Invariant ideals in Leavitt path algebras

Abstract

It is known that the ideals of a Leavitt path algebra $L_K(E)$ generated by $P_l(E)$, by $P_{\text{c}}(E)$, or by $P_{ec}(E)$ are invariant under isomorphism. Though the ideal generated by $P_{b^{\infty }}(E)$ is not invariant we find its “natural” replacement (which is indeed invariant): the one generated by the vertices of $P_{b_p^{\infty }}$ (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the $DCC$ topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if $H$ is a hereditary saturated subset of vertices providing an invariant ideal, its exterior $\text{ext}(H)$ in the $DCC$ topology of $E^0$ generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance $P_l$, etc.). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain ${\text{Soc}}^{(1)}()\subseteq {\text{Soc}}^{(2)}()\subseteq \cdots$, all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative.