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Degenerations of Poisson-type algebras

Hani Abdelwahab, Amir Fernández Ouaridi, Ivan Kaygorodov

Abstract

The degenerations of Poisson-type algebras are studied in the following varieties in dimension two: Leibniz--Poisson algebras, transposed Leibniz--Poisson algebras, Novikov--Poisson algebras, commutative pre-Lie algebras, anti-pre-Lie Poisson algebras and pre-Poisson algebras. For these varieties, the algebraic and geometric classifications are given. Also, the complete graph of degenerations is obtained, together with the description of the orbit closures of each of its algebras and parametric families up to isomorphism.

Degenerations of Poisson-type algebras

Abstract

The degenerations of Poisson-type algebras are studied in the following varieties in dimension two: Leibniz--Poisson algebras, transposed Leibniz--Poisson algebras, Novikov--Poisson algebras, commutative pre-Lie algebras, anti-pre-Lie Poisson algebras and pre-Poisson algebras. For these varieties, the algebraic and geometric classifications are given. Also, the complete graph of degenerations is obtained, together with the description of the orbit closures of each of its algebras and parametric families up to isomorphism.
Paper Structure (57 sections, 40 theorems, 26 equations)

This paper contains 57 sections, 40 theorems, 26 equations.

Table of Contents

  1. Preliminaries: the algebraic and geometric classifications
  2. The algebraic classification of algebras
  3. The algebraic classification of $2$-dimensional commutative associative algebras
  4. Degenerations and the geometric classification of algebras
  5. Leibniz--Poisson algebras
  6. The algebraic classification of $2$-dimensional Leibniz algebras
  7. The algebraic classification of $2$-dimensional Leibniz--Poisson algebras
  8. Leibniz--Poisson algebras defined on ${\rm A}_{01}$
  9. Leibniz--Poisson algebras defined on ${\rm A}_{02}$
  10. Leibniz--Poisson algebras defined on ${\rm A}_{03}$
  11. Leibniz--Poisson algebras defined on ${\rm A}_{04}$
  12. The algebraic classification of $2$-dimensional Leibniz--Poisson algebras
  13. The algebraic classification of $2$-dimensional generic Poisson algebras
  14. Degenerations of $2$-dimensional Leibniz--Poisson algebras
  15. Transposed Leibniz--Poisson algebras
  16. ...and 42 more sections

Key Result

Lemma 3

Let $\left( {\rm A},\cdot \right)$ be a commutative associative algebra and $\theta ,\vartheta \in {\rm Z}^{2}\left( {\rm A},{\rm A}\right)$. Then $\left( {\rm A},\cdot ,\bullet_{\theta }\right)$ and $\left( \rm{A},\cdot ,\bullet_{\vartheta }\right)$ are isomorphic if and only if there is a $\phi \i

Theorems & Definitions (80)

  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Theorem 4
  • Lemma 5
  • Definition 6
  • Definition 7
  • Lemma 8
  • Remark 9
  • ...and 70 more