On the cohomology of restricted Heisenberg Lie algebras

We show that the Heisenberg Lie algebras over a field F of characteristic p > 0 admit a family of restricted Lie algebras, and we classify all such non-isomorphic restricted Lie algebra structures. We use the ordinary 1and 2-cohomology spaces with trivial coefficients to compute the restricted 1and 2-cohomology spaces of these restricted Heisenberg Lie algebras. We describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and <middle dot> ([ p ])-operators. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.