A Poisson analog of the Dixmier-Moeglin equivalence is established for any affine Poisson algebra R on which an algebraic torus H acts rationally, by Poisson automorphisms, such that R has only finitely many prime Poisson H-stable ideals. In this setting, an additional characterization of the Poisson primitive ideals of R is obtained – they are precisely the prime Poisson ideals maximal in thei...

We study the nonstandard q-deformation U ′ q (so 4) of the universal enveloping algebra U (so 4) obtained by deforming the defining relations for skew-symmetric generators of U (so 4). This algebra is used in quantum gravity and algebraic topology. We construct a homomor-phism φ of U ′ q (so 4) to the certain nontrivial extension of the Drinfeld–Jimbo quantum algebra U q (sl 2) ⊗2 and show that...

We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some ”Kos...

The canonical trace on the reduced C∗-algebra of a discrete group gives rise to a homomorphism from the K-theory of this C∗-algebra to the real numbers. This paper addresses the range of this homomorphism. For torsion free groups, the Baum-Connes conjecture and Atiyah’s L-index theorem implies that the range consists of the integers. If the group is not torsion free, Baum and Connes conjecture ...

In this note a proof of a differential analog of Chevalley’s theorem [C] on homomorphism extensions is given. An immediate corollary is a condition of finitenes of extensions of differential algebras and several equivalent definitions of a differentially closed field, including Kolchin’s Nullstellensatz. In this note I give a proof of the following differential analog of Chevalley’s theorem [C]...

In the present work the properties of Cartan subalgebras and their connection with regular elements in finite dimensional Lie algebras are extended to the case of Leibniz algebras. It is shown that Cartan subalgebras and regular elements of a Leibniz algebra correspond to Cartan subalgebras and regular elements of a Lie algebra by a natural homomorphism. Conjugacy of Cartan subalgebras of Leibn...

1.1. Linear algebra is assumed as a prerequisite to these notes. However, this section serves to review the language of abstract linear algebra (particularly tensor products) for use in the remainder of these notes. We assume the reader familiar with these notions, and therefore do not provide a complete and detailed treatment. In particular, we assume the reader knows what a homomorphism of al...