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arXiv · 2607.28254

The geometric classification of left-symmetric algebras and superalgebras

Kobiljon Abdurasulov·Nigora Daukeyeva·Azamat Saydaliyev
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Abstract

We investigate the geometric classification of complex left-symmetric algebras and left-symmetric superalgebras in low dimensions. Building on the algebraic classifications of Bai and Zhang for 3-dimensional left-symmetric algebras and for 2- and 3-dimensional left-symmetric superalgebras, we determine all degenerations and non-degenerations among their isomorphism classes. We prove that the variety of 3-dimensional left-symmetric algebras has dimension 10 and decomposes into 31 irreducible components, ten of which are rigid. For left-symmetric superalgebras, we describe the irreducible components, determine their dimensions, and identify the rigid superalgebras in the varieties of dimensions (1,1), (1,2), and (2,1).

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Kobiljon Abdurasulov, Nigora Daukeyeva, Azamat Saydaliyev. 2026-07-30. The geometric classification of left-symmetric algebras and superalgebras. https://arxiv.org/abs/2607.28254

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