Representations of affine superalgebras and mock theta functions III

We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers type modifier is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $mathfrak{g}$ is non-degenerate, to that for the subalgebra $ mathfrak{g}^! $ of $ mathfrak{g}, $ orthogonal to a maximal isotropic set of roots of $ mathfrak{g}. $

Cyclic elements in semisimple Lie algebras

A theory of cyclic elements in semisimple Lie algebras is developed. It is applied to an explicit construction of regular elements in Weyl groups.

On classification of Poisson vertex algebras

We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators).

On exceptional nilpotents in semisimple Lie algebras

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.

On rationality of W-algebras

We study the problem of classification of triples ($mathfrak{g}, f, k$), where $mathfrak{g}$ is a simple Lie algebra, $f$ its nilpotent element and $k in CC$, for which the simple $W$-algebra $W_k (mathfrak{g}, f)$ is rational.