Wreath products by a Leavitt path algebra

Abstract

We introduce ring theoretic constructions that are similar to the construction of wreath product of groups. In particular, for a given graph $\Gamma=(V,E)$ and an associate algebra $A,$ we construct an algebra $B=A\, wr\, L(\Gamma)$ with the following property: $B$ has an ideal $I$,which consists of (possibly infinite) matrices over $A$, $B/I\cong L(\Gamma)$, the Leavitt path algebra of the graph $\Gamma$. \medskip \par Let $W\subset V$ be a hereditary saturated subset of the set of vertices [1], $\Gamma(W)=(W,E(W,W))$ is the restriction of the graph $\Gamma$ to $W$, $\Gamma/W$ is the quotient graph [1]. Then $L(\Gamma)\cong L(W)$ wr $L(\Gamma/W)$.