Let g be a Lie algebra of dimension n over a field K . Then g determines a multiplication table relative to each basis {e1, . . . , en}. If [ei, ej] = ∑n k=1 γ k i,jek , then (γ k i,j) ∈ K n is called a structure for g and the γ i,j the structure constants of g. The elements of Ln(K) are exactly the Lie algebra structures. They form an affine algebraic variety and the group GLn(K) acts on Ln(K)...

It is shown that projectivized irreducible components of nilpotent cones of complex symmetric spaces are projective self-dual algebraic varieties. Other properties equivalent to their projective self-duality are found. 1. Let g be a semisimple complex Lie algebra, let G be the adjoint group of g, and let θ ∈ Aut g be an element of order 2. We set k := {x ∈ g | θ(x) = x}, p := {x ∈ g | θ(x) = −x...

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.