Deformations of the Lie algebra o(5) in characteristics 3 and 2
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 o(5) is one of two simple modular Lie algebras with Cartan matrix that do have deformations (the Brown algebras br(2; α) appear in this family of deformations of the 10-dimensional Lie algebras, and therefore are not listed separately); moreover, the 29-dimensional Brown algebra 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 o(5) on the space H2(o(5); 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 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.