相关论文: Degenerate Odd Poisson Bracket on Grassmann Variab…
Transposed Poisson structures on complex Galilean type Lie algebras and superalgebras are described. It was proven that all principal Galilean Lie algebras do not have non-trivial $\frac{1}{2}$-derivations and as it follows they do not…
We introduce a search algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. To demonstrate the algorithm, we look at two classes of examples: (1) the model filiform Lie algebras and…
We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by…
The supermultiplet model, based on the reduction chain $\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)$, is revisited through the lens of commutants within universal enveloping algebras of Lie algebras. From this…
Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson…
In this paper, we study Lie superalgebras of $2\times 2$ matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional…
We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived…
Investigating a recent positive solution of a conjecture of Grunewald and O'Halloran for complex finite dimensional nilpotent Lie algebras, we are in the position to find results of existence and uniqueness for the construction of complex…
The three dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress and Miller have proved that in the case of non degenerate…
The aim of this paper is to present a short introduction to supergeometry on pure odd supermanifolds. (Pseudo)differential forms, Cartan calculus (DeRham differential, Lie derivative, "inner" product), metric, inner product, Killing's…
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three…
We establish a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gel'fand-Dikii and the fermionic theory in our earlier work. Certain interesting new algebraic structures are found in connection…
We consider the pairs of general weakly non-local Poisson brackets of Hydrodynamic Type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that under the requirement of non-degeneracy of the corresponding "first"…
A commutative Poisson subalgebra of the Poisson algebra of polynomials on the Lie algebra of n x n matrices over ${\Bbb C}$ is introduced which is the Poisson analogue of the Gelfand-Zeitlin subalgebra of the universal enveloping algebra.…
The algebraic and geometric classification of all complex $3$-dimensional transposed Poisson algebras is obtained. Also, we discuss strong special $3$-dimensional transposed Poisson algebras.
A generalization $\mathfrak{Gal}_{\ell}(p,q)$ of the conformal Galilei algebra $\mathfrak{g}_{\ell}(d)$ with Levi subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(p,q)$ is introduced and a virtual copy of the latter…
We introduce the notion of Poisson superbialgebra as an analogue of Drinfeld's Lie superbialgebras. We extend various known constructions dealing with representations on Lie superbialgebras to Poisson superbialgebras. We introduce the…
It is proved that given a divergence operator on the structural sheaf of graded commutative algebras of a supermanifold, it is possible to construct a generating operator for the Krashil'shchik-Schouten bracket. This is a particular case of…
It is shown that the fourth order real self-adjoint difference operator on the Tivalent Tree admits nontrivial deformations preserving one energy level and therefore defines a nontrinial hierarhy of the completely integrable nonlinear…
We introduce the PBW degeneration for basic classical Lie superalgebras and construct for all type I, $\mathfrak{osp}(1,2n)$ and exceptional Lie superalgebras new monomial bases. These bases are parametrized by lattice points in convex…