Locally finite weighted Leavitt path algebras

Abstract

A group graded $K$-algebra $A=\underset{g\in G}{⨁}{A}_g$ is called "locally finite" if ${\dim }_K{A}_g<\infty$ for every $g\in G$. We characterise the weighted graphs $(E,w)$ for which the weighted Leavitt path algebra $L_K(E,w)$ is locally finite with respect to its standard grading. We also prove that the locally finite weighted Leavitt path algebras are precisely the Noetherian ones and that $L_K(E,w)$ is locally finite iff $(E,w)$ is finite and the Gelfand-Kirillov dimension of $L_K(E,w)$ equals $0$ or $1$. Further it is shown that a locally finite weighted Leavitt path algebra is isomorphic to a locally finite Leavitt path algebra and therefore is isomorphic to a finite direct sum of matrix algebras over $K$ and $K[X,X^{-1}]$.

References in corpus (1)

• Applications of normal forms for weighted Leavitt path algebras: simple rings and domains