Fusion products and toroidal algebras

Abstract

We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we construct in different ways objects in that category and prove them to be isomorphic. As a consequence we obtain generators and relations for certain types of fusion products including the $N$--fold fusion product of $V(\lambda )$. This result shows that the fusion product of these types is independent of the chosen parameters, proving a special case of a conjecture by Feigin and Loktev. Moreover, we prove a conjecture by Chari, Fourier and Sagaki on truncated Weyl modules for certain classes of dominant integral weights and show that they are realizable as fusion products. In the last section we consider the case $\mathfrak{g}={\mathfrak{sl}}_2$ and compute a PBW type basis for truncated Weyl modules of the associated current algebra.

References in corpus (6)

• Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas
• Weyl modules for the hyperspecial current algebra
• The PBW Filtration, Demazure Modules and Toroidal Current Algebras
• Twisted Demazure modules, fusion product decomposition and twisted Q--systems
• Demazure Modules, Chari-Venkatesh Modules and Fusion Products
• A Steinberg type decomposition theorem for higher level Demazure modules

Cited by in corpus (6)

• Generalized Demazure modules and fusion products
• Wild Local Structures of Automorphic Lie Algebras
• Graded decompositions of fusion products in rank two
• Bases for local Weyl modules for the hyper and truncated current ${\mathfrak{sl}}_2$-algebras
• Spectral Characters of a class of Integrable Representations of Toroidal Lie Algebras
• On Truncated Weyl Modules