On the Broken-Circuit Complex of Graphs

First, we construct a bijection between the set of h-vectors and the set of socle-vectors of Artinian algebras. As a corollary, we find the minimum codimension that an Artinian algebra with a given socle-vector can have. Then, we study the main problem in the article: Determining when there is a unique socle-vector for a given h-vector. We solve the problem completely if the codimension is at most 3.     

Flux Hamiltonians, Lie algebras, and root lattices with minuscule decorations

We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra, which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagomé and pyrochlore can be given this Lie algebraic interpretation, and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.     

Toric degeneration of branching algebras

For each classical symmetric pair (G,H), there is a naturally defined multi-graded algebra , called the branching algebra for (G,H), which encodes the branching rule from G to H. This algebra has a natural family of subalgebras, depending on integer parameters. For a certain range of the parameters, the subalgebras have a particularly simple structure and are called stable branching algebras.In this paper, we show that the stable branching algebras for eight out of the ten families of classical symmetric pairs are flat deformations of the semigroup algebras of explicitly described lattice cones.     

On varieties parametrizing graded complex Lie algebras

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by , is defined by a complex n × n-matrix , i, j = 1, . . . , n. In order for to be a Lie algebra, should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X _R . We compute the first homology group of an irreducible component M of X _R , under some assumptions on M. We look into algebraic properties of graded Lie algebras where .