Solvability and nilpotency for algebraic supergroups

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• 作者：Akira Masuoka^a ; ^{akira@math.tsukuba.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Alexandr N. Zubkov^b ; ^{a.zubkov@yahoo.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14L15 ; 14M30 ; 16T05
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：221
• 期：2
• 页码：339-365
• 全文大小：607 K

文摘

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field K   of characteristic $\mathrm{char}K\ne 2$. Our first main theorem tells us that an algebraic supergroup G$\mathbb{G}$ is solvable if the associated algebraic group G_ev${\mathbb{G}}_{ev}$ is trigonalizable. To prove it we determine the algebraic supergroups G$\mathbb{G}$ such that dim⁡Lie(G)₁=1$\dim \mathrm{Lie}{(\mathbb{G})}_1=1$; their representations are studied when G_ev${\mathbb{G}}_{ev}$ is diagonalizable. The second main theorem characterizes nilpotent connected algebraic supergroups. A super-analogue of the Chevalley Decomposition Theorem is proved, though it must be in a weak form. An appendix is given to characterize smooth Noetherian superalgebras as well as smooth Hopf superalgebras.