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- W4302560565 abstract "Let $G$ be a linear algebraic group over an algebraically closed field of characteristic $pgeq 0$. We show that if $H_1$ and $H_2$ are connected subgroups of $G$ such that $H_1$ and $H_2$ have a common maximal unipotent subgroup and $H_1/R_u(H_1)$ and $H_2/R_u(H_2)$ are semisimple, then $H_1$ and $H_2$ are $G$-conjugate. Moreover, we show that if $H$ is a semisimple linear algebraic group with maximal unipotent subgroup $U$ then for any algebraic group homomorphism $sigmacolon Urightarrow G$, there are only finitely many $G$-conjugacy classes of algebraic group homomorphisms $rhocolon Hrightarrow G$ such that $rho|_U$ is $G$-conjugate to $sigma$. This answers an analogue for connected algebraic groups of a question of B. Kulshammer. In Kulshammer's original question, $H$ is replaced by a finite group and $U$ by a Sylow $p$-subgroup of $H$; the answer is then known to be no in general. We obtain some results in the general case when $H$ is non-connected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of non-connected linear algebraic groups. When $G$ is reductive, we formulate Kulshammer's question and related conjugacy problems in terms of the nonabelian 1-cohomology of unipotent radicals of parabolic subgroups of $G$, and we give some applications of this cohomological approach. In particular, we analyse the case when $G$ is a semisimple group of rank 2." @default.
- W4302560565 created "2022-10-06" @default.
- W4302560565 creator A5053197687 @default.
- W4302560565 creator A5068776388 @default.
- W4302560565 date "2016-10-30" @default.
- W4302560565 modified "2023-09-30" @default.
- W4302560565 title "On a question of Kulshammer for homomorphisms of algebraic groups" @default.
- W4302560565 doi "https://doi.org/10.48550/arxiv.1610.09665" @default.
- W4302560565 hasPublicationYear "2016" @default.
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