Abstract: In this talk, we recall some challenging problems in algebra, such as the characterization problem of polynomial rings, the automorphism groups of certain algebras, and the Dixmier property of algebras. We then explain how the concept of Aut-stable subspaces can be used as a tool to approach these problems.
Abstract: A compatible Lie algebra is a vector space equipped with two Lie products such that any linear combination of them is also a Lie product. These algebras arose from the related class of compatible Poisson algebras in the context of mathematical physics and Hamiltonian mechanics. In this talk, we start by stating some basic definitions and results about compatible Lie algebras. We then present counterexamples to analogues of some of the most important theorems in Lie algebra theory, namely the theorems of Weyl and Levi, highlighting how compatible Lie algebras can behave very differently from Lie algebras. We then move on to studying the representation theory of a family of simple two-dimensional compatible Lie algebras. We construct a family of irreducible representations for each algebra of this family, and thereafter, we focus on one specific simple two-dimensional compatible Lie algebra in order to make the computations simpler and results easier to state and prove. In this setting, we prove a Clebsch-Gordan formula for the irreducible representations previously described, and we also exhibit a second family of representations, this time "breaking" Weyl's theorem (i.e., reducible and indecomposable representations over the field of complex numbers). Time permitting, we finish by discussing the failure of further central results from Lie algebra theory in this broader context, including the characterization of semisimple algebras and the Whitehead Lemmas.
Abstract: In this talk, we will discuss non-weight modules over the Lie algebra sl(2). More precisely, we focus on modules that are free of finite rank over the universal enveloping algebra U(h) of a Cartan subalgebra h of sl(2). In particular, we will present a new family of simple U(h)-free modules of rank 2. The talk is based on joint work with K. Nguyen and K. Zhao.
Abstract: Quasigroupoids and weak Hopf quasigroups are non-associative generalizations of groupoids and weak Hopf algebras. In this talk, we will show that the category of finite quasigroupoids is equivalent to the one of pointed cosemisimple weak Hopf quasigroups over a given field K. As a consequence, we obtain a categorical equivalence between the categories of quasigroups, in the sense of Klim and Majid (i.e., loops with the inverse property), and the category of pointed cosemisimple Hopf quasigroups over K. On the other hand, in this talk, we introduce the notion of exact factorization of a quasigroupoid and the notion of a matched pair of quasigroupoids with a common base. We prove that if (A, H) is a matched pair of quasigroupoids, it is possible to construct a new quasigroupoid called the double cross product of A and H. Moreover, we show that if a quasigroupoid B admits an exact factorization, there exists a matched pair of quasigroupoids (A, H) and an isomorphism of quasigroupoids between the double cross product of A and H and B. Finally, we show that every matched pair of quasigroupoids (A, H) induces, thanks to the quasigroupoid magma construction, a pair (K[A], K[H]) of weak Hopf quasigroups and a double crossed product of weak Hopf quasigroups isomorphic as weak Hopf quasigroups to the quasigroupoid magma of the double cross product groupoid of A and H.
Abstract: In this talk, we discuss the problem of automatic additivity for multiplicative maps between rings, following the classical results of Martindale and Jodeit-Lam. We present a complete description of Jordan multiplicative self-maps on full matrix algebras M_n(F), where F is a field of characteristic not equal to 2. Without imposing additional assumptions, we show that such maps are either constant (and equal to a fixed idempotent), or additive (and hence Jordan monomorphisms). This result is joint work with Ilja Gogić.