UC Riverside

In 2021, authors Biswal, Chari, Shereen, and Wand showed for the type $A$ current algebra that under suitable conditions on pairs of dominant integral weights $(\nu,\lambda)$ that the fusion product of a local Weyl module with a level 2 Demazure module $D(2,\lambda) * W_{\loc}(\nu)$ is isomorphic to the module $M(\nu,\lambda)$ first introduced by Wand in his 2015 PhD thesis. The work of this thesis is to remove the condition on $(\nu,\lambda)$ by defining new modules $N(\nu,\lambda,\gamma)$ which in the case when $\gamma=0$ are isomorphic to $M(\nu,\lambda)$. We then construct short exact sequences that show under suitable conditions on $(\nu,\lambda,\gamma)$ that the modules $N(\nu,\lambda,\gamma)$ are isomorphic to the fusion product $W_{\loc}(\nu)*D(2,\lambda)*D(2,\gamma)$. In particular, when $\gamma=0$, this gives us the desired isomorphism for $M(\nu,\lambda)$ for all pairs $(\nu,\lambda)$. This allows us to define the fusion product of a local Weyl module and a level 2 Demazure module via generators and relations.