Degenerations for nonstandard algebras

- Algebras and Nonstandard Analysis

Suppose B is an ultraproduct of finite dimensional C∗-algebras. We consider mapping and injectability properties for separable C∗algebras into B. In the case of approximately finite C∗-algebras, we obtain a classification of these mappings up to inner conjugacy. Using a Theorem of Voiculescu, we show that for nuclear C ∗-algebras injectability into an ultraproduct of finite dimensional C ∗-alge...

Degenerations of nilpotent Lie algebras

In this paper we study degenerations of nilpotent Lie algebras. If λ, μ are two points in the variety of nilpotent Lie algebras, then λ is said to degenerate to μ , λ→deg μ , if μ lies in the Zariski closure of the orbit of λ . It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one-parameter subgroup. We construct degenerations between characte...

Nonstandard Poincare and Heisenberg Algebras

New deformations of the Poincare group Fun(P (1 + 1)) and its dual enveloping algebra U(p(1 + 1)) are obtained as a contraction of the hdeformed (Jordanian) quantum group Fun(SLh(2)) and its dual. A nonstandard quantization of the Heisenberg algebra U(h(1)) is also investigated. Ref. SISSA: 85/96/FM Off late, considerable interest has been generated towards the nonstandard quantization of Lie g...

Degenerations of 7-dimensional Nilpotent Lie Algebras

We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimen...

Degenerations for Modules over Representation-finite Biserial Algebras