#### Algebra & Combinatorics Seminar

##### Venue: LH-1, Mathematics Department

The study of Leavitt path algebras has two primary sources, the work of W.G. Leavitt in the early 1960’s on the module type of a ring, and the work by Kumjian, Pask, and Raeburn in the 1990’s on Cuntz-Krieger graph $C^*$-algebras. Given a directed graph $\Gamma$ and a field $F$, the Leavitt path algebra $L_F(\Gamma)$ is an $F$-algebra essentially built from the directed paths in the graph $\Gamma$. Reasonable necessary and sufficient graph-theoretic conditions for two directed graphs to have isomorphic Leavitt path algebras do not seem to be known. In this talk I will discuss a recent construction, due to Zhengpan Wang and myself, of a semigroup $LI(\Gamma)$ associated with a directed graph $\Gamma$, that we call the Leavitt inverse semigroup of $\Gamma$. The semigroup $LI(\Gamma)$ is closely related to the corresponding Leavitt path algebra $L_F(\Gamma)$ and the graph inverse semigroup $I(\Gamma)$ of $\Gamma$. Leavitt inverse semigroups provide a certain amount of structural information about Leavitt path algebras. For example if $LI(\Gamma) \cong LI(\Delta)$, then $L_F(\Gamma) \cong L_F(\Delta)$, but the converse is false. I will discuss some topological aspects of the structure of graph inverse semigroups and Leavitt inverse semigroups: in particular, I will provide necessary and sufficient conditions for two graphs $\Gamma$ and $\Delta$ to have isomorphic Leavitt inverse semigroups.

This is joint work with Zhengpan Wang, Southwest University, Chongqing, China.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 16 Oct 2019