Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)

TL;DR

This paper explores the harmonic analysis and free field realizations of the Takiff supergroup of gl(1|1), revealing the structure of its modules and their implications for conformal field theories.

Contribution

It provides the first detailed harmonic analysis of the Takiff supergroup of gl(1|1) and constructs free field realizations for related conformal field theories.

Findings

Every simple module appears as a submodule of an infinite-dimensional indecomposable module

Constructed free field realizations for the associated conformal field theory

Identified the module structure and representation theory of the Takiff supergroup

Abstract

Takiff superalgebras are a family of non semi-simple Lie superalgebras that are believed to give rise to a rich structure of indecomposable representations of associated conformal field theories. We consider the Takiff superalgebra of gl(1|1), especially we perform harmonic analysis for the corresponding supergroup. We find that every simple module appears as submodule of an infinite-dimensional indecomposable but reducible module. We lift our results to two free field realizations for the corresponding conformal field theory and construct some modules.