The Lie algebra preserving a degenerate bilinear form

TL;DR

This paper characterizes the structure of Lie algebras that preserve degenerate bilinear forms on vector spaces over arbitrary fields, revealing their composition as semidirect products and analyzing their radicals and quotients.

Contribution

It provides a detailed structural analysis of Lie algebras preserving degenerate bilinear forms, including radicals, derived series, and semisimple quotients, extending to graded derivations.

Findings

Lie algebras are semidirect products of classical Lie algebras and representations.

Determination of radicals, derived series, and semisimple quotients.

Application to graded Lie algebras of derivations of certain graded algebras.

Abstract

Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of matrices that preserve the subspace spanned by $M$. We show that these Lie algebras are semidirect products of classical Lie algebras and certain representations, and determine their radicals, derived series and semisimple quotients. Our main motivation and application is to determine the structure of the graded Lie algebra of derivations of each commutative or graded commutative algebra with Hilbert polynomial $1+dt+t^{2}$. Some of our results apply to more general bilinear forms and graded algebras.