The Bianchi variety

TL;DR

This paper provides an explicit description of the Bianchi variety, which is the algebraic variety of all Lie algebra structures on a 3-dimensional vector space, highlighting the role of compatibility among these structures.

Contribution

It offers a detailed characterization of Lie(V) for dim V=3, utilizing the concept of compatibility of Lie algebra structures, advancing understanding of the Bianchi variety.

Findings

Explicit description of Lie(V) for dim V=3

Introduction of compatibility notion for Lie algebra structures

Enhanced understanding of the algebraic structure of the Bianchi variety

Abstract

The totality Lie(V) of all Lie algebra structures on a vector space V over a field F is an algebraic variety over F on which the group GL(V) acts naturally. We give an explicit description of Lie(V) for dim V=3 which is based on the notion of compatibility of Lie algebra structures.