Related papers: Poisson orders on large quantum groups
To every Poisson algebraic variety X over an algebraically closed field of characteristic zero, we canonically attach a right D-module M(X) on X. If X is affine, solutions of M(X) in the space of algebraic distributions on X are Poisson…
Our first collection of results parametrize (filtered) actions of a quantum Borel $U_q(\mathfrak{b}) \subset U_q(\mathfrak{sl}_2)$ on the path algebra of an arbitrary (finite) quiver. When $q$ is a root of unity, we give necessary and…
Parabosonic $P_{B}^{(n)}$ and parafermionic $P_{F}^{(n)}$ algebras are described as quotients of the tensor algebras of suitably choosen vector spaces. Their (super-) Lie algebraic structure and consequently their (super-) Hopf structure is…
We give an exposition of graded and microformal geometry, and the language of $Q$-manifolds. $Q$-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in…
All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Its classification is fully performed by relating these PL groups with the corresponding Lie bialgebra…
A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a…
We study different algebraic and geometric properties of Heisenberg invariant Poisson polynomial quadratic algebras. We show that these algebras are unimodular. The elliptic Sklyanin-Odesskii-Feigin Poisson algebras $q_{n,k}(\mathcal E)$…
A.A. Kirillov introduced the family algebras in 2000. In this paper we study the noncommutative Poisson bracket P on the classical family algebra. We show that P is the first-order deformation from the classical family algebra to the…
We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley-Hamilton algebras in the sense of Procesi. We show that every…
Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional''…
A two-parameter quantum deformation of the affine Lie super algebra $osp(2|2)^{(2)}$ is introduced and studied in some detail. This algebra is the first example associated with nonsimply-laced and twisted root systems of a quantum current…
For a given finite dimensional Hopf algebra $H$ we describe the set of all equivalence classes of cocycle deformations of $H$ as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the…
In this work, we conduct a systematic study of Hamiltonian and quasi-Hamiltonian systems within the framework of nondecomposable generalized Poisson geometry. Our focus lies on the interplay between the algebraic structure of…
For a generalized Weyl Poisson algebra $A$, explicit sets of generators and defining relations are presented for its Poisson enveloping algebra $\CU (A)$. Simplicity criteria are given for the algebra $\CU (A)$ and algebra of Poisson…
The problem of introducing a dependence of elements of quantum group on classical parameters is considered. It is suggested to interpret a homomorphism from the algebra of functions on quantum group to the algebra of sections of a sheaf of…
In this paper, we introduce the notion of differential graded Poisson algebra and study its universal enveloping algebra. From any differential graded Poisson algebra $A$, we construct two isomorphic differential graded algebras: $A^e$ and…
Let $A=F[x,y]$ be the polynomial algebra on two variables $x,y$ over an algebraically closed field $F$ of characteristic zero. Under the Poisson bracket, $A$ is equipped with a natural Lie algebra structure. It is proven that the maximal…
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.
Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation…
The N-dimensional Cayley-Klein scheme allows the simultaneous description of $3^N$ geometries (symmetric orthogonal homogeneous spaces) by means of a set of Lie algebras depending on $N$ real parameters. We present here a quantum…