BMW Algebras Research Articles

Morita equivalences on Brauer algebras and BMW algebras of simply-laced types

The Morita equivalences of classical Brauer algebras and classical Birman–Murakami–Wenzl (BMW) 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 BMW algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of Coxeter groups, respectively.

Handlebody diagram algebras

In this paper we study handlebody versions of some classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer, BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.

Inductions and restrictions on towers of cellularly stratified diagram algebras

Hartmann et al. defined the concept of cellularly stratified algebras that combine the features of both cellular algebras and stratified algebras. Many important diagram algebras in mathematics and physics, such as some Brauer, partition and BMW algebras, are cellularly stratified algebras, and each of these forms a tower of algebras. This article gives the concept of towers of cellularly stratified algebras in an axiomatic manner, and studies it in terms of induction and restriction functors. In particular, for certain towers of cellularly stratified algebras, we provide a criterion for semi-simplicity by using the cohomology groups of cell modules.

Back to Baxterisation

In the continuity of our previous paper (Crampe et al. in Commun Math Phys 349:271, 2017, arXiv:1509.05516 ), we define three new algebras, $${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$$ , $${\mathcal{B}_{\mathfrak{n}}}$$ and $${\mathcal{C}_{\mathfrak{n}}}$$ , that are close to the braid algebra. They allow to build solutions to the Yang-Baxter equation with spectral parameters. The construction is based on a baxterisation procedure, similar to the one used in the context of Hecke or BMW algebras. The $${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$$ algebra depends on three arbitrary parameters, and when the parameter a is set to zero, we recover the algebra $${\mathcal{M}_{\mathfrak{n}}(b,c)}$$ already introduced elsewhere for purpose of baxterisation. The Hecke algebra (and its baxterisation) can be recovered from a coset of the $${\mathcal{A}_{\mathfrak{n}}(0,0,c)}$$ algebra. The algebra $${\mathcal{A}_{\mathfrak{n}}(0,b,-b^2)}$$ is a coset of the braid algebra. The two other algebras $${\mathcal{B}_{\mathfrak{n}}}$$ and $${\mathcal{C}_{\mathfrak{n}}}$$ do not possess any parameter, and can be also viewed as a coset of the braid algebra.

Fusion Procedure for Cyclotomic BMW Algebras

In this paper, 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. A consequence of the constructions is a one-parameter family of fusion procedures for the cyclotomic Hecke algebra and its degenerate analogue.

Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations**Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.

Commutative sets of Jucys–Murphy elements for affine braid groups of types were defined. Construction of R-matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the -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 the -type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras (BMW algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements in the double affine Hecke algebra of type A.

The Brauer category and invariant theory

A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group O (V) or the symplectic group Sp (V) over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain presentations for the endomorphism algebras of the module V^{\otimes r} , which are new in the classical symplectic case and in the orthogonal and symplectic quantum case, while in the orthogonal classical case, the proof we give here is more natural than in our earlier work. These presentations are obtained by appending to the standard presentation of the Brauer algebra of degree r one additional relation. This relation stipulates the vanishing of a single element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if dim V=2n , the element is precisely the central idempotent in the Brauer subalgebra of degree n+1 , which corresponds to its trivial representation. Since this is the Brauer algebra of highest degree which is semisimple, our generator is an exact analogue for the Brauer algebra of the Jones idempotent of the Temperley-Lieb algebra. In the orthogonal case the additional relation is also a quasi-idempotent in the integral Brauer algebra. Both integral and quantum analogues of these results are given, the latter of which involve the BMW algebras.

Morita equivalence for cyclotomic BMW algebras

In this paper, we determine explicitly the singular parameters for cyclotomic BMW algebras Br,n over an arbitrary field. Equivalently, we give a criterion for Br,n being Morita equivalent to the direct sum of some Ariki–Koike algebras. This generalizes König and Xi's work on Brauer algebras [14, Theorem 7.3] and our previous work on Brauer and Birman–Murakami–Wenzl algebras [20, Proposition 2.15].

Affine and degenerate affine BMW algebras: the center

The degenerate affine and affine BMW algebras arise naturally in the context of Schur--Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we make an exact parallel between the degenerate affine and affine cases via a new algebra which takes the role of the affine braid group for the degenerate setting. A main result of this paper is an identification of the centers of the affine and degenerate affine BMW algebras in terms of rings of symmetric functions which satisfy a ``cancellation property'' or ``wheel condition'' (in the degenerate case, a reformulation of a result of Nazarov). Miraculously, these same rings also arise in Schubert calculus, as the cohomology and K-theory of isotropic Grassmannians and symplectic loop Grassmannians. We also establish new intertwiner-like identities which, when projected to the center, produce the recursions for central elements given previously by Nazarov for degenerate affine BMW algebras, and by Beliakova--Blanchet for affine BMW algebras.

Decomposition numbers of cyclotomic NW and BMW algebras

Motivated by Martin and Cox–De Visscher’s work Martin [18], Cox and De Visscher [5] on the decomposition matrices for Brauer algebras and walled Brauer algebras over the complex field, we give an algorithm to compute the decomposition matrices on the cyclotomic Nazarov–Wenzl algebras and cyclotomic Birman–Murakami–Wenzl algebras over the field κ with some additional assumptions. Our result shows that the multiplicity-free theorem holds for such algebras.

On the cellularity of the cyclotomic Birman-Murakami-Wenzl algebras

The cyclotomic Birman–Murakami–Wenzl (or BMW) algebras ℬ n k were introduced by Häring-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. In the PhD thesis of the second author and previous work of the authors, these algebras are shown to be free of rank kn (2n−1)!! over ground rings with parameters satisfying the so-called ‘admissibility conditions’, the most general conditions under which these results can hold. These conditions are build from the representation theory of ℬ 2 k and are necessary and sufficient in order for these results to hold. (A review of these admissibility conditions and their differences with others used in the literature is provided in this paper.) Furthermore, ℬ n k may be topologically realized, in terms of affine/cylindrical tangles, as a cyclotomic version of the Kauffman tangle algebra, and bases that can be explicitly described both algebraically and diagrammatically were obtained. In this paper, it is shown that these bases combined with a carefully constructed lifting of an arbitrary cellular basis of the Ariki–Koike algebras yield a cellular basis of the cyclotomic BMW algebras, in the sense of Graham and Lehrer, thereby establishing cellularity of the algebra under the most general conditions available in the literature.

Remarks on cyclotomic and degenerate cyclotomic BMW algebras

We relate the structure of cyclotomic and degenerate cyclotomic BMW algebras, for arbitrary parameter values, to that for admissible parameter values. In particular, we show that these algebras are cellular. We characterize those parameter sets for affine BMW algebras over an algebraically closed field that permit the algebras to have non-trivial cyclotomic quotients.

The Cyclotomic BMW Algebra Associated with the Two String Type B Braid Group

The cyclotomic Birman–Murakami–Wenzl (or BMW) algebras , introduced by R. Häring–Oldenburg, are extensions of the cyclotomic Hecke algebras of Ariki–Koike, in the same way as the BMW algebras are extensions of the Iwahori–Hecke algebras of type A. In this article, we produce a basis of the two strand algebra and construct its left regular representation. In the process we obtain relations amongst the parameters which play an important role in the n strand case.

The Birman-Murakami-Wenzl algebras of type E n

The Birman-Murakami-Wenzl algebras (BMW algebras) of type E_ n for n = 6, 7, 8 are shown to be semisimple and free over the integral domain Z[δ^(±1),l^(±1),m]/(m1−δ)(l−l^(−1)) of ranks 1,440,585; 139,613,625; and 53,328, 069,225. We also show they are cellular over suitable rings. The Brauer algebra of type E_n is a homomorphic ring image and is also semisimple and free of the same rank as an algebra over the ring Z[δ^(±1)]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. The generalized Temperley-Lieb algebra of type E_n turns out to be a subalgebra of the BMW algebra of the same type. So, the BMW algebras of type E_n share many structural properties with the classical ones (of type A_n) and those of type D_n .

Non-vanishing Gram determinants for cyclotomic Nazarov–Wenzl and Birman–Murakami–Wenzl algebras

In this paper, we use the method in Rui and Si (2011) [35] to give a necessary and sufficient condition on non-vanishing Gram determinants for cyclotomic NW and cyclotomic BMW algebras over an arbitrary field. Equivalently, we give a necessary and sufficient condition for each cell module of such algebras being equal to its simple head over an arbitrary field.

Admissibility Conditions for Degenerate Cyclotomic BMW Algebras

We study admissibility conditions for the parameters of degenerate cyclotomic Birman–Wenzl–Murakami (BMW) algebras. We show that the u–admissibility condition of Ariki, Mathas, and Rui is equivalent to a simple module theoretic condition.

On cellular algebras with Jucys Murphy elements

We study analogues of Jucys–Murphy elements in cellular algebras arising from repeated Jones basic constructions. Examples include Brauer and BMW algebras and their cyclotomic analogues.

The Representations of Cyclotomic BMW Algebras, II

In this paper, we give a recursive formula to compute the Gram determinant associated to each cell module of the cyclotomic BMW algebras over an integral domain. As a by-product, we determine explicitly when is semisimple over a field. This generalizes our previous result on Birman-Murakami-Wenzl algebras in Rui and Si (J Reine Angew Math 631:153–180, 2009).

Cellularity and the Jones basic construction

We establish a framework for cellularity of algebras related to the Jones basic construction. Our framework allows a uniform proof of cellularity of Brauer algebras, ordinary and cyclotomic BMW algebras, walled Brauer algebras, partition algebras, and others. Our cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for the class of algebras that we study, we show that the cellular structures are compatible with restriction and induction of modules.

Blocks of Birman-Murakami-Wenzl Algebras

Birman, Wenzl [3], and independently Murakami [19] introduced a class of finite dimensional algebras , which are known as the Birman–Murakami–Wenzl algebras or BMW algebras. When the ground field κ contains invertible r and q such that o(q 2 ), the multiplicative order of q 2 , is strictly greater than n, and char(κ), the characteristic of κ is not 2, we determine the structure of the cell module Δ(1, λ) of , where λ is any partition of n −2. In particular, we compute the dimension of the simple head of Δ(1, λ). Cox, De Visscher, and Martin have classified the blocks of Brauer algebras B n in characteristic zero [5]. We solve the similar problem for over the aforementioned field κ. As a by-product, we give a criterion for each module of being equal to its simple head over an arbitrary field.