On geometric degenerations and Gerstenhaber formal deformations

Abstract

We study the degeneration relations on the varieties of associative and Lie algebra structures on a fixed finite dimensional vector space and give a description of them in terms of Gerstenhaber formal deformations. We use this result to show how the orbit closure of the $3$-dimensional Lie algebras can be determined using homological algebra. For the case of finite dimensional associative algebras, we prove that the $N$-Koszul property is preserved under the degeneration relation for all $N\ge 2$.

11 pages

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• The geometric classification of nilpotent algebras
• The algebraic and geometric classification of antiassociative algebras
• The geometric classification of nilpotent commutative $\mathfrak{CD}$-algebras
• The geometric classification of $2$-step nilpotent algebras and applications