The Variety of Two-dimensional Algebras Over an Algebraically Closed Field

Abstract

Yury Volkov (wolf86 666@list.ru).Abstract. The work is devoted to the variety of 2-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of certain algebra series in the variety of 2-dimensional algebras. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible, and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

The algebraic and geometric classification of nilpotent anticommutative algebras

“…: e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 1 e 5 = e 6 , e 2 e 3 = e 5 , e 2 e 5 = e 6 , e 3 e 4 = −e 6 ; A 41 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 1 e 5 = e 6 , e 2 e 3 = e 5 , e 3 e 4 = e 6 ; A 42 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 2 e 4 = e 6 , e 3 e 5 = e 6 ; A 43 (α) : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 2 e 5 = αe 6 , e 3 e 4 = e 6 ; A 44 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 3 e 5 = e 6 ; A 45 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 4 e 5 = e 6 .…”

The algebraic classification of nilpotent Tortkara algebras

“…Another interesting direction in classifications of algebras is geometric classification. We refer the reader to [27][28][29]32] for results in this direction for Jordan, Lie, Leibniz, Zinbiel and other algebras. In the present paper, we give algebraic classification of nilpotent algebras of a new class of non-associative algebras introduced by Dzhumadildaev in [15].…”

Degenerations of Filippov algebras

“…There are also works in which the full information about degenerations was found for some variety of algebras. Here one can mention the descriptions of degenerations of low dimensional associative, Lie, pre-Lie, Malcev, Leibniz (see [7,24,25]) and all 2-dimensional (see [26]) algebras. There is an obvious connection between the notions of degeneration and deformation.…”