WEEKLY ACTIVITIES

Time: 9:30 -- 11:00, April 12th, 2023.

Venue: Room 612, A6, Institute of Mathematics, VAST

Abstract: The notion of abelian sandpile models was introduced in 1987 by the physicists Bak, Tang and Wiesenfeld. The book of Bak "How Nature Works, Oxford University Press, Oxford, 1997." describes how events in nature apparently follow this type of behaviour. In 1990, Dhar systematically associated a finite abelian monoid/group to a sandpile model, now called a sandpile monoid/group, and championed the use of this monoid/group as an invariant which proved to capture many properties of the model. The abelian sandpile model was independently discovered in 1991 by combinatorialists Bjorner, Lovasz and Shor.

In a different realm, the notion of Leavitt path algebras L_K(E) of directed graphs E over a field K, was introduced in 2005. These are a generalisation of algebras L_K(1, 1 + k) introduced in 1962 by W. Leavitt as a universal ring A of type (1, 1 +k), i.e., A cong A^{1 + k} as right A-modules. In fact, Leavitt established much more in the 1962 article: he proved that for any n, k, there exists a universal ring A of type (n, n+k), denoted L_K(n, n+k), for which A^n cong A^{n+k} as right A-modules. When n ge 2, this universal ring is not realizable as a Leavitt path algebra. With this in mind, the notion of weighted Leavitt path algebras L_K(E, w of )weighted graphs (E, w) was introduced in 2011 by R. Hazrat. The weighted Leavitt path algebras provide a natural context in which all of Leavitt's algebras can be realised as a specific example.

In 2022, Abrams and Hazrat showed that the notions of sandpile monoids and weighted Leavitt path algebras are quite naturally related, via the non-stable K-theory. This relationship allows us to associate a weighted Leavitt path algebra to the theory of sandpile models, thereby opening up an avenue by which to investigate sandpile models via the structure of weighted Leavitt path algebras, and vice versa. This talk takes the first step to relate the structure of sandpile monoids to the algebraic structure of weighted Leavitt path algebras. This is joint work with Roozbeh Hazrat, Western Sydney University.