Invariants ideals in Leavitt Path algebras

TL;DR

This paper identifies new invariant ideals in Leavitt path algebras, introduces topological and categorical methods to construct them, and generalizes existing invariant ideal concepts.

Contribution

It finds a natural invariant replacement for the non-invariant ideal generated by (E), introduces the ps topology, and develops categorical procedures for constructing invariant ideals.

Findings

The ideal generated by vertices with pure infinite bifurcations is invariant.

The ps topology relates to algebraic annihilators and helps generate invariant ideals.

Hereditary and saturated functors produce invariant ideals, generalizing the socle chain.

Abstract

It is known that the ideals of a Leavitt path algebra $L_K(E)$ generated by $\Pl(E)$, by $\Pc(E)$ or by $\Pec(E)$ are invariant under isomorphism. Though the ideal generated by $\Pb(E)$ is not invariant we find its \lq\lq natural\rq\rq\ replacement (which is indeed invariant): the one generated by the vertices of $\Pbp$ (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 $\tops$ 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 $\ext(H)$ in the $\tops$ 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 $\Pl$, 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 $\soc^{(1)}(\ )\subset\soc^{(2)}(\ )\subset\cdots$ 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.