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.

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Last updated: 06 Aug 2024