Hamming graphs in Nomura algebras

Let A be an association scheme on q≥3 vertices. We show that the Bose-Mesner algebra of the generalized Hamming scheme H(n,A), for n?2, is not the Nomura algebra of any type II matrix.This result gives examples of formally self-dual Bose-Mesner algebras that are not the Nomura algebras of type II matrices.     

On Spin Models,Triply Regular Association Schemes,and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.     

Bose-Mesner Algebras Related to Type II Matrices and Spin Models

A type II matrix is a square matrixW with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebra N(W) , which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebrasN(W) . Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.     

On Even Generalized Table Algebras

Generalized table algebras were introduced in Arad, Fisman and Muzychuk (Israel J. Math. 114 (1999), 29–60) as an axiomatic closure of some algebraic properties of the Bose-Mesner algebras of association schemes. In this note we show that if all non-trivial degrees of a generalized integral table algebra are even, then the number of real basic elements of the algebra is bounded from below (Theorem 2.2). As a consequence we obtain some interesting facts about association schemes the non-trivial valencies of which are even. For example, we proved that if all non-identical relations of an association scheme have the same valency which is even, then the scheme is symmetric.     

On structures of modular adjacency algebras of Johnson schemes

In this paper, we consider algebras over a field of characteristic p, which are generated by adjacency algebras of Johnson schemes. If the algebra is semisimple, the structure is the same as that of the well-known Bose-Mesner algebras. We determine the structure of the algebra when it is not semisimple.     

Association Schemes and Fusion Algebras (An Introduction)

We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition (ST)³ = S ². We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H(d, q) has the modular invariance property.     

Type-II Matrices Attached to Conference Graphs

We determine the Nomura algebras of the type-II matrices belonging to the Bose-Mesner algebra of a conference graph.     

On the Brauer-Wielandt-Harada theorem for group-like regular association schemes

We consider a generalization of the Brauer-Wielandt-Harada Theorem to group-like regular association schemes. As an application, we give a necessary condition for commutative association schemes to be regular. Moreover, we derive the number of irreducible characters of multiplicity 1 from the product of all adjacency matrices and all valencies for a commutative regular association scheme.     

Association schemes,fusion rings,C-algebras,and reality-based algebras where all nontrivial multiplicities are equal

Multiplicities corresponding to irreducible characters are defined for reality-based algebras. These algebras with a distinguished basis include fusion rings, C-algebras, and the adjacency algebras of finite association schemes. The definition of multiplicity generalizes that for schemes. For a broad class of these structures, which includes the adjacency algebras, it is proved that if all the nontrivial multiplicities are equal then the algebra is commutative, and is a C-algebra if its dimension is larger than two.     

Strongly regular graphs and spin models for the Kauffman polynomial

We study spin models for invariants of links as defined by Jones [22]. We consider the two algebras generated by the weight matrices of such models under ordinary or Hadamard product and establish an isomorphism between them. When these algebras coincide they form the Bose-Mesner algebra of a formally self-dual association scheme. We study the special case of strongly regular graphs, which is associated to a particularly interesting link invariant, the Kauffman polynomial [27]. This leads to a classification of spin models for the Kauffman polynomial in terms of formally self-dual strongly regular graphs with strongly regular subconstituents [7]. In particular we obtain a new model based on the Higman-Sims graph [17].     

Commutative Association Schemes Whose Symmetrizations Have Two Classes

If a symmetric association scheme of class two is realized as the symmetrization of a commutative association scheme, then it either admits a unique symmetrizable fission scheme of class three or four, or admits three fission schemes, two of which are class three and one is of class four. We investigate the classification problem for symmetrizable (commutative) association schemes of two-class symmetric association schemes. In particular, we give a classification of association schemes whose symmetrizations are obtained from completely multipartite strongly regular graphs in the notion of wreath product of two schemes. Also the cyclotomic schemes associated to Paley graphs and their symmetrizable fission schemes are discussed in terms of their character tables.     

Classification of commutative association schemes with almost commutative Terwilliger algebras

We classify the commutative association schemes such that all non-primary irreducible modules of their Terwilliger algebras are one-dimensional.     

Polynomial functions on upper triangular matrix algebras

There are two kinds of polynomial functions on matrix algebras over commutative rings: those induced by polynomials with coefficients in the algebra itself and those induced by polynomials with scalar coefficients. In the case of algebras of upper triangular matrices over a commutative ring, we characterize the former in terms of the latter (which are easier to handle because of substitution homomorphism). We conclude that the set of integer-valued polynomials with matrix coefficients on an algebra of upper triangular matrices is a ring, and that the set of null-polynomials with matrix coefficients on an algebra of upper triangular matrices is an ideal.     

可换环上辛代数与一般线性李代数之间的中间李代数

本文研究了含幺可换环上一般线性李代数的子代数结构.通过构造特殊矩阵并利用这些矩阵进行计算, 得到了任意含幺可换环上辛代数与一般线性李代数之间的所有中间李代数的形式.并且有利于研究可换环上相应的典型群的子群结构.     

Representations of nongraded Lie algebras of block type

From his classification of quadratic conformal algebras corresponding to certain Hamiltonian pairs in integrable systems, Xu found a family of simple Lie algebras related to pairs of locally-finite derivations on certain commutative associative algebras. In this paper, we construct a large family of irreducible modules with four parameters for Xu's two-devivation algebras via the corresponding algebras of Weyl type. When the derivations are graded operators, we obtain a large family of uniformly-bounded irreducible weight modules for the Block algebras.     

Algébre de lie de type witt

Let ρ be an abelian group and k a commutative field. Inspired by the example of the Witt algebra, we introduce a family of ρ-graded Lie algebras (called Witt type Lie algebras) V = ⊕α?r Vα. where the dimension of each homogeneous component is 1. We characterise those which are simple and we give a complete classification of these Lie algebras. When P is free and the characteristic of k is zero, we show that the universal central extension of simple Witt type Lie algebras is 1-dimensional, and the corresponding cocycle is similar to the cocycle of the Virasoro algebra.     

可换环上一般线性李代数的李三导子

本文研究了含幺可换环上一般线性李代数的李三导子.通过构造特殊矩阵并利用这些矩阵进行运算,得到了任意含幺可换环上一般线性李代数的任意一个李三导子的具体形式,从而推广了导子的概念.     

Commutative Standard Table Algebras with at Most One Nontrivial Multiplicity

We classify commutative standard table algebras (STA) with at most one nontrivial multiplicity. The main result shows that there exists exactly one nontrivial multiplicity if and only if the table basis is the wreath product of a two-dimensional subalgebra and an abelian group. The theorem applies to adjacency algebras of commutative association schemes with exactly one primitive idempotent matrix of rank greater than one. A theorem of Seitz that characterizes finite groups with exactly one irreducible representation of degree greater than one is another corollary of the main theorem.     

Characterizations of wreath products of one-class association schemes

In this paper we show that the wreath product of one-class association schemes is characterized by the algebraic structure of its Bose-Mesner algebra.