Free nilpotent and nilpotent quadratic Lie algebras

TL;DR

This paper establishes an equivalence between categories of t-nilpotent quadratic Lie algebras and symmetric invariant bilinear forms, leading to classifications for specific cases over fields of characteristic zero.

Contribution

It introduces a new categorical equivalence and classifies t-nilpotent quadratic Lie algebras with d generators for certain parameters.

Findings

Classification of t-nilpotent quadratic Lie algebras for d=2, t<6

Classification for d=3, t<4

Establishment of categorical equivalence

Abstract

In this paper we introduce an equivalence between the category of the t-nilpotent quadratic Lie algebras with d generators and the category of some symmetric invariant bilinear forms on the t-nilpotent free Lie algebra with d generators. Taking into account this equivalence, t-nilpotent quadratic Lie algebras with d generators are classified (up to isometric isomorphism, and over any field of characteristic zero), in the following cases: d=2 and t<6, d=3 and t<4.