Geometry and classifications of some $\omega$-Lie algebras

TL;DR

This paper investigates the geometric structure and classification of finite-dimensional $ ext{ extomega}$-Lie algebras using group actions, orbit methods, and computational algebra, providing a complete classification of 3-dimensional cases.

Contribution

It establishes a geometric framework for understanding $ ext{ extomega}$-Lie algebras and offers a complete classification of all 3-dimensional $ ext{ extomega}$-Lie algebras over algebraically closed fields.

Findings

The variety of 3-dimensional $ ext{ extomega}$-Lie algebras is a 6-dimensional irreducible affine variety.

This variety is a complete intersection.

A full classification of 3-dimensional $ ext{ extomega}$-Lie algebras over algebraically closed fields is achieved.

Abstract

Using group actions and orbit-stabilizer methods, we study the geometry of isomorphism classes of finite-dimensional $\omega$-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 $\omega$. We also apply techniques from computational ideal theory to explore the geometric structure of the affine variety of all 3-dimensional $\omega$-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 $\omega$-Lie algebras over an algebraically closed field of characteristic $\neq 2$, up to $\omega$-Lie algebra isomorphism.