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.