Birman Murakami Wenzl Algebras Research Articles

The 5-CB algebra and fused SU(2) lattice models

We study the fused SU(2) models put forward by Date et al, that are a series of models with arbitrary number of blocks, which is the degree of the polynomial equation obeyed by the Boltzmann weights. We demonstrate by a direct calculation that a version of Birman–Murakami–Wenzl (BMW) algebra [, ] is obeyed by five, six and seven blocks models, conjecturing that the BMW algebra is a part of the algebra valid for any model with more than two blocks. To establish this conjecture, we assume that a certain ansatz holds for the baxterization of the models. We use the Yang–Baxter equation to describe explicitly the algebra for five blocks, obtaining 19 additional non-trivial relations. We name this algebra 5-CB (conformal braiding) algebra. Our method can be utilized to describe the algebra for any solvable model of this type and for any number of blocks.

Quantum matrix algebras of BMW type: Structure of the characteristic subalgebra

A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the components of a ‘quantum’ matrix M possess certain properties which resemble structure theorems of the ordinary matrix theory. It turns out that such structure results are naturally derived in a more general framework of the QM-algebras. In this work we consider a family of Birman–Murakami–Wenzl (BMW) type QM-algebras. These algebras are defined with the use of R-matrix representations of the BMW algebras. Particular series of such algebras include orthogonal and symplectic types RTT- and RE-algebras, as well as their super-partners.For a family of BMW type QM-algebras, we investigate the structure of their ‘characteristic subalgebras’ — the subalgebras where the coefficients of characteristic polynomials take values. We define three sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. We also define an associative ⋆-product for the matrix M of generators of the QM-algebra which is a proper generalization of the classical matrix multiplication. We determine the set of all matrix ‘descendants’ of the quantum matrix M, and prove the ⋆-commutativity of this set in the BMW type.

On SO(N) spin vertex models

We describe the Boltzmann weights of the Dk algebra spin vertex models. Thus, we find the SO(N) spin vertex models, for any N, completing the Bk case found earlier. We further check that the real (self–dual) SO(N) models obey quantum algebras, which are the Birman–Murakami–Wenzl (BMW) algebra for three blocks, and certain generalizations, which include the BMW algebra as a sub–algebra, for four and five blocks. In the case of five blocks, the B4 model is shown to satisfy additional twenty new relations, which are given. The D6 model is shown to obey two additional relations.

The 4-CB algebra and solvable lattice models

We study the algebras underlying solvable lattice models of the type fusion interaction round the face (IRF). We propose that the algebras are universal, depending only on the number of blocks, which is the degree of polynomial equation obeyed by the Boltzmann weights. Using the Yang-Baxter equation and the ansatz for the Baxterization of the models, we show that the three blocks models obey a version of Birman-Murakami­Wenzl (BMW) algebra. For four blocks, we conjecture that the algebra, which is termed 4-CB (Conformal Braiding) algebra, is the BMW algebra with a different skein relation, along with one additional relation, and we provide evidence for this conjecture. We connect these algebras to knot theory by conjecturing new link invariants. The link invariants, in the case of four blocks, depend on three arbitrary parameters. We check our result for G2 model with the seven dimensional representation and for SU(2) with the isospin 3/2 representation, which are both four blocks theories.

On the algebraic approach to solvable lattice models

We treat here interaction round the face (IRF) solvable lattice models. We study the algebraic structures underlining such models. For the three block case, we show that the Yang Baxter equation is obeyed, if and only if, the Birman-Murakami-Wenzl (BMW) algebra is obeyed. We prove this by an algebraic expansion of the Yang Baxter equation (YBE). For four blocks IRF models, we show that the BMW algebra is also obeyed, apart from the skein relation, which is different. This indicates that the BMW algebra is a sub-algebra for all models with three or more blocks. We find additional relations for the four block algebra using the expansion of the YBE. The four blocks result, that is the BMW algebra and the four blocks skein relation, is enough to define new knot invariant, which depends on three arbitrary parameters, important in knot theory.

Random walks on the BMW monoid: an algebraic approach

We consider Metropolis-based systematic scan algorithms for generating Birman–Murakami–Wenzl (BMW) monoid basis elements of the BMW algebra. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We also consider the Brauer algebra and use Metropolis-based scans to generate Brauer diagrams, giving rise to random walks on perfect matchings. Taking an algebraic perspective, we translate these walks into left multiplication operators in the BMW algebra and so give an algebraic interpretation of the Metropolis algorithm in this setting.

Three blocks solvable lattice models and Birman–Murakami–Wenzl algebra

Birman–Murakami–Wenzl (BMW) algebra was introduced in connection with knot theory. We treat here interaction round the face solvable (IRF) lattice models. We assume that the face transfer matrix obeys a cubic polynomial equation, which is called the three block case. We prove that the three block theories all obey the BMW algebra. We exemplify this result by treating in detail the SU(2)2×2 fused models, and showing explicitly the BMW structure. We use the connection between the construction of solvable lattice models and conformal field theory. This result is important to the solution of IRF lattice models and the development of new models, as well as to knot theory.

The Efficient Computation of Fourier Transforms on Semisimple Algebras

We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a semisimple algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path algebra and construction of Gel’fand–Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley–Lieb, and Birman–Murakami–Wenzl algebras.

Recollements of self-injective algebras, and classification of self-injective diagram algebras

Diagram algebras, in particular Brauer algebras, Birman–Murakami-Wenzl algebras and partition algebras, are used in representation theory and invariant theory of orthogonal and symplectic groups, in knot theory, in mathematical physics and elsewhere. Classifications are known when such algebras are semisimple, of finite global dimension or quasi-hereditary. We obtain a characterisation of the self-injective case, which is shown to coincide with the (previously also unknown) symmetric case. The main tool is to show that indecomposable self-injective algebras in general are derived simple, that is, their bounded derived module categories admit trivial recollements only. As a consequence, self-injective algebras are seen to satisfy a derived Jordan–Holder theorem.

Quantum teleportation and Birman–Murakami–Wenzl algebra

In this paper, we investigate the relationship of quantum teleportation in quantum information science and the Birman---Murakami---Wenzl (BMW) algebra in low-dimensional topology. For simplicity, we focus on the two spin-1/2 representation of the BMW algebra, which is generated by both the Temperley---Lieb projector and the Yang---Baxter gate. We describe quantum teleportation using the Temperley---Lieb projector and the Yang---Baxter gate, respectively, and study teleportation-based quantum computation using the Yang---Baxter gate. On the other hand, we exploit the extended Temperley---Lieb diagrammatical approach to clearly show that the tangle relations of the BMW algebra have a natural interpretation of quantum teleportation. Inspired by this interpretation, we construct a general representation of the tangle relations of the BMW algebra and obtain interesting representations of the BMW algebra. Therefore, our research sheds a light on a link between quantum information science and low-dimensional topology.

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.

Decomposition matrices of Birman–Murakami–Wenzl algebras

In this paper, we calculate decomposition matrices of the Birman–Murakami–Wenzl algebras over C.

Topological basis realization for BMW algebra and Heisenberg XXZ spin chain model

In this paper, we study three-dimensional (3D) reduced Birman–Murakami–Wenzl (BMW) algebra based on topological basis theory. Several examples of BMW algebra representations are reviewed. We also discuss a special solution of BMW algebra, which can be used to construct Heisenberg XXZ model. The theory of topological basis provides a useful method to solve quantum spin chain models. It is also shown that the ground state of XXZ spin chain is superposition state of topological basis.

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].

Idempotents for Birman-Murakami-Wenzl algebras and reflection equation

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 byproduct 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–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D n 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!! =1·3…(2n − 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ℤ[δ±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D n is a subalgebra of the BMW algebra of the same type.

SYMMETRIZERS AND ANTISYMMETRIZERS FOR THE BMW-ALGEBRA

Let n ∈ ℕ and Bn(r, q) be the generic Birman–Murakami–Wenzl algebra with respect to indeterminants r and q. It is known that Bn(r, q) has two distinct linear representations generated by two central elements of Bn(r, q) called the symmetrizer and antisymmetrizer of Bn(r, q). These generate for n ≥ 3 the only one-dimensional two sided ideals of Bn(r, q) and generalize the corresponding notion for Hecke algebras of type A. The main result, Theorem 3.1, in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of Bn(r, q).

Affine and degenerate affine BMW algebras: actions on tensor space

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 of the affine and degenerate affine BMW algebras. 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 actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements—the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.

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.

Singular parameters for the Birman–Murakami–Wenzl algebra

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.