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Geometry and classifications of some $ω$-Lie algebras

Yin Chen, Shan Ren, Runxuan Zhang

Abstract

Using group actions and orbit-stabilizer methods, we study the geometry of isomorphism classes of finite-dimensional $ω$-Lie algebras over a field $\mathbb{K}$ of characteristic $\neq 2$ and establish a one-to-one correspondence between the set of isomorphism classes and the orbit space of a stabilizer of $ω$. We also apply techniques from computational ideal theory to explore the geometric structure of the affine variety of all 3-dimensional $ω$-Lie algebras over $\mathbb{K}$, showing that this variety is a 6-dimensional irreducible affine variety and a complete intersection. As an application, we derive a complete classification of all 3-dimensional $ω$-Lie algebras over an algebraically closed field of characteristic $\neq 2$, up to $ω$-Lie algebra isomorphism.

Geometry and classifications of some $ω$-Lie algebras

Abstract

Using group actions and orbit-stabilizer methods, we study the geometry of isomorphism classes of finite-dimensional -Lie algebras over a field of characteristic and establish a one-to-one correspondence between the set of isomorphism classes and the orbit space of a stabilizer of . We also apply techniques from computational ideal theory to explore the geometric structure of the affine variety of all 3-dimensional -Lie algebras over , showing that this variety is a 6-dimensional irreducible affine variety and a complete intersection. As an application, we derive a complete classification of all 3-dimensional -Lie algebras over an algebraically closed field of characteristic , up to -Lie algebra isomorphism.
Paper Structure (9 sections, 21 theorems, 110 equations)

This paper contains 9 sections, 21 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Varieties of $\upomega$-Lie Algebras
  3. Geometric Structure of $\mathcal{A}_3(\mathbb{K})$
  4. Classification of Three-dimensional $\upomega$-Lie Algebras
  5. The case $a_3\neq 0$ and $\Delta \neq 0$
  6. The case $a_3\neq0$ and $\Delta=0$
  7. The case $a_3=0$
  8. Proof of Theorem \ref{['thm3']}
  9. Two Remarks

Key Result

Theorem 1.1

Let $\upomega$ be a nonzero skew-symmetric bilinear form on an $n$-dimensional vector space $V$ over a field $\mathbb{K}$ of characteristic $\neq 2$ and $\mathcal{L}_\upomega(\mathbb{K})$ be the affine variety of all $\upomega$-Lie algebras on $V$ with respect to $\upomega$. Suppose $G_\upomega$ den

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • ...and 39 more