A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras

Year 2014, Volume: 43 Issue: 6, 943 - 951, 01.12.2014

A. Koç S Esin İ. Güloğlu M. Kanuni

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. We shall consider the direct sum of finite dimensional
full matrix rings over a field K. All such finite dimensional semisimple
algebras arise as finite dimensional Leavitt path algebras. For this
specific finite dimensional semisimple algebra A over a field K, we define
a uniquely determined specific graph - called a truncated tree associated
with A - whose Leavitt path algebra is isomorphic to A. We define an
algebraic invariant κ(A) for A and count the number of isomorphism
classes of Leavitt path algebras with the same fixed value of κ(A).
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 we also determine the number of distinct Leavitt path algebras of
line graphs with a given number of vertices. 

Keywords

Finite dimensional semisimple algebra , Leavitt path algebra , Truncated trees , Line graphs

References

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