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 G is solvable if the associated algebraic group G_ev is trigonalizable. To prove it we determine the algebraic supergroups G such that dim⁡Lie(G)₁=1; their representations are studied when 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.