Finite-dimensional Solvable Lie Algebras Research Articles

Two-Variable Identities in Groups and Lie Algebras

We study two-variable Engel-like relations and identities characterizing finite-dimensional solvable Lie algebras and, conjecturally, finite solvable groups and introduce some invariants of finite groups associated with such relations. Bibliography: 29 titles.

Spectra for solvable Lie algebras of bundle endomorphisms

The aim of the paper is to investigate spectral properties of the Lie algebras correspond- ing to the symmetry groups of certain flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of C ∗ -algebras. The flag manifolds in C ∗ -algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles. Mathematics Subject Classification (2000): 47A13, 22E65, 58B25, 46L10

An Example of Bernstein Duality

We study local duality in a non-commutative framework which extends well known local dualities in commutative Gorenstein rings and in enveloping algebras of finite dimensional solvable Lie algebras. We show examples of non-commutative algebras having a local duality in our sense. In order to study these examples we will need a non-commutative version of Groebner bases.

Goldie conditions for constants of algebraic derivations of semiprime algebras

Relations between Goldie conditions of a semiprime algebraR and its subalgebraR d of constants under an algebraic derivation are studied. The results obtained are applied to actions of finite dimensional solvable Lie algebras on associative algebras with no non-zero nilpotent elements.

Classical localizability in solvable enveloping algebras and Poincaré-Birkhoff-Witt extensions

In several classes of noncommutative noetherian rings, Jategaonkar's intersection conditions are verified for finite unions of cliques of prime ideals, and it follows that these finite unions of cliques are classically localizable if they satisfy incomparability. Among the rings in question are the enveloping algebra of any finite-dimensional solvable Lie algebra g over any field k of characteristic zero, any twisted smash product R # U(g) where R is a commutative noetherian k-algebra and g acts on R via k-linear derivations, and certain iterated differential operator rings over R. In contrast to previously verified cases, k is allowed to be countable. One tool used here—a lying over theorem for cliques—has some independent interest. Namely, if R ⊆ S is a flat finite centralizing extension of noetherian rings satisfying the second layer condition, and if X is a clique in Spec( R), there exists a clique Y in Spec( S) such that { P ∩ R| P ϵ Y} = X.

Localization and ideal theory in iterated differential operator rings

In this paper we study primality, hypercentrality, simplicity, and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings. We give sufficient conditions for the skew enveloping algebra of a nilpotent Lie algebra with coefficient ring containing the rational numbers to be a simple ring, and we give necessary and sufficient conditions in the case that the Lie algebra is Abelian. Our main results show that if L is a finite dimensional solvable Lie algebra over a field k of characteristic zero and R is an Artinian or a commutative Noetherian algebra over k, then the skew enveloping algebra R# U( L) satisfies the second layer condition. We discuss consequences of this for localization and use the localization theory to state a classical Krull dimension versus global dimension inequality when k is uncountable.