \mathrm_n(\ell,m) is generated by G_1^,G_2^,\dots, G_^,E_1,E_2,\dots,E_ and relations: :G_iG_j = G_jG_i, \mathrm \left\vert i-j \right\vert \geq 2, :G_i G_ G_i = G_ G_i G_,         E_i E_ E_i = E_i, :G_i + ^ = m(1+E_i), :G_ G_i E_ = E_i G_ G_i = E_i E_,      G_ E_i G_ =^ E_ ^, :G_ E_i E_ = ^ E_,      E_ E_i G_ =E_ ^, :G_i E_i= E_i G_i = l^ E_i,     E_i G_ E_i = l E_i. These relations imply the further relations:
:E_i E_j=E_j E_i, \mathrm \left\vert i-j \right\vert \geq 2,
:(E_i)^2 = (m^(l+l^)-1) E_i,
:^2 = m(G_i+l^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 - ^=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^(l-l^)+1) E_i, # (Braid relations) #: G_iG_j=G_jG_i, \text \left\vert i-j \right\vert \geqslant 2, \text G_i G_ G_i=G_ G_i G_, # (Tangle relations) #: E_i E_ E_i=E_i \text G_i G_ E_i = E_ E_i, # (Delooping relations) #: G_i E_i= E_i G_i = l^ E_i \text E_i G_ E_i =l E_i.


    Properties

    *The dimension of \mathrm_n(\ell,m) is (2n)!/(2^nn! ). *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 S_n is a quotient of the Birman–Murakami–Wenzl algebra \mathrm_n. *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: B_n \hookrightarrow \mathrm_n.


    Isomorphism between the BMW algebras and Kauffman's tangle algebras

    It is proved by that the BMW algebra \mathrm_n(\ell,m) 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 \mathrm_n. The isomorphism \phi \colon \mathrm_n \to \mathrm_n is defined by
    and


    Baxterisation of Birman–Murakami–Wenzl algebra

    Define the face operator as :U_i(u) = 1- \frac(e^ G_i -e^^), where \lambda and \mu are determined by :2\cos \lambda = 1+(l-l^)/m and :2\cos \lambda = 1+(l-l^)/(\lambda \sin \mu). 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 ...
    . : U_(v) U_i(u+v) U_(u) = U_i(u) U_(u+v) U_i(v) Now E_i=U_i(\lambda) with : \rho(u)=\frac . In the limits u \to \pm i \infty, 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 F on a certain associative algebra. In 1989, constructed a two-parameter family of algebras \mathrm_n(\ell,m) with the Kauffman polynomial K_n(\ell,m) as trace after appropriate renormalization.


    References

    * * * {{DEFAULTSORT:Birman-Wenzl algebra Representation theory Knot theory Diagram algebras