On graded irreducible representations of Leavitt path algebras

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• 作者：Roozbeh Hazrat^a ; ^{r.hazrat@WesternSydney.edu.au" class="auth_mail" title="E-mail the corresponding author} ; Kulumani M. Rangaswamy^b ; ^{krangasw@uccs.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16D70
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：450
• 期：Complete
• 页码：458-486
• 全文大小：572 K

文摘

Using the E-algebraic branching systems, various graded irreducible representations of a Leavitt path K-algebra L of a directed graph E are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by the Laurent vertices or the line points leading to a detailed description of the graded socle of L  . Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducible representation V_[p]$V_{[p]}$ induced by infinite paths tail-equivalent to an infinite path p (we call this a Chen simple module) is graded if and only if p is an irrational path. We also show that every one-sided ideal of L is graded if and only if the graph E contains no cycles. Supplementing the theorem of one of the co-authors that every Leavitt path algebra L is graded von Neumann regular, we show that L is graded self-injective if and only if L is a graded semi-simple algebra, made up of matrix rings of arbitrary size over the field K   or the graded field K[xⁿ,x⁻ⁿ]$K[x^n,x^{-n}]$ where n∈N$n\in \mathbb{N}$.