Invariant ideals in Leavitt path algebras.

Abstract

It is known that the ideals of a Leavitt path algebra LK (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 “natural” replacement (which is indeed invariant): the one generated by the vertices of Pb∞ p (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 ext(H) in the DCC topology of E0 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)( ) ⊆ Soc(2)( ) ⊆ · · · 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