2025/06/13 by Mathevet, Hugo
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2506.11562
Given a natural number m and a Lie algebra g, the m th generalized Takiff Lie algebra of g is the Lie algebra gm := g ⊗ C[T ]/T m+1 . For n ≥ m, we define the (m, n)-modality of an adjoint orbit Ωm in gm to be the minimum codimension of an adjoint orbit in the pullback of Ωm in gn. In this paper, we study this family of invariants in generalized Takiff Lie algebras associated to a quadratic Lie algebra g. We show that this family of invariants satisfies some concavity and hereditary properties. From which we deduce that (n -m)χ(g) is a lower bound, where χ(g) is the index of g. We prove that this lower bound is in fact an equality for a dense set of orbits, and that if g is reductive, it is always an equality when m = 0 (and also some special orbits). We conjecture that equality holds for all m when g is reductive.