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We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog and…
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables…
By applying the Hamiltonian reduction scheme we recover the R-matrix of the trigonometric and elliptic Calogero-Moser system.
For any cocommutative Hopf algebra $H$ and a left $H$-module $V$, we construct an operad $\mathcal{P}^{cl}_H(V)$, which in the special case when $H$ is the algebra of polynomials in one variable reduces to the classical operad…
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related:…
In this paper, we consider compatible Hom-associative algebras as a twisted version of compatible associative algebras. Compatible Hom-associative algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie…
We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology…
A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra…
We consider nonholonomic geodesic flows of left-invariant metrics and left-invariant nonintegrable distributions on compact connected Lie groups. The equations of geodesic flows are reduced to the Euler-Poincare-Suslov equations on the…
The $\lambda$-differential operators and modified $\lambda$-differential operators are generalizations of classical differential operators. This paper introduces the notions of $\lambda$-differential Poisson ($\lambda$-DP for short)…
We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS$_{3}$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis…
We introduce the notion of G-algebroid, generalising both Lie and Courant algebroids, as well as the algebroids used in $E_{n(n)}\times\mathbb{R}^+$ exceptional generalised geometry for $n\in\{3,\dots,6\}$. Focusing on the exceptional case,…
We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a setting for the calculation of a Gr\"obner--Shirshov basis in a module over an associative conformal algebra and apply this…
In this paper we introduce the concept of morphisms of K\"ahler-Poisson algebras and study their algebraic properties. In particular, we find conditions, in terms of the metric, for two algebras to be isomorphic, and we introduce direct…
Poisson superpair is a pair of Poisson superalgebra structures on a super commutative associative algebra, whose any linear combination is also a Poisson superalgebra structure. In this paper, we first construct certain linear and quadratic…
The connections between Euler's equations on central extensions of Lie algebras and Euler's equations on the original, extended algebras are described. A special infinite sequence of central extensions of nilpotent Lie algebras constructed…
We construct an action of the braid group B_N on the twisted quantized enveloping algebra U'_q(o_N) where the elements of B_N act as automorphisms. In the classical limit q -> 1 we recover the action of B_N on the polynomial functions on…
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe…
The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable…
Enveloping algebras of Hom-Lie and Hom-Leibniz algebras are constructed.