The socle of a Leavitt path algebra.
Suppose that \(E\) is a directed countable graph such that for any vertex \(v\) there exist finitely many edges entering \(v\). For any edge \(e\) denote by \(s(e)\) its source and by \(r(e)\) its end. If \(k\) is a field then one can define the Leavitt path \(k\)-algebra \(L(E)\) as the associative algebra with a unit element generated by the set of orthogonal idempotents which are vertices \(v\) from \(E\) and by double copies \(\{e,e^*\}\) of the set of edges in \(E\). These generators are connected by defining relations \[ s(e)e=er(e)=e,\quad r(e)e^*=e^*s(e)= e^*,\quad e^*e'=\delta_{e,e'}r(e),\quad v=\sum_{\{e\mid s(e)=v\}}ee^*. \] Denote by \(P_1(E)\) the set of all vertices in \(E\) such that \(s^{-1}(v)\) contains at most one element and \(v\) is contained in a cycle at some vertex from a tree of \(v\). It is shown that if \(u\) is a vertex then \(L(E)u\) is a minimal left ideal in \(L(E)\) if and only if \(u\in P_1(E)\). Any minimal left ideal has this form. Hence the socle of \(L(E)\) is the ideal generated by \(P_1(E)\).
scientific article