Related papers: W-algebras at the critical level

Let $\Sigma_{\infty, 1}$ be the inductive limit of compact oriented surfaces with one boundary component. We prove the center of the Goldman Lie algebra of the surface $\Sigma_{\infty,1}$ is spanned by the constant loop. A similar statement…

We study the representations of the W-algebra W(g) associated to an arbitrary finite-dimensional simple Lie algebra g via the quantized Drinfeld-Sokolov reductions. The characters of irreducible representations with non-degenerate highest…

We produce explicit generators of the classical W-algebras associated with the principal nilpotents in the simple Lie algebras of all classical types and in the exceptional Lie algebra of type $G_2$. The generators are given by determinant…

Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the…

We show that quantum Casimir W-algebras truncate at degenerate values of the central charge c to a smaller algebra if the rank is high enough: Choosing a suitable parametrization of the central charge in terms of the rank of the underlying…

We classify the finite dimensional irreducible representations with integral central character of finite $W$-algebras $U(\mathfrak g,e)$ associated to standard Levi nilpotent orbits in classical Lie algebras of types B and C. This…

We consider the centre of the affine vertex algebra at the critical level associated with the orthosymplectic Lie superalgebra. It is well-known that the centre is a commutative superalgebra, and we construct a family of its elements in an…

We give a presentation for the finite W-algebra associated to a nilpotent matrix inside the general linear Lie algebra over C. In the special case that the nilpotent matrix consists of n Jordan blocks each of the same size l, the…

Let U(g,e) be the finite W-algebra associated with a nilpotent element e in a simple Lie algebra g and assume that e is induced from a nilpotent element e_0 in a Levi subalgebra l of g. We show that if the finite W-algebra U(l,e_0) has a…

Let $H(R, \phi, z)$ be a generalized Weyl algebra associated with a ring $R$, its central element $z\in Z(R)$ and an automorphism $\phi,$ such that for some $l \geq 1$, $\phi^l(z)-z$ is nilpotent and $(z,\phi^i(z))=R$ for all $0<i<l$. We…

This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Maffei's theorem relating…

We prove that any unitary highest weight module over a universal minimal quantum affine $W$-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine…

In this note we study simple modules for a reduced enveloping algebra U_chi(g) in the critical case when chi element of g^* is ``nilpotent''. Some dimension formulas computed by Jantzen suggest modified versions of Weyl's dimension formula,…

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple of g. Denote by $\mathfrak{g}^{e}$ the…

Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra, and let $\mu \in \mathfrak{g}^{*}$. In the first part of the paper, the relation is discussed between the derived algebra of the stabilizer of $\mu$ and the set of…

We show that the higher rank triplet W-algebra is simple when p is bigger than or equal to h-1. Furthermore, we show that the Feigin-Tipunin's module over the higher rank triplet W-algebra introduced in arXiv:1002.5047 is simple if the…

Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie algebra…

We classify all pairs (m,e), where m is a positive integer and e is a nilpotent element of a semisimple Lie algebra, which arise in the classification of simple rational W-algebras.

We consider the lower central filtration of the free associative algebra $A_n$ with $n$ generators as a Lie algebra. We consider the associated graded Lie algebra. It is shown that this Lie algebra has a huge center which belongs to the…

Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a simple Lie algebra $\frak g$ over an algebraically closed field of characteristic 0. We show that if an analogue of the Gelfand-Kirillov conjecture holds for such a…