Alladi Ramakrishnan Hall

Cohn-Leavitt path algebras of bi-separated graphs

Raj Mohan

ISI Bangalore

We say a unital ring R has Invariant Basis Number (IBN) in case, for each pair of positive integers i,j if the left R-modules $R^i$ and $R^j$ are isomorphic, then i=j. The first examples of non IBN rings were studied by William Leavitt in 1950's and he defined (what is now known as) Leavitt algebras which are 'universal' with non IBN property. Later Paul Cohn studied stronger properties which imply IBN and defined Cohn algebras. In early 1970's George Bergman generalized Cohn's and Leavitt's constructions to solve `realization problem for hereditary rings' in positive.

In early 2000's the algebraic structures arising from quivers known as Leavitt path algebras (LPA for short) were initially developed as a first step towards a solution for `realization problem for von Neumann regular rings'. LPA are defined as quotients of quiver algebras and they generalize a particular class of Leavitt algebras.
During the intervening decade various attempts were made to generalize all Leavitt algebras and Cohn algebras as quotients of quiver algebras. The goal of this talk is to introduce the notion of Leavitt path algebras and their various generalizations as well as to present a unified framework to study these generalizations called Cohn-Leavitt path algebras of bi-separated graphs.

(Joint work with B.N. Suhas)