Representations of Quivers over F1

2010/06/04 by Matthew Szczesny, Szczesny, Matthew
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1006.0912

openalex publication_date 2010/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra \HQ of \RepQ, which behaves in some ways like the specialization at q=1 of the Hall algebra of \onRep(\Q, F_q). We prove the existence of a Hopf algebra homomorphism of ρ': \U(\n₊) → \HQ, from the enveloping algebra of the nilpotent part \n₊ of the Kac-Moody algebra with Dynkin diagram \Q - 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.

Citations

• Non-Additive Prolegomena (to any future Arithmetic that will be able to present itself as a Geometry)

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