Deformations of the Lie algebra o(5) in characteristics 3 and 2

TL;DR

This paper investigates the deformations of the orthogonal Lie algebra o(5) over fields of characteristics 2 and 3, providing explicit descriptions of global deformations and extending previous classifications.

Contribution

It explicitly describes global deformations of o(5) in characteristics 2 and 3, expanding understanding of modular Lie algebra deformations beyond known classifications.

Findings

o(5) has non-trivial deformations in characteristic 3

Explicit descriptions of deformations in characteristic 2 are provided

Extends previous work on automorphism orbits and deformation classes

Abstract

The finite dimensional simple modular Lie algebras with Cartan matrix cannot be deformed if the characteristic p of the ground field is equal to 0 or greater than 3. If p=3, the orthogonal Lie algebra o(5)is one of the two simple modular Lie algebras with Cartan matrix that have deformations (the Brown algebras br(2; a) are among these 10-dimensional deforms and hence are not counted separately); the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra with Cartan matrix that has deformations. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of the group O(5) of automorphisms of o(5) on the space H^2(o(5);o(5)) and produced representatives of the isomorphism classes. Here we explicitly describe global deforms of o(5) and of the simple analog of this orthogonal Lie algebra in characteristic 2.