A complete system of pairwise orthogonal minimal idempotents for Birman-Murakami-Wenzl algebras is obtained by a consecutive evaluation of a rational function in several variables on sequences of quantum contents of up-down tableaux. A by-product of the construction is a one-parameter family of fusion procedures for Hecke algebras. Classical limits to two different fusion procedures for Brauer algebras are described.
The Birman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys--Murphy elements. We show that the set of Jucys--Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters $q$ and $r$, for B--M--W alg…
Commutative sets of Jucys-Murphyelements for affine braid groups of $A^{(1)},B^{(1)},C^{(1)},D^{(1)}$ types were defined. Construction of $R$-matrix representations of the affine braid group of type $C^{(1)}$ and its distinguish commutative subgroup generated by the $C^{(1)}$-type Jucys--Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary conditions for Sklyanin's type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of…
We determine the Zariski closure of the representations of the braid groups that factorize through the Birman-Wenzl-Murakami algebra, for generic values of the parameters $α,s$. For $α,s$ of modulus 1 and close to 1, we prove that these representations are unitarizable, thus deducing the topological closure of the image when in addition $α,s$ are algebraically independent.
The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We show that the cyclotomic BMW algebras are free modules over any (admissible, integral) ground ring, and that they are isomorphic to cyclotomic versions of the Kauffman tangle algebras
We introduce a BMW type algebra for every Coxeter group. These new algebras are introduced as deformations of the Brauer type algebras introduced by the author, they have the corresponding Hecke algebras as quotients.
An explicit isomorphism is constructed between the Birman-Wenzl algebra, defined algebraically by J. Birman and H. Wenzl using generators and relations, and the Kauffman algebra, constructed geometrically by H. R. Morton and P. Traczyk in terms of tangles. The isomorphism is obtained by constructing an explicit basis in the Birman-Wenzl algebra, analogous to a basis previously constructed for the Kauffman algebra using 'Brauer connectors'. The geometric isotopy arguments for the Kauffman algebra are systematically replaced by algebraic versions using the Birman-Wenzl relations.
In this paper, we compute the Gram determinants associated to each cell module of the Birman-Wenzl algebras. As a by-product, we give the necessary and sufficient condition for semisimple Birman-Wenzl algebras over an arbitrary field.
The cyclotomic Birman-Wenzl-Murakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We study admissibility conditions on the ground ring for these algebras, and show that the algebras defined over an admissible integral ground ring $S$ are free $S$--modules and isomorphic to cyclotomic Kauffman tangle algebras. We also determine the representation theory in the generic semisimple case, obtain a recursive formula for the weights of the Markov trace, and give a sufficient condition for semisimplicity.
We construct a new inductive basis of the Birman-Murakami-Wenzl algebra. Using it, we provide a new proof of the existence of the Markov trace on the BMW algebras affording the two-variable Kauffman polynomial. We prove also that all the transverse Markov traces on the BMW algebras are determined by the self-linking number, the HOMFLY-PT polynomial and the two-variable Kauffman polynomial.
In this paper we first present a Birman-Murakami-Wenzl type algebra for every Coxeter system of rank 2 (corresponding to dihedral groups). We prove they have semisimple for generic parameters, and having natural cellular structures. And classcify their irreducible representations. Among them there is one serving as a generalization of the Lawrence-Krammer representation with quite neat shape and the "correct" dimension. We conjecture they are isomorphic to the generalized Lawrence-Krammer representaions defined by I.Marin as monodromy of certain KZ connections. We prove these representations a…
A representation of the Birman-Wenzl-Murakami algebra BW_{t}(-q^{2n},q) exists in the centraliser algebra End_{U_q(osp(1|2n))}(V^{\otimes t}), where V is the fundamental (2n+1)-dimensional irreducible U_{q}(osp(1|2n))-module. This representation is defined using permuted R-matrices acting on V^{\otimes t}. A complete set of projections onto and intertwiners between irreducible U_{q}(osp(1|2n))-summands of V^{\otimes t} exists via this representation, proving that End_{U_q(osp(1|2n))}V^{\otimes t}) is generated by the set of permuting R-matrices acting on V^{\otimes t}. We also show that a repr…
The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey-Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley-Lieb, Birman-Murakami-We…
----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman…
The Morita equivalences of classical Brauer algebras and classical Birman-Murakami-Wenzl algebras have been well studied. Here we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and Birman-Murakami-Wenzl algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of some Coxeter groups, respectively.
The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image af an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the polynomial ring Z with delta and its inverse adjoined. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of…
The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann that the BMW algebras are isomorphic to algebras of tangles in (disc $\times$ interval), modulo Kauffman skein relations. This isomorphism allows one to see very clearly certain properties of the BMW algebras -- for example existence of Markov traces and conditional expectations an…
We classify the Markov traces factoring through the Birman-Wenzl-Murakami (BMW) algebras. For this purpose, we define a common `cover' for the two variations of the BMW-algebra originating from the quantum orthogonal/symplectic duality, which are responsible for the so-called `Dubrovnik' variation of the Kauffman polynomial. For generic values of the defining parameters of the BMW algebra, this cover is isomorphic to the BMW algebra itself, and this fact provides a shorter defining relation for it, in the generic case. For a certain 1-dimensional family of special values however, it is a non-t…
Inspired by the work [IMOg2], in this note, we prove that the pairwise orthogonal primitive idempotents of generic cyclotomic Birman-Murakami-Wenzl algebras can be constructed by consecutive evaluations of a certain rational function. In the appendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras.
The diagram algebra introduced by Brauer that describes the centralizer algebra on tensor products of the natural representation of an orthogonal group has a presentation by generators and relations that only depends on the graph of type An on n nodes. Here we describe an algebra depending on an arbitrary graph of type M. We study its structure when the type is An, Dn, E6, E7, E8. We determine the representations and find the dimensions. The algebra is generically semisimple and contains the group algebra of the Coxeter type M as a subalgebra. It is a ring homomorphism of the Birman-Murakami-W…
In this paper, we go on Rui-Xu's work on cyclotomic Birman-Wenzl algebras $\W_{r, n}$ in \cite{RX}. In particular, we use the representation theory of cellular algebras in \cite{GL} to classify the irreducible $\W_{r, n}$-modules for all positive integers $r$ and $n$. By constructing cell filtrations for all cell modules of $\W_{r, n}$, we compute the discriminants associated to all cell modules for $\W_{r, n} $. Via such discriminats together with induction and restriction functors given in section~5, we determine explicitly when $\W_{r, n}$ is semisimple over a field. This generalizes our pr…
This paper extends the Birman-Wenzl category by including a spin representation and then developing the recoupling theory, following Cvitanovic. In particular this gives a q-analogue of the chromatic evaluation of a spin network. The recoupling theory is developed up to an evaluation of the q-analogues of the Fierz coefficients.
The cyclotomic Birman-Murakami-Wenzl (or BMW) algebras B_n^k, introduced by R. Haring-Oldenburg, are extensions of the cyclotomic Hecke algebras of Ariki-Koike, in the same way as the BMW algebras are extensions of the Hecke algebras of type A. In this paper we focus on the case n=2, producing a basis of B_2^k and constructing its left regular representation.
Let $n\in\mathds{N}$ and $B_n(r,q)$ be the generic Birman-Murakami-Wenzl algebra with respect to indeterminants $r$ and $q$. It is known that $B_n(r,q)$ has two distinct linear representations generated by two central elements of $B_n(r,q)$ called the symmetrizer and antisymmetrizer of $B_n(r,q)$. These generate for $n\geq 3$ the only one dimensional two sided ideals of $B_n(r,q)$ and generalize the corresponding notion for Hecke algebras of type $A$. The main result in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of $B_n(r,q)$.
The affine and degenerate affine Birman-Murakami-Wenzl (BMW) algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce acti…
