On geometric degenerations and Gerstenhaber formal deformations

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 and quasi-affine lines. We use this result to show how the orbit closure of the three-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 ⩾ 2 .

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