Related papers: The PBW property for associative algebras as an in…
Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition…
We study the PBW-filtration on the highest weight representations $V(\la)$ of the Lie algebras of type ${\tt A}_{n}$ and ${\tt C}_{n}$. This filtration is induced by the standard degree filtration on $\U(\fn^-)$. In previous papers, the…
We study the PBW filtration on irreducible finite--dimensional representations for the Lie algebra of type $\tt B_n$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite--dimensional…
Inspired by the recent work of Chen-Sti\'enon-Xu on Atiyah classes associated to inclusions of Lie algebroids, we give a very simple criterium (in terms of those classes) for relative Poincar\'e-Birkhoff-Witt type results to hold. The tools…
The survey is devoted to associative $\Z_{\ge0}$-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such algebras…
A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a Poincar\'e-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions…
For any differential graded (DG for short) Poisson algebra $A$ given by generators and relations, we give a "formula" for computing the universal enveloping algebra $A^e$ of $A$. Moreover, we prove that $A^e$ has a Poincar\'e-Birkhoff-Witt…
We will introduce an $\mathbb{N}$-filtration on the negative part of a quantum group of type $A_n$, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation…
Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of…
In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of…
Poincare-Birkhoff-Witt (PBW) Theorems have attracted significant attention since the work of Drinfeld (1986), Lusztig (1989), and Etingof-Ginzburg (2002) on deformations of skew group algebras $H \ltimes {\rm Sym}(V)$, as well as for other…
This note has two purposes. First we establish that the map defined in [L, $\S 40.2.5$ (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincar\'e--Birkhoff--Witt (PBW) type for…
We study the PBW filtration on the irreducible highest weight representations of simple complex finite-dimensional Lie algebras. This filtration is induced by the standard degree filtration on the universal enveloping algebra. For certain…
In this paper we study the PBW filtration on irreducible integrable highest weight representations of affine Kac-Moody algebra $\gh$. The $n$-th space of this filtration is spanned with the vectors $x_1... x_s v$, where $x_i\in\gh$, $s\le…
This expository paper focuses on free Lie $K$-algebras and the basic PBW theorem. We argue in various ways that the basic PBW theorem is a quite close consequence of the Magnus-Witt theorems concerning free Lie algebras.
We construct a linear basis of a free GDN superalgebra over a field of characteristic $\neq 2$. As applications, we prove a PBW theorem, that is, any GDN superalgebra can be embedded into its universal enveloping commutative associative…
We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of…
The aim of this paper is to establish necessary and sufficient algorithmic conditions to guarantee that an algebra is actually a 3-dimensional skew polynomial algebra in the sense of Bell and Smith.
Let $\alpha$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $\mathbb{C}$. Then Kontsevich [K97] gives a formula for a quantization $f\star g$ of the algebra $S(V)^*$. We give a construction of an algebra with…
We prove in a very general framework several versions of the classical Poincar\'e-Birkhoff-Witt Theorem, which extend results from [BeGi, BrGa, CS, HvOZ, WW]. Applications and examples are discussed in the last part of the paper.