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 c ( E ), or by P ec ( E ) are invariant under isomorphism. Though the ideal generated by P b ∞ ( E ) is not invariant we find its “natural” replacement (which is indeed invariant): the one generated by the vertices of P b ∞ p (vertices with pure in-finite 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 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 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.