A Combinatorial Discussion on Finite Dimensional Leavitt Path Algebras

Yıl 2014, Cilt: 43 Sayı: 6, 943 - 951, 01.12.2014

A. Koç S Esin İ. Güloğlu M. Kanuni

Öz

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. 

Anahtar Kelimeler

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

Kaynakça

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