Related papers: W-algebras at the critical level

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…

We study the specializations $\mathcal{L}_{g,n}^\epsilon$ at roots of unity $\epsilon$ of odd order of the graph algebras, associated to a simply-connected complex semi-simple algebraic group $G$ and a compact oriented surface…

The equivariant $\mathcal{W}$-algebra of a simple Lie algebra $\mathfrak{g}$ is a BRST reduction of the algebra of chiral differential operators on the Lie group of $\mathfrak{g}$. We construct a family of vertex algebras $A[\mathfrak{g},…

Let $\mathbb K$ be a field of characteristic zero, $A$ an integral domain over $\mathbb K$ with the field of fractions $R = \text{Frac}(A),$ and $\text{Der}_{\mathbb{K}}A$ the Lie algebra of all $\mathbb K$-derivations on $A$. Let…

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic 0 and O be a nilpotent orbit in g. Then Orb is a symplectic algebraic variety and one can ask whether it is possible to quantize $\Orb$ (in an…

In this paper, we extend Feigin-Frenkel duality at the critical level to complex rank by identifying two seemingly unrelated constructions in complex rank. On the affine side, we interpolate Molev's construction of higher Segal-Sugawara…

We present a quantization of an isomorphism of Mirkovi\'c and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in $\mathfrak{gl}_N$ , to a slice between spherical Schubert varieties in the affine…

In this note we present an explicit realization of the affine vertex algebra $V^{cri}(\frak{gl}(1 \vert 1)) $ inside of the tensor product $F\otimes M$ where $F$ is a fermionic verex algebra and $M$ is a commutative vertex algebra. This…

In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras) can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum…

Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of…

In the first part of this paper, we discuss the classical W-algebra $\mathcal{W}(\mathfrak{g}, F)$ associated with a Lie superalgebra $\mathfrak{g}$ and the nilpotent element $F$ in an $\mathfrak{sl}_2$-triple. We find a generating set of…

With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W-algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian…

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy…

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;\alpha)$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group…

We give a geometric proof of inverse Hamiltonian reduction for all affine W-algebras in type A at generic level, a certain embedding of the affine W-algebra corresponding to an arbitrary nilpotent in $\mathfrak{gl}_N$ into that…

We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld-Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on…

An explicit realization of the affine Lie algebra \hat{sl}_2(C) at the critical level is constructed using a mixture of bosons and parafermions. Subsequently a representation of the associated Lepowsky-Wilson Z-algebra is given on a space…

In this paper, we introduce and study two cyclotomic level maps defined respectively on the set of nilpotent orbits $\underline{\mathcal{N}}$ in a complex semi-simple Lie algebra $\mathfrak{g}$ and the set of conjugacy classes…

Let G be a semisimple algebraic group over an algebraically closed field of characteristic p>0, and let g be its Lie algebra. The crucial Lie algebra representations to understand are those associated with the reduced enveloping algebra…

Let g be a complex simple Lie algebra and e a nilpotent element in g. To a certain nilpotent subalgebra m attached to e, called an admissible subalgebra of g, we associate an endomorphism algebra H. When m is constructed from a good grading…