Deformations of the Lie Algebra $\mathfrak{o}(5)$ in Characteristics~$3$ and~$2$

S. Bouarroudj ; A. V. Lebedev ; F. Vagemann

Matematičeskie zametki, Tome 89 (2011) no. 6, pp. 808-824

Résumé

All finite-dimensional simple modular Lie algebras with Cartan matrix fail to have deformations, even infinitesimal ones, if the characteristic $p$ of the ground field is equal to $0$ or exceeds $3$. If $p=3$, then the orthogonal Lie algebra $\mathfrak o(5)$ is one of two simple modular Lie algebras with Cartan matrix that do have deformations (the Brown algebras $\mathfrak{br}(2;\alpha)$ appear in this family of deformations of the $10$-dimensional Lie algebras, and therefore are not listed separately); moreover, the $29$-dimensional Brown algebra $\mathfrak{br}(3)$ is the only other simple Lie algebra which has a Cartan matrix and admits a deformation. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of an algebraic group $O(5)$ of automorphisms of the Lie algebra $\mathfrak o(5)$ on the space $H^2(\mathfrak o(5);\mathfrak o(5))$ of infinitesimal deformations and presented representatives of the isomorphism classes. We give here an explicit description of the global deformations of the Lie algebra $\mathfrak o(5)$ and describe the deformations of a simple analog of this orthogonal algebra in characteristic $2$. In characteristic $3$, we have found the representatives of the isomorphism classes of the deformed algebras that linearly depend on the parameter.

Détail

Keywords: finite-dimensional simple modular Lie algebra, Brown algebra, infinitesimal deformation, global deformation, Jacobi identity, Massey bracket, Chevalley basis.
Mots-clés : Cartan matrix, Maurer–Cartan equation