Representing k-graphs as Matrix Algebras

Abstract

For any commutative unital ring R and finitely aligned k-graph Λ with |Λ| < ∞ without cycles, we can realise Kumjian-Pask algebra KPR (Λ) as a direct sum of of matrix algebra over some vertices v with properties {ν} = νΛ, i.e: ⊕νΛ={ν} M|Λv|(R). When there is only a single vertex ν ∈ Λ° such that {ν} = νΛ, we can realise the Kumjian-Pask algebra as the matrix algebra M|ΛV|(R). Hence the matrix algebra M|vΛ|(R) can be regarded as a representation of the k-graph Λ. In this talk we will figure out the relation between finitely aligned k-graph and matrix algebra.