Purely imaginary roots of kac-moody algebras

In this paper, we introduce a new class of imaginary roots, called purely imaginary roots and give a necessary and sufficient condition for an imaginary root to be purely imaginary,thereby giving a complete classification of those Kac-Moody algebras with the purely imaginary property, namely, all their imaginary roots are purely imaginary. We also define a new class of non-hyperbolic indefinite Kac-Moody algebras, called the extended hyperbolic kac-Moody algebras possessing the purely imaginary property.     

Extended Rees algebras and mixed multiplicities

Partially supported by the general research fund at the University of Kansas     

Weight sets of irreducible integrable representations of kac-moody algebras of indefinite type

In this paper, the weight sets of some irreducible and integrable representations, which are not highest or lowest weight representations, of rank two Kac-Moody algebras of indefinite type are completely determined.     

On multiplicities of cocharacters for algebras with superinvolution

In this paper we deal with finitely generated superalgebras with superinvolution, satisfying a non-trivial identity, whose multiplicities of the cocharacters are bounded by a constant. Along the way, we prove that the codimension sequence of such algebras is polynomially bounded if and only if their colength sequence is bounded by a constant.     

Young modules and filtration multiplicities for Brauer algebras

We define permutation modules and Young modules for the Brauer algebra B _k (r,δ), and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of Young modules, respecting an ordering condition similar to that for symmetric groups. Moreover, we determine precisely in which cases cell module filtration multiplicities are well-defined, as done by Hemmer and Nakano for symmetric groups. Supported by the European Community through Marie Curie fellowship MCFI 2002-01325 Supported by EPSRC grant GR/S18151/01     

On modules of bounded multiplicities for the symplectic algebras

Simple infinite dimensional highest weight modules having
bounded weight multipicities are classified as submodules of a tensor product. Also, it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible.

     

On some root multiplicities for Nichols algebras of diagonal type over arbitrary fields

In this article, our main aim is to determine the multiplicities of a class of roots for Nichols algebra of diagonal type over fields of arbitrary characteristic, which is a generalization of the results on the multiplicities of these roots over fields of characteristic zero obtained by Heckenberger and the author.     

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Root closed function algebras on compacta of large dimension

Let be a Hausdorff compact space and let be the algebra of all continuous complex-valued functions on , endowed with the supremum norm. We say that is (approximately) -th root closed if any function from is (approximately) equal to the -th power of another function. We characterize the approximate -th root closedness of in terms of -divisibility of the first Cech cohomology groups of closed subsets of . Next, for each positive integer we construct an -dimensional metrizable compactum such that is approximately -th root closed for any . Also, for each positive integer we construct an -dimensional compact Hausdorff space such that is -th root closed for any .

     

On the simplicity of some kac-moody groups

Recently, Marcuson extended the classical construction of Tits systems in Steinberg groups to include the Kac-Moody Steinberg groups associated with the infinite dimensional versions of the great Lie algebras. If these Lie algebras and their Kac-Moody groups are viewed as limits of their finite dimensional counterparts, more direct methods may be employed. In fact, the Kac-Moody Chevalley groups of these Lie algebras are seen to be simple.     

On the simplicity of some kac-moody groups

Recently, Marcuson extended the classical construction of Tits systems in Steinberg groups to include the Kac-Moody Steinberg groups associated with the infinite dimensional versions of the great Lie algebras. If these Lie algebras and their Kac-Moody groups are viewed as limits of their finite dimensional counterparts, more direct methods may be employed. In fact, the Kac-Moody Chevalley groups of these Lie algebras are seen to be simple.     

Formulas for multiplicities of occurence of irreducible components in the tensor product of representations of simple lie algebras

New combinatorial formulas for multiplicities in the decomposition of the tensor product of representations of simple Lie algebras into irreducible components are obtained. A connection of the constructions arising with the Bethe equations and the representation theory of Yangians is established. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 30–37. Translated by B. M. Bekker.