1 Feb~7 Mei 2010: A Course on Leavitt path algebras 

MA 6211 Advanced Topic in Algebra

• Prof.Gonzalo Aranda Pino (Universitas Malaga, Spanyol) (Feb 1 – Feb 26, 2010) 

• Dr. Intan Muchtadi (March 1 -  May 7, 2010) 

every Tuesday 11.00-12.40 and every Thursday 09.00-10.40 

This course is, in part, an adaptation of Enrique Pardo and Mercedes Siles Molina’s notes of the course “Algebras and Graphs” that was held at the University of Málaga during Winter 2009 (and will be held again in Winter 2010) as part of the Interdepartmental Master’s Degree in Mathematics. 

Leavitt path algebras are a specific type of path K-algebras associated to a graph E, modulo some relations, and are denoted by L(E). They can be considered, on the one hand, as natural generalizations of Leavitt algebras L(1,m) of type (1,m), introduced and investigated by Leavitt in order to give examples of algebras not satisfying the IBN property (i.e., of algebras that can have bases of different cardinals). On the other hand, they are the algebraic version of Cuntz-Krieger graph C*-algebras, a class of algebras intensively investigated by analysts for more than two decades. 

Besides the classical Leavitt algebras L(1,m), many other well-known algebras can be realized as the Leavitt path algebra of a graph, some of those include matrix algebras of finite or infinite size, Laurent polynomials, Toeplitz algebras, etc. Further, constructions such as direct sums, direct limits and combinations of all of the above can be also obtained. 

In this sense Leavitt path algebras provide a pictorial, and hence very convenient and useful, representation of abstract rings and algebras and thus the focus of this course is try to give an introduction to the subject by characterizing some of the most basic properties of L(E) in terms of graph theoretic properties of E such as finite-dimensionality, simplicity and purely infinite simplicity. 

REFERENCES: 

[1] G. Abrams, G. Aranda Pino, “The Leavitt path algebra of a graph”, J. Algebra 293 (2005), 319-334. 

[2] G. Abrams, G. Aranda Pino, M. Siles Molina, “Finite-dimensional Leavitt path algebras”, J. Pure Appl. Algebra 209 (3) (2007), 753–762. 

[3] P. Ara, M.A. Moreno, E. Pardo, “Nonstable K-theory for graph algebras”, Algebras & Representation Theory, 10 (2007), 157-178. 

[4] I. Raeburn, “Graph algebras”. CBMS Regional Conference Series in Mathematics, 103, American Mathematical Society, Providence, 2005. ISBN 0-8218-3660-9.