Program
Courses (5 x 1h talks)
Jocelyn Vilela and Alfilgen Sebandal
"Introduction to Path Algebras and Graph Monoids"
In this course, we introduce two algebraic structures obtained from a directed graph, namely the path algebras and graph monoids. We shall see basic examples of path algebras and with additional relations, the construction of the Leavitt path algebra; of how the addition of the relations bring more structure to the algebra. On the framework of the monoids, we shall be focused on the talented monoids and of how the Leavitt path algebra relates directly to this monoid.
Giang Nam Tran
"Solvable and simple Lie algebras arising from Leavitt path algebras"
I will talk about the following topics:
1. Lie solvable Leavitt path algebras: In this talk, we give necessary and sufficient conditions on the graph E and the field K for which the Leavitt path algebra L_K(E) is solvable. Consequently, we obtain a complete description of Lie nilpotent Leavitt path algebras. Moreover, we compute the solvable index of a Lie solvable Leavitt path algebra.
2. Simple Lie Algebras arising from Steinberg algebras of Hausdorff ample groupoids: In this talk, we identify the fields K and Hasdorff ample groupoids G for which the simple Steinberg algebra A_K(G) yields a simple Lie algebra [A_K(G), A_K(G)]. Consequently, we obtain easily computable necessary and sufficient conditions to determine which Lie algebras of the form [L_K(E), L_K(E)] are simple, where E is a arbitrary graph and the Leavitt path algebra $L_K(E)$ is simple, which generalize Abrams and Mesyan's earlier results.
María Alejandra Alvarez
"Free Lie algebras"
The aim of this course is to provide an introduction to free Lie algebras. We will review the basics on free Lie algebras starting with enveloping algebras and the Poincaré-Birkhoff-Witt Theorem. We will study algebraic properties and some of the constructions of bases for free Lie algebras.
Roozbeh Hazrat
"Weighted Leavitt path algebras and chips firing"
The notion of sandpile models or chips firing encapsulates a process by which objects may spread and evolve along a grid. The models were conceived in 1987 in the seminal paper by Bak, Tang and Wiesenfeld as an example of self-organized criticality, or the tendency of physical systems to organise themselves without any input from outside the system, toward critical but barely stable states. The models were used to describe phenomena such as forest fires, traffic jams, stock market fluctuations, etc. In subsequent major work (1990), Dhar championed the use of an abelian group naturally associated to a sandpile model as an invariant which was shown to capture many properties of the model. This abelian group is paired with a naturally-arising monoid that arises from the grid.
In a different realm, the notion of Leavitt path algebras L_K(E) associated to directed graphs E, with coefficients in a field K, were introduced in 2005. These are a generalisation of algebras (denoted by L_K(1, 1+k)) introduced by William Leavitt in 1962; these ``Leavitt algebras'' arise as the universal ring of type (1, 1+k) (i.e., A_1 \cong A_{1+k} as right A-modules, where k in N). In fact Leavitt established much more in the 1962 article: he showed that for any n, k in N, there is 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>1, 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) associated to weighted graphs (E, w) were introduced in 2011. The weighted Leavitt path algebras L_K(E, w) provide a natural (but extremely broad) context in which all of Leavitt’s algebras (corresponding to any pair n, k in N) can be realized as a specific example.
The study of the commutative monoid V(B) of isomorphism classes of finitely generated projective right modules of a unital ring B (with operation \oplus) goes back to the work of Grothendieck and Serre. For a Leavitt path algebra L_K(E), the monoid V(L_K (E)) has received substantial attention since the introduction of the topic. Furthermore, the monoid V(L_K (E, w)) has been described in later works. In this course we’ll show how the notions of sandpile monoids and weighted Leavitt path algebras are quite naturally related, via the V-monoid. This relationship allows us to associate an algebra, a ``sandpile algebra'', to the theory of sandpile models, thereby opening up an avenue by which to investigate sandpile models via the structure of the sandpile algebras, and vice versa. The sandpile algebras provide a natural (but significantly more focused) context in which all of Leavitt’s algebras can be realized as a specific example.
Samuel Lopes
"From symmetry to non-associativity and back"
Groups are the abstract concept underlying symmetry and are the basis for the fundamental notion of invariant. In Hermann Weyl’s words, ``all geometric facts are expressed by the vanishing of invariants'', a paradigm of the latter coming from the action of a group on a (non-commutative) algebra.
The aim of this course is to show: firstly, how almost all ``important'' groups arise as symmetry groups; secondly, how the study of the symmetry groups of combinatorial, algebraic, geometric, differential or topological structures can be tackled with the use of derivations; thirdly, how non-associative algebras arise in this context; lastly, how to compute and relate all of the above. We will try to cover applications and computational aspects of the theory.
Stephane Launois
"Quantum algebras and beyond"
The aim of this course is to provide an introduction to quantum algebras (mainly quantised coordinate rings) and their semiclassical limits. We will introduce recent algorithmic methods to study the representation theory of these algebras and will explain their link with various combinatorial objects (including totally nonnegative matrices).
Research talks (1h talks)
Iryna Kashuba
"A moment map for Jordan algebras"
We study the variety of complex n-dimensional Jordan algebras Jor(n) using techniques from Geometric Invariant Theory. The variety Jor(n) is a rational representation of a complex reductive linear algebraic group GL(n, C), however the action of GL(n, C) on Jor(n) is very unpleasant from the point of view of invariant theory since every point of Jor(n) is unstable, which makes it very difficult to study the quotient space Jor\(n)/GL(n, C). Nevertheless, Kirwan and Ness have showed that the moment map associated to this action can be used to study the orbit space of the set of unstable vectors. We use it to construct a Morse-type stratification of Jor(n) into finitely many invariant smooth subvarieties, with respect to the energy functional associated to the canonical moment map.
This is joint work with Claudio Gorodski and Maria Eugenia Martin.