Related papers: On $\mathcal{O}$-operators on modules over Lie alg…
The class of $\D$-locally nilpotent algebras (introduced in the paper) is a wide generalization of the algebras of differential operators on commutative algebras. Examples includes all the rings $\CD (A)$ of differential operators on…
The purpose of this paper is to develop a deformation theory controlled by pre-Lie algebras with divided powers over a ring of positive characteristic. We show that every differential graded pre-Lie algebra with divided powers comes with…
All coboundary Lie bialgebras and their corresponding Poisson--Lie structures are constructed for the oscillator algebra generated by $\{\aa,\ap,\am,\bb\}$. Quantum oscillator algebras are derived from these bialgebras by using the…
The authors define a Category $\mathcal{O}$ for any quasi-reductive Lie superalgebra $\mathfrak{g}$ with respect to a triangular decomposition. This much needed approach unifies many important constructions in the existing literature in a…
Let $\FRAK{g}$ be a classical simple Lie superalgebra. To every nilpotent orbit $\cal O$ in $\FRAK{g}_0$ we associate a Clifford algebra over the field of rational functions on $\cal O$. We find the rank, $k(\cal O)$ of the bilinear form…
We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C*-algebra may be regarded as a result of a quantization procedure.…
The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson-Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic…
In this paper we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras respectively to the alternative setting. Constructions in terms bimodules, splitting of…
Let ${\cal S}(\mathcal{H})$ denote the set of all self-adjoint operators (not necessarily bounded) on a Hilbert space $\mathcal{H}$, which is the set of all physical quantities on a quantum system $\mathcal{H}$. We introduce a binary…
The notions of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras are introduced. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. The notion of Manin triple of transposed Hom-Poisson…
This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the…
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…
An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of…
Weighted Rota-Baxter operators on associative algebras are closely related to modified Yang-Baxter equations, splitting of algebras, weighted infinitesimal bialgebras, and play an important role in mathematical physics. For any $\lambda \in…
This is a companion article to my papers on Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebras gl(m|n) (much revised!) and q(n). The goal is to develop the general theory of tilting modules for Lie superalgebras,…
In this paper, first we introduce the notion of reflections on quadratic Rota-Baxter Lie algebras of weight $\lambda$, and show that they give rise to solutions of the classical reflection equation for the corresponding triangular Lie…
We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras naturally compatible with a reductive decomposition, we extend the description of the moduli space of…
We study the algebraic structure of the Poisson algebra P(O) of polynomials on a coadjoint orbit O of a semisimple Lie algebra. We prove that P(O) splits into a direct sum of its center and its derived ideal. We also show that P(O) is…
Lie conformal algebras are useful tools for studying vertex operator algebras and their representations. In this paper, we establish close relations between Poisson conformal algebras and representations of Lie conformal algebras. We also…
An averaging operator on an associative algebra $A$ is an algebraic abstraction of the time average operator on the space of real-valued functions defined in time-space. In this paper, we consider relative averaging operators on a bimodule…