相关论文: On finite dimensional representations of finite W-…

We consider the finite W-superalgebras for a basic classical Lie superalgebra g associated with an even nilpotent element in g both over the field of complex numbers field and and over a filed of positive characteristic. We present the PBW…

We consider the finite $W$-superalgebra $U(\mathfrak{g_\bbf},e)$ for a basic Lie superalgebra ${\ggg}_\bbf=(\ggg_\bbf)_\bz+(\ggg_\bbf)_\bo$ associated with a nilpotent element $e\in (\ggg_\bbf)_{\bar0}$ both over the field of complex…

We consider the finite $W$-algebra $U(\g,e)$ associated to a nilpotent element $e \in \g$ in a simple complex Lie algebra $\g$ of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem, we verify a…

W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete…

Let $\frak g$ be a simple finite-dimensional Lie algebra over an algebraically closed field $\mathbb F$ of characteristic 0. We denote by $\operatorname{U}(\frak g)$ the universal enveloping algebra of $\frak g$. To any nilpotent element…

In this paper we formulate a conjecture about the minimal dimensional representations of the finite $W$-superalgebra $U(\mathfrak{g}_\bbc,e)$ over the field of complex numbers and demonstrate it with examples including all the cases of type…

A finite W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. In this survey we review recent developments in the representation theory of W-algebras. We emphasize various interactions…

Let $\mathfrak{g}=\mathfrak{gl}_{M|N}(\mathbb{k})$ be the general linear Lie superalgebra over an algebraically closed field $\mathbb{k}$ of characteristic zero. Fix an arbitrary even nilpotent element $e$ in $\mathfrak{g}$ and let…

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1}$ be a basic Lie superalgebra over $\mathbb{C}$, and $e$ a minimal nilpotent element in $\mathfrak{g}_{\bar 0}$. Set $W_\chi'$ to be the refined $W$-superalgebra associated with…

We prove that all finite W-algebras associated with nilpotent elements e in a complex semisimple Lie algebra g have finite-dimensional representations. In order to obtain this result we establish a connection between primitive ideals of…

$W$-algebras are certain algebraic structures associated to a finite dimensional Lie algebra $\mathfrak g$ and a nilpotent element $f$ via Hamiltonian reduction. In this note we give a review of a recent approach to the study of (classical…

A W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of these algebras. Under some conditions on a nilpotent element…

Finite W-algebras are certain associative algebras arising in Lie theory. Each W-algebra is constructed from a pair of a semisimple Lie algebra g (our base field is algebraically closed and of characteristic 0) and its nilpotent element e.…

We consider finite W-algebras U(g,e) associated to even multiplicity nilpotent elements in classical Lie algebras. We give a classification of finite dimensional irreducible U(g,e)-modules with integral central character in terms of the…

We classify the finite dimensional irreducible representations of rectangular finite $W$-algebras, i.e., the finite $W$-algebras $U(\mathfrak{g}, e)$ where $\mathfrak{g}$ is a symplectic or orthogonal Lie algebra and $e \in \mathfrak{g}$ is…

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(\mathfrak g,e)$…

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in \mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\mathcal E(\mathfrak…

Let $\mathtt{k}$ be an algebraically closed field of characteristic zero. Let $\mathfrak{g} $ be a finite dimensional classical simple Lie superalgebra over $\mathtt{k}$ or $\mathfrak{g} l(m,n)$. In the case that $\mathfrak{g} $ is a…

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As…

In recent years, the finite W-algebras associated to a semisimple Lie algebra and its nilpotent element have been studied intensively from different viewpoints. In this lecture series, we shall present some basic constructions, connections,…