On the representations of Leavitt path algebras

Gonçalves, Daniel; Royer, Danilo

doi:10.1016/j.jalgebra.2011.02.034

Cited by 38 publications

(61 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Given an arbitrary directed graph E, Gonçalves and Royer defined in [16] and [17] a branching system using a measure space (X, µ) and indicated a method of constructing a large number of representations of the graph C *algebra C * (E) in the space of bounded linear operators on L 2 (X, µ). As an algebraic analogue of this theory, they defined the concept of an E-algebraic branching system in [18] to construct various representations of Leavitt path algebras (which are algebraic analogues of graph C*-algebras). Following this, X.W.…”

Section: Introductionmentioning

confidence: 99%

On graded irreducible representations of Leavitt path algebras

Hazrat

Rangaswamy

2016

Journal of Algebra

View full text Add to dashboard Cite

Using the E-algebraic branching systems, various graded irreducible representations of a Leavitt path K-algebra L of a directed graph E are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by the Laurent vertices or the line points leading to a detailed description of the graded socle of L. Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducble representation V [p] induced by infinite paths tail-equivalent to an infinite path p (we call this a Chen simple module) is graded if and only if p is an irrational path. We also show that every one-sided ideal of L is graded if and only if the graph E contains no cycles. Supplementing the theorem of one of the co-authors that every Leavitt path algebra L is graded von Neumann regular, we show that L is graded self-injective if and only if L is a graded semi-simple algebra, made up of matrix rings of arbitrary size over the field K or the graded field K[x n , x −n ] where n ∈ N.

show abstract

Section: Introductionmentioning

confidence: 99%

On graded irreducible representations of Leavitt path algebras

Hazrat

Rangaswamy

2016

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…In the previous work [6,7,8,9,10], motivated by the connection between wavelet theory and representations of the Cuntz-Krieger algebra, see [4], the study of representations of graph algebras via branching systems has been initiated and developed. Branching systems arise in many areas of mathematics such as the Perron-Frobenius operator from ergodic theory (see [7,9]).…”

Section: Introductionmentioning

confidence: 99%

Faithful Representations of Graph Algebras via Branching Systems

Li¹,

Royer²

2016

Can. math. bull.

Self Cite

View full text Add to dashboard Cite

Abstract. We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz-Krieger uniqueness theorem for graph algebras by means of branching systems.

show abstract

“…Graph algebras have been studied both in pure algebra and in operator theory (see, e.g., [7], [11], and [9]). Similarly, branching systems arise in neighboring disciplines, such as random walks, symbolic dynamics, and scientific computing (see, e.g., [10], [15], [5], [3], and [18]).…”

Section: Introductionmentioning

confidence: 99%