Current Trends in Algebra

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Coordinateurs administratifs et scientifiques

Programme scientifique

Cours 2: "introductory course - 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.", María Alejandra ALVAREZ (University of Antofagasta, Chile)

Cours 3: "introductory course - Quandle theory is now a well-established theory. It lies at the intersection of low dimensional topology and non-associative algebraic structures. We will review the motivations of the theory, then discuss an algebraic approach based on non-associative structures and a topological approach based on Knot theory. The lectures will be self-contained.", Mohamed ELHAMDADI (University of South Florida, United States)

Cours 4: "introductory course - In this course, we offer an introduction to partial actions and their associated skew group rings, focusing on the interactions between the (topological) dynamical aspects of partial actions and the algebraic properties of the associated skew rings. We also show how to realize (one-sided) shift algebras as partial skew group rings and use this realization to describe conjugacy between shift spaces in terms of isomorphisms between the associated algebras.", Daniel GONÇALVES (Federal University of Santa Catarina, Brazil)

Cours 5: "introductory course - Axial algebras are a new class of non-associative algebra, introduced recently by Hall, Rehren and Shpectorov, which generalise some properties found in vertex operator algebras and the Griess algebra and have a strong connection to groups. Axial algebras are generated by axes which are idempotents whose adjoint action decomposes the algebra as a direct sum of eigenspaces. The multiplication of eigenvectors is controlled by a so-called fusion law. When this is graded, it leads naturally to a subgroup of automorphisms of the algebra called the Miyamoto group. The prototypical example is the Griess algebra which has the Monster sporadic group as its Miyamoto group. Other examples include a wide number of Jordan algebras and also Matsuo algebras, which are defined from $3$-transposition groups.", Pilar PÁEZ-GUILLÁN (University of Santiago de Compostela, Spain)

Cours 6: "advanced course - 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 $k \in \mathbb{N}$). In fact Leavitt established much more in the 1962 article: he showed that for any $n, k \in\mathbb{N}$ 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 $n \ge 2$, this universal ring is not realizable as a Leavitt path algebra. With this in mind, the notion of weighted Leavitt path algebras $L_K(E, w)$ 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 $n, k \in\mathbb{N}$) 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.", Roozbeh HAZRAT (Western Sydney University, Australia)

Cours 7: "advanced course - 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.", Samuel LOPES (University of Porto, Portugal)

Cours 8: "advanced course - The Gelfand-Kirillov dimension is a main tool to quantify the growth of noncommutative algebras. In this mini course, we will discuss the Gelfand-Kirillov dimension of associative algebras. Examples (eg. Ore extensions, Leavitt path algebras, generalized Weyl algebras, etc.) will be given to show the calculation and application of this dimension.", Xiangui ZHAO (Huizhou University, China)

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Date limite d'inscription : octobre 27, 2023