Semi-direct products of Lie algebras, their invariants and representations

摘要: The goal of this paper is to extend the standard invariant-theoretic design, well-developed in reductive case, setting representation certain non-reductive groups. This concerns following notions and results: existence generic stabilisers isotropy groups for finite-dimensional representations; structure fields algebras invariants; quotient morphisms their fibres. One main tools obtaining Lie semi-direct product construction. We observe that there are surprisingly many whose adjoint has a polynomial algebra invariants. results Takiff, Geoffriau, Rais-Tauvel, Levasseur-Stafford concerning Takiff wider class products. includes $Z_2$-contractions simple generalised algebras.