Recent Trends in Non-Commutative Algebras

Recent developments in non-commutative algebra include parallel theories of graph C*-algebras and Leavitt path algebras. In this research school, we mainly focus on theories of Leavitt path algebras and their connections with other areas of mathematics.

Leavitt path algebras arose out of the work of Leavitt (1962) and the graph C*-algebras trace their origin to the work of Cuntz (1977). These algebras were introduced about ten years ago by Abrams – Aranda Pino, and Ara – Moreno – Pardo in two separate papers. It is an algebraic analogue of C*-algebras, a subject which originated in the late 1990s in the work of Kumjian – Pask – Raeburn – Renault, and has been pioneered by the Australian researchers, including Bates, an Huef, Pask, Raeburn, Sims, Szymanski et.al.

Although the two areas — Leavitt path algebras and graph C*-algebras — sprang from independent works, they are closely related. There are many striking parallels between the two subjects, and many of the key results first arising in either one of the two fields turn out to have an exact analogue in the other. In terms of research, the main goal of the school is to explore K-theoretic techniques, and the graded structures of both the Leavitt path algebras and the graph C* algebras as tools for understanding the similarities and the differences between the two areas.

Apart from the above mentioned topics, we will also discuss some recent developments in the direction of homological algebra and deformation theory. Algebraic deformation theory is primarily concerned with the interplay between homological algebra and the perturbations of algebraic structures. On hopes to explore some future developments of the theory of Leavitt path algebras in this direction.

There will be some introductory courses on Lie algebras, Basic K-theory and graph C*-algebras. One each day of the research school there will be some tutorial sessions for the beginners.