A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras

TL;DR

This paper characterizes finite dimensional semisimple Leavitt path algebras over a field by associating them with specific graphs called truncated trees, introduces an invariant to classify them, and analyzes their dimensions and counts.

Contribution

It introduces a unique graph construction for finite dimensional semisimple Leavitt path algebras and defines an invariant to classify and count these algebras.

Findings

Defined a truncated tree associated with each algebra

Introduced an invariant {ta}(A) for classification

Determined maximum and minimum dimensions for given vertex counts

Abstract

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph - which we name as a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant {\kappa}(A) for A and count the number of isomorphism classes of Leavitt path algebras with {\kappa}(A)=n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.