Equivalent constructions of nilpotent quadratic Lie algebras

TL;DR

This paper establishes an equivalence among classical methods for constructing quadratic nilpotent Lie algebras, simplifying their classification by relating it to the study of trivectors and invariant forms.

Contribution

It provides an equivalent characterization of the double extension, T*-extension, and free nilpotent Lie algebra methods, streamlining classification processes.

Findings

Reduces classification of quadratic 2-step nilpotent Lie algebras to trivectors.

Provides rules for switching among construction methods.

Establishes equivalence among classical construction techniques.

Abstract

The double extension and the T*-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.