邓邦明^1，*， 阿布都吾甫^2，， 庞雅丽^3，

（  1、北京师范大学；       2、新疆大学；       3、华东师范大学；  ）

Bangming Deng^1,*， Abdukadir Obul^2,， Yali Pang^3,

（  1、Beijing Normal University；       2、Xinjiang University；       3、East China Normal University； ）

Abstract： Let ${mathbb F}_q$ be the finite field of $q$ elements and $k$ be its algebraic closure. Let $Q$ be a quiver with automorphism $sigma$. In this survey we focus on the study of modules over the ${mathbb F}_q$-algebra ${frak A}(Q,sigma;q)$ associated with the pair $(Q,sigma)$ in terms of $F$-stable representations of $Q$ over $k$, and we also discuss several polynomials obtained by counting numbers of indecomposable ${frak A}(Q,sigma;q)$-modules. In case $Q$ is a tame quiver, we present the formula for the number of isoclasses of indecomposable ${frak A}(Q,sigma;q)$-modules with a fixed dimension vector.
Keywords： quiver with automorphism，hereditary algebra, representation

 

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