Hebing Rui, Mei Si
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.
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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.
Explicit formulas for the symmetrizer and the antisymmetrizer of the Birman-Wenzl-Murakami algebras BWM(r,q)_n are given.
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.
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 show the equivalence of admissibility conditions proposed by Wilcox and Yu and by Rui and Xu for the parameters of cyclotomic BMW algebras.
We show that the cyclotomic Birman-Wenzl-Murakami algebras are cellular by producing a cellular basis of affine tangle diagrams.
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.
In this paper, we calculate decomposition matrices of the Birman-Murakami-Wenzl algebras over $\mathbb C$.
In this paper, we classify the singular parameters for the Birman-Murakami-Wenzl algebra over an arbitrary field. Equivalently, we give a criterion for the Birman-Murakami-Wenzl algebra being Morita equivalent to the direct sum of the Hecke algebras associated to certain symmetric groups.
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.
We show that the affine BMW algebras are affine cellular algebras.
We give skein theoretic formulas for minimal idempotents in the Birman-Murakami-Wenzl algebras. These formulas are then applied to derive various known results needed in the construction of quantum invariants and modular categories. In particular, an elementary proof of the Wenzl formula for quantum dimensions is given. This proof does not use the representation theory of quantum groups and the character formulas.
We introduce a reduced form of a Birman-Murakami-Wenzl Algebra associated to the braid group of Coxeter type B and investigate its semisimplicity, Bratteli diagram and Markov trace. Applications in knot theory and physics are outlined.
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…
We introduce an Ariki-Koike like extension of the Birman-Murakami-Wenzl Algebra and show it to be semi-simple. This algebra supports a faithful Markov trace that gives rise to link invariants of closures of Coxeter type B braids.
----- 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…
We show that to every local representation of the Birman-Murakami-Wenzl algebra defined by a skew-invertible R-matrix $R\in Aut(V\otimes V)$ one can associate pairings $V\otimes V\to C$ and $V^*\otimes V^*\to C$, where V is the representation space. Further, we investigate conditions under which the corresponding quantum group is of SO or Sp type.
In this article we consider conditions under which projection operators in multiplicity free semi-simple tensor categories satisfy Temperley-Lieb like relations. This is then used as a stepping stone to prove sufficient conditions for obtaining a representation of the Birman-Murakami-Wenzl algebra from a braided multiplicity free semi-simple tensor category. The results are found by utalising the data of the categories. There is considerable overlap with the results found in arXiv:1607.08908, where proofs are shown by manipulating diagrams.
Explicit expressions for three series of $R$ matrices which are related to a ``dilute'' generalisation of the Birman--Wenzl--Murakami are presented. Of those, one series is equivalent to the quantum $R$ matrices of the $D^{(2)}_{n+1}$ generalised Toda systems whereas the remaining two series appear to be new.
A ``dilute'' generalisation of the Birman--Wenzl--Murakami algebra is considered. It can be ``Baxterised'' to a solution of the Yang--Baxter algebra. The $D^{(2)}_{n+1}$ vertex models are examples of corresponding solvable lattice models and can be regarded as the dilute version of the $B^{(1)}_{n}$ vertex models.
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…
We consider integrable open spin chains related to the quantum affine algebras U_q(o(3)) and U_q(A_2^{(2)}). We discuss the symmetry algebras of these chains with the local C^3 space related to the Birman-Wenzl-Murakami algebra. The symmetry algebra and the Birman-Wenzl-Murakami algebra centralize each other in the representation space, and this defines the structure of the spin system spectra. Consequently, the corresponding multiplet structure of the energy spectra is obtained.
We construct a representation of the Birman-Wenzl-Murakami algebra acting on a space of polynomials in n variables vanishing when three points coincide. These polynomials are closely related to the Pfaffian state of the Quantum Hall Effect and to the components the transfer matrix eigenvector of a O(n) crossing loop model.
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…
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…
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.
Bubbles in the affine Brauer and Kauffman categories
本文提出了一种生成函数方法，用于研究仿射Brauer和Kauffman范畴，通过该方法高效恢复重要关系并推导范畴作用的限制，同时恢复了循环BMW代数及其退化版本的可接受结果。
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 $B_n(r,q)$.
We use the Jones-Wenzl idempotents to construct a basis of Temperley-Lieb algebra TL_n. This allows a short calculation for a Gram determinant of Lickorish's bilinear form on the Temperley-Lieb algebra.
We establish isomorphisms between certain specializations of Birman-Murakami-Wenzl algebras and the symmetric squares of Temperley-Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an application, we compute the closed images of the irreducible braid group representations factoring over these specialized BMW algebras.
Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors
本文研究带彩色区域的广义Temperley-Lieb分类，确定其半单性并分解张量积的简单对象，利用双变量Chebyshev多项式和彩色Jones-Wenzl投影子证明了一个猜想。
Let $L$ be a link and $Φ^{A}_{L}(q)$ its link invariant associated with the vector representation of the quantum (super)algebra $U_{q}(A)$. Let $F_{L}(r,s)$ be the Kauffman link invariant for $L$ associated with the Birman--Wenzl--Murakami algebra $BWM_{f}(r,s)$ for complex parameters $r$ and $s$ and a sufficiently large rank $f$. For an arbitrary link $L$, we show that $Φ^{osp(1|2n)}_{L}(q) = F_{L}(-q^{2n},q)$ and $Φ^{so(2n+1)}_{L}(-q) = F_{L}(q^{2n},-q)$ for each positive integer $n$ and all sufficiently large $f$, and that $Φ^{osp(1|2n)}_{L}(q)$ and $Φ^{so(2n+1)}_{L}(-q)$ are identical up t…
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