Cohomology of restricted twisted Heisenberg Lie algebras

Abstract Over an algebraically closed field $\mathbb F$ of characteristic $p \gt 0$ , the restricted twisted Heisenberg Lie algebras are studied. We use the Hochschild–Serre spectral sequence relative to its Heisenberg ideal to compute the trivial cohomology. The ordinary 1- and 2-cohomology spaces are used to compute the restricted 1- and 2-cohomology spaces and describe the restricted one-dimensional central extensions, including explicit formulas for the Lie brackets and $-^{[p]}$ -operators.