Let G be a complex simple Lie group and let g be its Lie algebra. Then G has the adjoint action on g. The orbit Ox of a nilpotent element x ∈ g is called a nilpotent orbit. A nilpotent orbit Ox admits a non-degenerate closed 2-form ω called the Kostant-Kirillov symplectic form. The closure Ōx of Ox then becomes a symplectic singularity. In other words, the 2-form ω extends to a holomorphic 2-fo...

Let G be a complex simple Lie group and let g be its Lie algebra. Then G has the adjoint action on g. The orbit Ox of a nilpotent element x ∈ g is called a nilpotent orbit. A nilpotent orbit Ox admits a non-degenerate closed 2-form ω called the Kostant-Kirillov symplectic form. The closure Ōx of Ox then becomes a symplectic singularity. In other words, the 2-form ω extends to a holomorphic 2-fo...

Our work is concerned with the problem on limit cycle bifurcation for a class of Z3-equivariant Lyapunov system of five degrees with three third-order nilpotent critical points which lie in a Z3-equivariant vector field. With the help of computer algebra system-MATHEMATICA, the first 5 quasiLyapunov constants are deduced. The fact of existing 12 small amplitude limit cycles created from the thr...

Let  be an n-dimensional Riemannian manifold, and  be its tangent bundle with the lift metric. Then every infinitesimal fiber-preserving conformal transformation  induces an infinitesimal homothetic transformation on .  Furthermore,  the correspondence   gives a homomorphism of the Lie algebra of infinitesimal fiber-preserving conformal transformations on  onto the Lie algebra of infinitesimal ...

Let G be a connected reductive group over an algebraically closed field $\Bbbk $ . Under mild restrictions on the characteristic of , we show that any G-module with good filtration also has as module for part centralizer nilpotent element x in its Lie algebra.

Let O be a nilpotent orbit in the Lie algebra n( ) and let V be an orbital variety contained in O. Let P be the largest parabolic subgroup of SL(n, ) stabilizing V. We describe nilpotent orbits such that all the orbital varieties in them have a dense P orbit and show that for n big enough the majority of nilpotent orbits do not fulfill this. Résumé Soit O une orbite nilpotente dans l’algèbre de...

The definition of index goes back to Dixmier [3, 11.1.6]. This notion is important in Representation Theory and also in Invariant Theory. By Rosenlicht’s theorem [12], generic orbits of an arbitrary action of a linear algebraic group on an irreducible algebraic variety are separated by rational invariants; in particular, ind g = tr.degK(g). The index of a reductive algebra equals its rank. Comp...

After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎wit...