Representations of quivers over \(\mathbb F_{1}\) and Hall algebras (Q2889570)

This paper is concerned with the category \(\mathrm{Rep}(Q,{\mathbb F}_1)\) of representations of a given quiver \(Q\) on \({\mathbb F}_1\) vector spaces. The latter are defined to be pointed sets \((M,m_0)\) where morphisms are pointed maps \(f:(M,m_0)\to(N,n_0)\) which are injective outside the marked fibre \(f^{-1}(n_0)\). This category possesses kernels, cokernels and finite products and behaves in many ways similar to an abelian category, except that it is not abelian. In the paper [Far East J. Math. Sci. (FJMS) 70, No. 1, 1--46 (2012; Zbl 1271.18016)] by the reviewer, such categories have been named \textit{belian categories}. They allow for a version of homological algebra and the construction of a Hall algebra \(H_Q\).NEWLINENEWLINEIn this nicely writte paper analogues of the Jordan-Hölder and Krull-Schmidt theorems are proven and a Hopf-algebra homomorphism from the enveloping algebra of the nilpotent part of the Kac-Moody algebra with Dynkin diagram \(Q\) to the Hall algebra is constructed and its properties are studied in important examples.