The kumjian-pask algebras of finite finitely aligned k-graphs without cycle as modules

In general, every algebra can be viewed as a module over itself. In this paper, we analyze the structure of Kumjian-Pask algebras of finite finitely aligned k-graph without cycles via module theory.


Introduction
For a directed graph E, it can be constructed, in general, the corresponding graph algebra.Actually, graph algebra is a C*-algebra satisfying some specific conditions.One of the most important notion in the construction of the algebra is the Cuntz-Krieger relation.Many important results on this topic have been published by researchers.One may refer to [1] for the detailed elaboration about it.
In 2005, Gene Abrams and Aranda Pino [2] introduced the Leavitt path algebra for a directed graph, which can be viewed as an algebraic analogue of the graph algebra.Since then, the study on this area goes on to obtain a complete description about this algebra.
For many reason, the concept directed graph is extended to the so-called k-graph which is the higher rank version of it.For a k-graph Λ, one may interested in the corresponding algebra and obtains many important results.Many researchers have studied this topic for the cases row-finite without sources, locally convex, and finitely aligned k-graph [3].Some of them concern in the algebraic analogue of it.Aranda Pino... [4] introduced this algebra and called it Kumjian-Pask Algebra, for the row-finite Λ without sources.This study is followed by L.O.Clark et.al [5], for the locally convex case.In [6], L.O.Clark and Pangalela published their results for finitely aligned kgraph.
In 2016, Larki studied the simplicity of Kumjian-Pask algebras and gave a sufficient condition for the algebra to be purely infinite.In particular, Larki obtained a characterization for a finite-dimensional Kumjian-Pask Algebra [7].

Preliminaries
We write N for the set of natural numbers, and let kN.The symbol N k means the set of k-tuples of non-negative integers.For m,nN k , m  n means mi  ni, for all 1  i  k, and m ∨ n means their coordinate-wise maximum.
Note that we can regard N k as a category with one object.
The set of objects (vertex) in  is denoted by 0  , and can be identified as the identity morphism at the object.Accordingly, the set of paths which is not a vertex is denoted by the vertices s(λ), r(λ) are called the source, the range, respectively, of  .
Here are some notations will be used throughout the paper.For E ⊂ Λ, v ∈ Λ 0 , set, For example, a directed graph E is actually a 1-graph.The corresponding degree map is the length function l : E 1 → N. It is clear that l satisfies the factorization property.
Let Λ be a k-graph.If vΛ n is finite for each vertex v and n ∈ N k , then Λ is called row-finite.The vertex v is called source in the case vΛ n = ∅ for some n ∈ N k .The k-graph Λ is said to be locally-convex if for each v ∈ Λ 0 , 1 ≤ i,j ≤ k with ij  , λ ∈ vΛ ei , µ ∈ vΛ ej , the sets s(µ)Λ ei , s(λ)Λ ej are nonempty.One may realize that a finitely aligned k-graph is more general than the rowfinite and the locally-convex ones.Now, we turn to the concept of Kumjian-Pask algebra.Assume that Λ is a finitely aligned k-graph, and R is a commutative ring with identity.For a path λ ∈ Λ, the ghost path λ * ∈ Λ is defined as that of path satisfying (KP3) Sλ * Sµ =  (ρ,θ)∈ Λmin (λ,µ) Sρ Sτ * for all λ,µ ∈ Λ; (KP4) Π λ∈ E (S r(E) − S λ S λ * ) = 0 for all E ∈ FE(Λ).

Method
The structure of algebra has a deep connection with that of ring and module.In this paper we use ring and module theory to analyze the structure of the Kumjian-Pask algebra.

Discussion
One may see that the structure of a Kumjian-Pask algebra is more complicated then the other algebras, even for the 2-graph case.But for the finite case without cycles, the algebra can be viewed as a direct sum of matrices spaces.The following is the result of Larki.Theorem 4.1.Let Λ be a finitely aligned k-graph and R be a unital commutative ring.Then KP R (Λ) is a finitedimensional R-algebra if and only if Λ is finite and contains no cycles.In particular, in the case we have Note that if Λ is finite and without cycles, there must exists, at least, a vertex v satisfying vΛ = {v}.Hence, we have the fact that for a single vertex v satisfying vΛ = {v} the KP R (Λ) is no more than the ring Mn (R) for some n ∈ N, together with the multiplication in the left by elements in R. Based on this fact, we shall use the related results in ring theory to study the KPR (Λ).
Here are some definitions and the corresponding facts.Note that C is simple, hence as a ring, Mn(C) is simple, and therefore, Mn(C) is primitive.
Assume that Λ is a finite and finitely aligned k-graph without cycles.By the Theorem of Larki, the corresponding KP C (Λ) is finite dimensional.Moreover, we can view KP C (Λ) as a C * -algebra.So, in general, the facts about the C * -algebra also apply in KP C (Λ).
If we assume that Λ has a source v satisfying {v} = vΛ, then we have the fact KPC(Λ)  M|Λv|(C).In addition, the simplicity of M|Λv|(C) implies the simplicity of KPC(Λ).Hence, there exists no non-trivial ideal in the KPC(Λ).

Definition 4 . 2 .Proposition 4 . 6 .
A ring R with identity is simple if it has no nontrivial proper ideals.Definition 4.3.[8, Definition 1.7].An R-module is semisimple if it is a direct sum of simple R-modules.Definition 4.4.[9, Definition 4.1] A ring R is primitive if it has faithful irreducible representation.Definition 4.5.[9, Definition 4.1] A ring R is semiprimitive if for any a 6= 0 in R there exists an irreducible representation ρ such that ρ(a)  0. Any simple ring is primitive.