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

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.