Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 067, 24 pages      arXiv:1411.1072      https://doi.org/10.3842/SIGMA.2015.067

Harmonic Analysis and Free Field Realization of the Takiff Supergroup of ${\rm GL}(1|1)$

Andrei Babichenko a and Thomas Creutzig b
a) Department of Mathematics, Weizmann Institut, Rehovot, 76100, Israel
b) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received May 28, 2015, in final form August 01, 2015; Published online August 06, 2015

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 $\mathfrak{gl}(1\vert 1)$, 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.

Key words: logarithmic CFT; Harmonic analysis; free field realization.

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