I will talk about our proof of the Kontsevich conjecture (1996) on non-commutative birational transformations. It deals with difficulties arising out of the fact that there are no canonical form for non-commutative rational expressions. Miraculous identities proved supposedly reflect some kind of non-commutative group actions.
Representation theory of Khovanov-Lauda-Rouquier (KLR) algebras in affine type A can be studied through the lens of Specht modules, associated with the cellular structure of cyclotomic KLR algebras, or through the lens of cuspidal modules, associated with categorified PBW bases for the quantum group of affine type A. Cuspidal ribbons provide a sort of combinatorial bridge between these approaches. I will describe some recent results on cuspidal ribbon tableaux, and some implications in the world of KLR representation theory, such as bounds on labels of simple factors of Specht modules, and the presentation of cuspidal modules.
Cluster algebras were invented by Fomin and Zelevinsky twenty years ago. Since then they have played an important role in a number of settings in combinatorics, geometry, representation theory and topology. We will introduce a notion of root of unity quantum cluster algebras which are PI algebras, and will show that they have large canonical central subalgebras isomorphic to the original cluster algebras. These are far reaching generalizations of the De Concini-Kac-Procesi central subalgebras that appear in the study of the irreducible representations of big quantum groups. We will describe a general theorem computing the discriminants of these algebras. In a special situation it yields a formula for the discriminants of the quantum unipotent cells at roots of unity associated to all symmetrizable Kac-Moody algebras.
We prove that in q-Weyl generators the multi-parameter quantizations of type An+ and Bn+ are quadratic-linear Koszul algebras.
Feigin homomorphisms map the 'upper triangular subalgebras' of quantum groups to some quantum (or twisted) polynomial algebras. They are important in the study of their skew fields of quotients. Several years ago, I gave a construction of these 'quantum upper triangular subalgebras' as subalgebras of quantum shuffle algebras. More recently, the construction of Feigin homomorphisms has been extended to the whole quantum shuffle algebras by D. Rupel, with a computational proof. I shall explain another, quite direct approach, stressing the universal property of the quantum shuffle algebra, and putting quantum polynomial algebras naturally in this framework. All necessary background will be recalled.
In 2012, Hernandez and Jimbo introduced a new tensor category of representations of a Borel subalgebra of a quantum loop algebra, and classified its simple objects. This category contains the finite-dimensional representations of the quantum loop algebra, together with some new infinite dimensional representations. The motivation of Hernandez and Jimbo came from mathematical physics, in particular from papers of Bazhanov et al. where some examples of these new representations were used to define analogues of Baxter’s Q-operators in conformal field theory. Recently, using this new category, Frenkel and Hernandez were able to prove a long-standing conjecture of Frenkel and Reshetikhin on the spectra of the transfer matrices of some quantum integrable systems associated with quantum loop algebras. In this talk, I will explain that the new category of Hernandez and Jimbo fits very well with cluster algebras. More precisely I will show that cluster structures occur naturally in its Grothendieck ring, and can be helpful in finding new interesting functional relations. This is a joint work with David Hernandez.
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