Twisted Demazure modules, fusion product decomposition and twisted Q--systems

TL;DR

This paper introduces a new family of finite-dimensional modules for twisted current algebras, simplifies their defining relations, and establishes their fusion product decompositions and connections to Demazure modules and twisted Q-systems.

Contribution

It provides a new family of modules, simplifies Demazure module relations, and links twisted Demazure modules to fusion products and twisted Q-systems.

Findings

Modules become isomorphic to Demazure modules at specific parameters.

Fusion product decomposition of twisted Demazure modules proved.

Twisted Demazure modules relate to graded modules of untwisted Demazure modules.

Abstract

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an $|R^+|$-tuple of partitions $\bxi=(\xi^{\alpha})_{\alpha\in R^+}$ satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of $\bxi$ these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the notion of fusion products for twisted modules, first defined in \cite{FL99} for untwisted modules, and use the simplified presentation to prove a fusion product decomposition of twisted Demazure modules. As a consequence we prove that twisted Demazure modules can be obtained by taking the associated graded modules of (untwisted) Demazure modules for simply-laced affine algebras. Furthermore we give a semi-infinite fusion product construction for the irreducible representations of twisted affine algebras. Finally, we prove that the twisted $Q$-sytem defined in \cite{HKOTT02} extends to a non-canonical short exact sequence of fusion products of twisted Demazure modules.