In this paper, Jordan–Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on lambda parameter a coalgebra, i.e., linear space dual to algebra. For some

Abstract We determine the 6-dimensional nilpotent metric Lie algebras such that algebra $${\mathfrak {n}}$$ n has a descending series of ideals invariant under all automorphisms and dimension consecutive members decreases by one. call them having framing determined ideals. classify isometry equivalence classe...

Abstract The problem of finding generators the subalgebra invariants under action a group automorphisms finite-dimensional Lie algebra on its universal enveloping is reduced to homogeneous same acting symmetric tensor algebra. This process applied prove constructive Hilbert–Nagata Theorem (including degree bounds) for in nilpotent relatively free associative endowed with an induced by represent...

Throughout this paper, let k be a field of characteristic 2. Let G be a classical group over k and g be its Lie algebra. Let g be the dual vector space of g. Fix a Borel subgroup B and a maximal torus T ⊂ B of G. Let B = TU be the Levi decomposition of B. Let b, t and n be the Lie algebra of B, T and U respectively. We have a natural action of G on g, g.ξ(x) = ξ(Ad(g)x) for g ∈ G, ξ ∈ g and x ∈...

In a recent article [Gi99], V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra g. He also obtained several results for more general pairs. As a next step, we considered in [Pa99] almost principal nilpotent pairs. The aim of this paper is to make a contribution to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pai...

A normal complex algebraic variety X is called a symplectic variety (cf. [Be]) if its regular locus Xreg admits a holomorphic symplectic 2-form ω such that it extends to a holomorphic 2-form on a resolution f : X̃ → X. Affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex Lie algebra (cf. [CM]), Slodowy slices to nilpotent orbits (cf...

With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W -algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian reduction. We observe that W is the invariant algebra for an action of a reductive group G with Lie algebra g on a quantized symplectic affine variety and ...

Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relativel...