We define and study the category of representations of a quiver in —the category of vector spaces “over .” is an -linear category possessing kernels, co-kernels, and direct sums. Moreover, satisfies analogues of the Jordan–Hölder and Krull–Schmidt theorems. We are thus able to define the Hall algebra HQ of , which behaves in some ways like the specialization at q=1 of the Hall algebra of Rep(Q,Fq). We prove the existence of a Hopf algebra homomorphism of , from the enveloping algebra of the nilpotent part of the Kac–Moody algebra with Dynkin diagram —the underlying unoriented graph of Q. We study ρ′ when Q is the Jordan quiver, a quiver of type A, the cyclic quiver, and a tree, respectively.
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