On the Second Cohomology of an Algebraic Group and of Its Lie Algebra in a Positive Characteristic

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic $p\ge 3h-3$, where $h$ stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups $\mathrm{SL}_2(k)$, $\mathrm{SL}_3(k)$, $\mathrm{SL}_4(k)$, $\mathrm{Sp}_4(k)$, and $G_2$, nontrivial isomorphisms of this kind exist for $\mathrm{SL}_4(k)$ and $G_2$ only. For $\mathrm{SL}_4(k)$, there are two simple modules with nontrivial second cohomology and, for $G_2$, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.