69^th Algebra Day

Moufang sets, i.e., split BN-pairs (= Tits systems) of rank one, are known since the work of deMedts and Weiss to be intimitely connected with Jordan division rings. Following Segev and Tent, I will use this connection to translate a natural question about Moufang sets into an as yet unsolved problem for Jordan algebras. Partial solutions to the problem will be presented, and speculations about an answer in general will be offered.

Given the weight lattice Λ of a crystallographic root system with Weyl group W, the exponent of the W-action on Λ measures how far is the the ring of W-invariants of Sym^* (Λ)^W from being a polynomial ring in the generator set of ℕ[Λ]^W. In this talk we extend this notion to the case of a formal group ring. This is joint work with K. Zainoulline.

Hecke algebras are well-known, in part, for their joint interpretations as both algebraic and geometric objects. For example, the affine Hecke algebra can be constructed geometrically via K-theory. In this talk I will discuss a general approach to Hecke algebras using axioms of oriented cohomology theories and related objects called formal group algebras. This approach leads to a general framework which specializes to known geometric constructions of the Hecke algebra and related algebraic objects. This is joint work with J. Malagon-Lopez, A. Savage, and K. Zainoulline.

In this talk I will revisit the classical topic of polynomial equations (in one variable) and Tschirnhaus transformations. I will discuss 19th century theorems of Hermite, Joubert and Klein, recent results in this area, and several open problems.

We find an interesting category of representations of the three simple finitary Lie algebras. The modules in question are weight modules for every splitting Cartan subalgebra. We describe the injective modules in this category, and show that the category is anti-equivalent to the category of locally unitary finite-dimensional modules over a direct limit of finite-dimensional Koszul algebras. The talk is based on joint work with Ivan Penkov and Vera Serganova.