Algebra Learning Seminar

Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo

“ELI5: Leavitt Path Algebras”

Abstract

If E is a directed graph, you can construct a ring L(E) generated by the vertices and edges of E subject to some relations. This is called a Leavitt path algebra. These rings were introduced around 2004, generalizing an earlier construction of Leavitt from the 80’s. L(E) has a very rich algebraic structure, being a locally unital, Z-graded, semiprimitive ring.

Changing the graph E results in changing the algebraic structure of L(E), and as of today this is a predictable process in many cases. For example, there are necessary-and-sufficient geometric conditions on E so that L(E) is finite-dimensional, simple, prime, and so on. There are also some “moves” you can perform on E which result in Morita equivalent Leavitt path algebras. But the converse is currently unknown — if two Leavitt path algebras are Morita equivalent, how can you relate their underlying graphs?

In this talk I hope to give an overview of the theory of Leavitt path algebras, including known results and some open questions. As the title suggests, you should bring your five year-old.

MC 5479