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Abstract: We define and study the category $\RepQ$ of representations of a quiver in$\VFun$ - the category of vector spaces -over $\Fun$-. $\RepQ$ is an$\Fun$-linear category possessing kernels, co-kernels, and direct sums.Moreover, $\RepQ$ satisfies analogues of the Jordan-H\-older and Krull-Schmidttheorems. We are thus able to define the Hall algebra $\HQ$ of $\RepQ$, whichbehaves in some ways like the specialization at $q=1$ of the Hall algebra of$\on{Rep}\Q, \mathbf{F} q$. We prove the existence of a Hopf algebrahomomorphism of $ ho-: \U + ightarrow \HQ$, from the enveloping algebraof the nilpotent part $ +$ of the Kac-Moody algebra with Dynkin diagram$\bar{\Q}$ - the underlying unoriented graph of $\Q$. We study $ ho-$ when$\Q$ is the Jordan quiver, a quiver of type $A$, the cyclic quiver, and a treerespectively.

Author: Matthew Szczesny


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