Inner derivations of simple Lie pencils of rank 1

We prove that simple Lie pencils of rank 1 over an algebraically closed field P of characteristic 0 with operators of left multiplication being derivations are of the form of a sandwich algebra M ₃(U,D′), where U is the subspace of all skew-symmetric matrices in M ₃(P) and D′ is any subspace containing 〈E〉 in the space of all diagonal matrices D in M ₃(P).     

Gradings of positive rank on simple Lie algebras

We complete the classification of positive rank gradings on Lie algebras of simple algebraic groups over an algebraically closed field k whose characteristic is zero or not too small, and we determine the little Weyl groups in each case. We also classify the stable gradings and prove Popov’s conjecture on the existence of a Kostant section.     

Lie sheaves of small dimensions

We obtain a classification of 2- and 3-dimensional Lie sheaves.     

Simple Lie sheaves of small dimension

We classify simple Lie sheaves of dimension at most 4.     

The spectrum of the Laplace operator on connected compact simple Lie groups of rank four

In the present article, we explicitly compute the spectrum of the Laplace operator on smooth real-valued and complex-valued functions on connected compact simple Lie groups of rank four with a bi-invariant Riemannian metrics that correspond to the root systems B ₄, C ₄, and D ₄. We also find a connection between the obtained formulas, number theory, and integral quadratic forms in two, three, and four variables.     

The spectra of the Laplace operators on connected compact simple Lie groups of rank 3

We expose explicit calculations of the spectra of the Laplace operators for smooth real or complex functions on all connected compact simple Lie groups of rank 3 with bi-invariant Riemannian metric and establish the relationship of the obtained formulas with number theory and integer-valued ternary and binary quadratic forms.     

Lie p-algebras of finite p-subalgebra rank

We say that a Lie p-algebra L has finite p-subalgebra rank if the minimal number of generators required to generate every finitely generated p-subalgebra is uniformly bounded by some integer r. This paper is concerned with the following problem: does L being of finite p-subalgebra rank force ad(L) to be finite-dimensional? Although this seems unlikely in general, we show that this is indeed the case for Lie p-algebras in a large class including all locally, residually, and virtually soluble Lie p-algebras.     

Simply connected coset complexes for rank 1 groups of Lie type

This work was partially supported by BSF 92-00320