The classification of 2-dimensional rigid algebras

Abstract

Using the algebraic classification of all 2-dimensional algebras, we give the algebraic classification of all 2-dimensional rigid, conservative and terminal algebras over an algebraically closed field of characteristic 0. We have the geometric classification of the variety of 2-dimensional terminal algebras, and based on the geometric classification of these algebras we formulate some open problems.

The Algebraic and Geometric Classification of Nilpotent Bicommutative Algebras

“…It is typical to focus on small dimensions, and there are two main directions for the classification: algebraic and geometric. Varieties as Jordan, Lie, Leibniz or Zinbiel algebras have been studied from these two approaches ( [1, 9, 12-15, 22, 27, 30, 37] and [3,5,6,9,11,22,24,25,31,32,[35][36][37][38], respectively). In the present paper, we give the algebraic and geometric classification of 4-dimensional nilpotent bicommutative algebras.…”

The algebraic and geometric classification of nilpotent anticommutative algebras

“…There are fewer works in which the full information about degenerations has been given for some variety of algebras. This problem was solved for 2-dimensional pre-Lie algebras [7], for 2-dimensional terminal algebras [11], for 3-dimensional Novikov algebras [8], for 3-dimensional Jordan algebras [25], for 3-dimensional Jordan superalgebras [6], for 3-dimensional Leibniz algebras [35], for 3-dimensional anticommutative algebras [35], for 3-dimensional nilpotent algebras [17], for 4-dimensional Lie algebras [9], for 4-dimensional Lie superalgebras [5], for 4-dimensional Zinbiel algebras [40], for 4-dimensional nilpotent Leibniz algebras [40], for 4-dimensional nilpotent commutative algebras [17], for 5-dimensional nilpotent Tortkara algebras [24], for 5-dimensional nilpotent anticommutative algebras [17], for 6-dimensional nilpotent Lie algebras [27,48], for 6-dimensional nilpotent Malcev algebras [41], for 2-step nilpotent 7-dimensional Lie algebras [4], and for all 2-dimensional algebras [42]. There are many results related to the algebraic and geometric classification of low-dimensional algebras in the varieties of Jordan, Lie, Leibniz and Zinbiel algebras; for algebraic classifications see, for example, [1, 11, 13-16, 22, 24, 32, 33, 35, 36, 39, 42]; for geometric classifications and descriptions of degenerations see, for example, [1, 3-6, 8, 9, 11, 20, 21, 23-25, 27, 28, 33-36, 38-48].…”

The geometric classification of nilpotent Tortkara algebras

“…There are many results related to both the algebraic and geometric classification of small dimensional algebras in the varieties of Jordan, Lie, Leibniz and Zinbiel algebras; for algebraic results see, for example, [1,11,18,19,23,[25][26][27]30]; for geometric results see, for example, [1, 3-6, 8, 10, 11, 19-31, 34]. Here we give a geometric classification of 6-dimensional nilpotent Tortkara algebras over C. Our main result is Theorem 3 which describes the rigid algebras in this variety.…”

“…There are fewer works in which the full information about degenerations has been found for some variety of algebras. This problem has been solved for 2-dimensional pre-Lie algebras in [7], for 2-dimensional terminal algebras in [11], for 3-dimensional Novikov algebras in [8], for 3-dimensional Jordan algebras in [20], for 3-dimensional Jordan superalgebras in [6], for 3-dimensional Leibniz algebras in [25], for 3-dimensional anticommutative algebras in [25], for 4-dimensional Lie algebras in [10], for 4-dimensional Lie superalgebras in [5], for 4-dimensional Zinbiel algebras in [28], for 3-dimensional nilpotent algebras [17], for 4-dimensional nilpotent Leibniz algebras in [28], for 4-dimensional nilpotent commutative algebras [17], for 5-dimensional nilpotent Tortkara algebras in [19], for 5-dimensional nilpotent anticommutative algebras in [17], for 6-dimensional nilpotent Lie algebras in [21,34], for 6-dimensional nilpotent Malcev algebras in [29], for 7-dimensional 2-step nilpotent Lie algebras in [4], and for all 2dimensional algebras in [30].…”