Representations of Smith algebras which are free over the Cartan subalgebra

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element $h$ and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity.Comment: 16 page

Intermediate Wakimoto modules for Affine sl(n+1)

We construct certain boson type realizations of affine sl(n+1) that depend on a parameter r. When r=0 we get a Fock space realization of Imaginary Verma modules appearing in the work of the first author and when r=n they are the Wakimoto modules described in the work of Feigin and Frenkel

Dynkin diagrams and spectra of graphs

Dynkin diagrams rst appeared in [20] in the connection with classication of simple Lie groups. Among Dynkin diagrams a special role is played by the simply laced Dynkin diagrams An, Dn, E6, E7 and E8. Dynkin diagrams are closely related to Coxeter graphs that appeared in geometry (see [8]). After that Dynkin diagrams appeared in many braches of mathematics and beyond, em particular em representation theory

Virasoro action on Imaginary Verma modules and the operator form of the KZ-equation

We define the Virasoro algebra action on imaginary Verma modules for affine sl(2) and construct the analogs of Knizhnik-Zamolodchikov equation in the operator form. Both these results are based on a free field realization of imaginary Verma modules

Systems of subspaces of a unitary space

For a given poset, we consider its representations by systems of subspaces of a unitary space ordered by inclusion. We classify such systems for all posets for which an explicit classification is possible.Comment: 20 page

Three Representation Types for Systems of Forms and Linear Maps

We consider systems of bilinear forms and linear maps as representations of a graph with undirected and directed edges. Its vertices represent vector spaces; its undirected and directed edges represent bilinear forms and linear maps, respectively. We prove that if the problem of classifying representations of a graph has not been solved, then it is equivalent to the problem of classifying representations of pairs of linear maps or pairs consisting of a bilinear form and a linear map. Thus, there are only two essentially different unsolved classification problems for systems of forms and linear maps