Related papers: Finite W-superalgebras for basic classical Lie sup…
The main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple Lie algebra. The first one, very classical, is the Goldie rank of an ideal. The second one…
First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the…
We show that each integral infinitesimal block of parabolic category O (including singular ones) for a semi-simple Lie algebra can be realized as a full subcategory of a "thick" category O over a finite W-algebra for the same Lie algebra.…
Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basic classical Lie superalgebra over $\mathbb{C}$, and $e=e_{\theta}\in\mathfrak{g}_{\bar0}$ with $-\theta$ being a minimal root of $\mathfrak{g}$. Set $U(\mathfrak{g},e)$…
In this paper we present a systematic study of $W$ algebras from the Hamiltonian reduction point of view. The Drinfeld-Sokolov (DS) reduction scheme is generalized to arbitrary $sl_2$ embeddings thus showing that a large class of W algebras…
Recently it has been discovered that the W-algebras (orbifold of) WD_n can be defined even for negative integers n by an analytic continuation of their coupling constants. In this letter we shall argue that also the algebras WA_{-n-1} can…
In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely…
Two series of W-algebras with two generators are constructed from chiral vertex operators of a free field representation. If $c = 1 - 24k$, there exists a W(2,3k) algebra for k in $Z_{+}/2$ and a W(2,8k) algebra for k in $Z_{+}/4$. All…
We show that there are exactly two anti-involution $\sigma_{\pm}$ of the algebra of differential operators on the circle that are a multiple of $p(t\partial_t)$ preserving the principal gradation ($p\in\CC[x]$ non-constant). We classify the…
In this short note we confirm an analog of a conjecture of James Wiegold for finite dimensional nilpotent Lie algebras.
Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional''…
In this paper we extend general results obtained by V. Kac and J. Liberati, in "Unitary quasifinite representations of $W_\infty$", (Letters Math. Phys., 53 (2000), 11-27), for quasifinite highest weight representations of $\Z$-graded Lie…
After some definitions, we review in the first part of this talk the construction and classification of classical $W$ (super)algebras symmetries of Toda theories. The second part deals with more recently obtained properties. At first, we…
We survey a number of results regarding the representation theory of $W$-algebras and their connection with the resent development of the four dimensional $N=2$ superconformal field theories in physics.
We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of…
We classify real and complex infinite-dimensional narrow positively graded Lie algebras ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\mathfrak…
Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain…
We classify integrable bounded simple weight modules over classical Lie superalgebras at infinity. We also study the categories of such modules, and we prove that for most of the classical Lie superalgebras at infinity the respective…
The coefficient algebra of a finite-dimensional Lie algebra on a finite-dimensional representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We explore connections between…
For q generic or a primitive l-th root of unity, q-Witt algebras are described by means of q-divided power algebras. The structure of the universal q-central extension of the q-Witt algebra, the q-Virasoro algebra, is also determined. q-Lie…