Abstract:To each symmetrizable Cartan matrix, we associate a finite free EI category. We prove that the corresponding category algebra is isomorphic to the algebra defined by Geiss-Leclerc-Schr\"{o}er (we call it a GLS's algebra for convenient), which is associated to another symmetrizable Cartan matrix. In certain cases, the algebra isomorphism provides an algebraic enrichment of the well-known correspondence between symmetrizable Cartan matrices and graphs with automorphisms. Let $(\Delta,\sigma)$ be a finite acyclic quiver $\Delta$ with an admissible automorphism $\sigma$ such that $\sigma$ has order $p^m$, where $p$ is a prime number and $m>0$. Let $k$ be a field with characteristic $p$. We prove that the skew group algebra $k\Delta\otimes_k k\langle\sigma\rangle$ is Morita equivalent to a GLS's algebra associated with a Cartan matrix $C$. In this case, we prove that $\tau$-locally free modules of the skew group algebra are exactly indecomposable induced modules, and the Morita equivalence of algebras provides an algebraic enrichment of the well-known correspondence between the $\sigma $-orbits of the positive roots of $\Delta$ and the positive roots of the semisimple complex Lie algebras associated with $C$. This is ongoing joint work with Xiao-Wu Chen.