Representing k-graphs as Matrix Algebras

Abstract

Abstract. For any commutative unital ring R and finitely aligned k-graph Λ with |Λ| < ∞ without cycles, we can realise Kumjian-Pask algebra ‫ܲܭ‬ ோ (Λ) as a direct sum of of matrix algebra over some vertices v with properties ‫}ݒ{‬ = ‫ݒ‬Λ , i.e: ⊕ ௩Λୀ{௩} ‫ܯ‬ |Λ௩| (ܴ) . When there is only a single vertex ‫ݒ‬ ∈ Λ such that ‫}ݒ{‬ = ‫ݒ‬Λ, we can realise the Kumjian-Pask algebra as the matrix algebra ‫ܯ‬ |Λ௩| (ܴ). Hence the matrix algebra ‫ܯ‬ |௩Λ| (ܴ) 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.