Classification and Ideal Lattices of Leavitt Path Algebras
Yvan Grinspan, Seth Yoo
TL;DR
This work analyzes how graph structure governs the ideal lattice of Leavitt path algebras $L_K(E)$ for finite graphs by providing a new proof characterizing graded and non-graded ideals and by developing the concept of $\lambda$-reducible ideals. It introduces a canonical generating set $\Lambda(I)$ for such ideals and a divisibility criterion governing containment, enabling a reduction to a graph-theoretic classification. In the two-vertex case, it yields a complete nine-class classification of $\lambda$-reducible ideal lattices and counts graphs up to isomorphism with a given number of edges. The results sharpen the link between graph combinatorics and noncommutative algebra, and they lay groundwork for extending the classification to larger graphs and for exploring Morita-equivalence implications.
Abstract
Leavitt path algebras are free algebras subject to relations induced by directed graphs. This paper investigates the ideals of Leavitt path algebras, with an emphasis on the relationship between graph-theoretic properties of a directed graph and the ideals of the associated Leavitt path algebra. We begin by presenting a new proof of a fundamental result characterizing graded and non-graded ideals of a Leavitt path algebra using a condition on the number of closed paths at each vertex in its directed graph. Appealing to this result, we then classify the Leavitt path algebras of directed graphs with two vertices up to isomorphism and determine all possible lattice structures of a class of well-behaved ideals possessed by such algebras.