Jordan–Kronecker Invariants of Semidirect Sums of the Form sl(n) + (ℝn)k and gl(n) + (ℝn)k

Abstract

We calculate Jordan-Kronecker invariants for semidirect sums of Lie algebras sl(n) and gl(n) with k copies of R n with respect to their standard representation for cases where k > n or n is a multiple of k.

Jordan–Kronecker Invariants for Lie Algebras of Small Dimensions

“…• In [29], [30], [31] and [32] the JK invarians are the sizes of blocks. This is not ideal, there is no need to lose the valuable information -which Jordan blocks have the same eigenvalue.…”

“…We can decompose each symplectic (S z , ω λ,z ) using Turiel's factorization theorem 4.7. We get coordinates (30) such that the matrices of all forms ω λ,z are block-diagonal:…”

“…In [30] it was shown that all Jordan blocks are 2 × 2. It was done by calculating dim g y for a singular y ∈ Sing 0 and using Proposition 10.11.…”

“…Apart from [5] and references therein, the Jordan-Kronecker invariants for various Lie algebras were calculated in [29], [30], [31], [32], [10], [11]. Yet the following question, asked in [1], [2] and [4] remains open.…”