Degenerations of nilpotent associative commutative algebras

Abstract

We give a complete description of degenerations of complex 5-dimensional nilpotent associative commutative algebras.

The geometric classification of 2-step nilpotent algebras and applications

“…is rigid in the variety of complex n-dimensional nilpotent associative algebras. Thanks to [17], all complex 5-dimensional (or 4-dimensional) nilpotent commutative associative algebras are degenerated from µ 5 0 (or µ 4 0 ). The variety of complex 4-dimensional 2-dimensional nilpotent algebras is defined by the following families of algebras The algebra A 4 05 satisfies the following conditions {A 1 A 2 ⊆ A 4 , c → A 4 06 (1) we have that the variety of complex 4-dimensional nilpotent associative algebras is defined by µ 4 0 , A 4 05 , N 2 (α) and N 3 (α).…”

“…It offers an insightful geometric perspective on the subject and has been the object of a lot of research. In particular, there are many results concerning degenerations of algebras of small dimensions in a variety defined by a set of identities (see, for example, [1,3,4,12,14,17,19] and references therein). One of the main problems of the geometric classification of a variety of algebras is a description of its irreducible components.…”

The algebraic classification of nilpotent commutative CD-algebras

“…The geometric classification of 5-dimensional nilpotent commutative CD-algebras. The geometric classification of 5-dimensional nilpotent commutative CD-algebras is based on some previous works: namely, all irreducible components of 5-dimensional nilpotent associative commutative algebras are given in [31] and all degenerations between these algebras are given in [27]; all irreducible components of 5-dimensional nilpotent Jordan algebras were described in [23]. In the proof of the present theorem we give all necessary arguments for the description of all irreducible components of the variety of 5-dimensional nilpotent commutative CD-algebras.…”

Degenerations of noncommutative Heisenberg algebras

“…Degenerations have also been used to study a level of complexity of an algebra [15]. There are many results concerning degenerations of algebras of small dimensions in a variety defined by a set of identities (see, for example, [1,4,6,12,16,20,21] and references therein). An interesting question is to study those properties which are preserved under degenerations.…”