Takiff superalgebras and Conformal Field Theory

TL;DR

This paper explores non-semisimple extensions of Lie superalgebras, demonstrating their connection to conformal field theory through a generalized Sugawara construction and analyzing their representation theory.

Contribution

It introduces Takiff superalgebras of affine Kac-Moody type and establishes their link to conformal field theory via a Virasoro algebra construction.

Findings

Classified irreducible modules of the extended superalgebra gl(1|1)

Computed supercharacters and verified a continuum Verlinde formula

Deduced partial results on fusion rings and indecomposable structures

Abstract

A class of non-semisimple extensions of Lie superalgebras is studied. They are obtained by adjoining to the superalgebra its adjoint representation as an abelian ideal. When the superalgebra is of affine Kac-Moody type, a generalisation of Sugawara's construction is shown to give rise to a copy of the Virasoro algebra and so, presumably, to a conformal field theory. Evidence for this is detailed for the extension of the affinisation of the superalgebra gl(1|1): Its highest weight irreducible modules are classified using spectral flow, the irreducible supercharacters are computed and a continuum version of the Verlinde formula is verified to give non-negative integer structure coefficients. Interpreting these coefficients as those of the Grothendieck ring of fusion, partial results on the true fusion ring and its indecomposable structures are deduced.