Analogues of Leavitt path algebras for higher-rank graphs

Directed graphs and their higher-rank analogues provide an intuitive framework to study a class of C*-algebras which we call graph algebras. The theory of graph algebras has been developed by a number of researchers and also influenced other branches of mathematics: Leavitt path algebras and Cohn path algebras, to name just two. Leavitt path algebras for directed graphs were developed independently by two groups of mathematicians using different approaches. One group, which consists of Ara, Goodearl and Pardo, was motivated to give an algebraic framework of graph algebras. Meanwhile, the motivation of the other group, which consists of Abrams and Aranda Pino, is to generalise Leavitt's algebras, in which the name Leavitt comes from. Later, Abrams and now with Mesyan introduced the notion of Cohn path algebras for directed graphs. Interestingly, both Leavitt path algebras and Cohn path algebras for directed graphs can be viewed as algebraic analogues of C*-algebras of directed graphs. In 2013, Aranda Pino, J. Clark, an Huef and Raeburn introduced a higher-rank version of Leavitt path algebras which we call Kumjian-Pask algebras. At their first appearance, Kumjian-Pask algebras were only defined for row-finite higher-rank graphs with no sources. Clark, Flynn and an Huef later extended the coverage by also considering locally convex row-finite higher-rank graphs. On the other hand, Cohn path algebras for higher rank graphs still remained a mystery. This thesis has two main goals. The first aim is to introduce Kumjian-Pask algebras for a class of higher-rank graphs called finitely-aligned higher-rank graphs. This type of higher-rank graph covers both row-finite higher-rank graphs with no sources and locally convex row-finite higher-rank graphs. Therefore, we give a generalisation of the existing Kumjian-Pask algebras. We also establish the graded uniqueness theorem and the Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras of finitely-aligned higher-rank graphs. The second aim is to introduce a higher-rank analogue of Cohn path algebras. We then study the relationship between Kumjian-Pask algebras and Cohn path algebras and use this to investigate properties of Cohn path algebras. Finally, we establish a uniqueness theorem for Cohn path algebras