Solvability and nilpotency for algebraic supergroups

We study solvability, nilpotency and splitting property for algebraic supergroups over an arbitrary field K   of characteristic charK≠2. Our first main theorem tells us that an algebraic supergroup GG is solvable if the associated algebraic group GevGev is trigonalizable. To prove it we determine the algebraic supergroups GG such that dim⁡Lie(G)1=1dim⁡Lie(G)1=1; their representations are studied when GevGev 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.