Third cohomology for Frobenius kernels and related structures

zbMath1418.17044arXiv1410.2322MaRDI QIDQ3178829

Daniel K. Nakano, Christopher P. Bendel, Cornelius Pillen

Publication date: 20 December 2016

Abstract: Let $G$ be a simple simply connected group scheme defined over ${mathbb F}_{p}$ and $k$ be an algebraically closed field of characteristic $p>0$. Moreover, let $B$ be a Borel subgroup of $G$ and $U$ be the unipotent radical of $B$. In this paper the authors compute the third cohomology group for $B$ and its Frobenius kernels, $B_{r}$, with coefficients in a one-dimensional representation. These computations hold with relatively mild restrictions on the characteristic of the field. As a consequence of our calculations, the third ordinary Lie algebra cohomology group for ${mathfrak u}= ext{Lie }U$ with coefficients in $k$ is determined, as well as the third $G_{r}$-cohomology with coefficients in the induced modules $H^{0}(lambda)$.

Full work available at URL: https://arxiv.org/abs/1410.2322

Mathematics Subject Classification ID

Representation theory for linear algebraic groups (20G05) Cohomology of Lie (super)algebras (17B56) Cohomology theory for linear algebraic groups (20G10) Modular Lie (super)algebras (17B50)

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