Gus Lehrer: Invariant theory-classical quantum and super III

(07 Juin 2022/ June 07, 2022) Série de conférences de la Chaire Aisenstadt.

Atelier/Workshop Algèbres non commutatives, théorie des représentations et fonctions spéciales/ Non-commutative algebras, representation theory and special functions 23 Mai- 10 Juin/May 23- June 1

Gus Lehrer: The first and second fundamental theorems of invariant theory respectively describe a set of generators, and a complete set of relations among these generators, for the space XU of invariants of a group, Lie algebra, associative algebra, or some other algebraic structure U, acting linearly on a space X. The subject has a very rich history, going back at least to Gauss.
In recent times, there has been significant progress, even in the classical cases of tensor representations of classical groups. This has been partly through the introduction of diagrammatic methods (which really go back to Brauer in 1937), the notion of quantum deformations, which has brought braid groups into the picture, the theory of cellular algebras, which are well adapted to the study of non-semisimple deformations of semisimple representations and developments in
the calculus of ribbons and more general tangles.
These are all applied in a categorical setting, when the invariants concerned can be interpreted in several different ways. I will explain some of the classical background, the new ideas, including algebraic geometric methods and Brauer, Temperley-Lieb and other diagram categories with a braid-like origins.
In the third (and last) lecture, I shall describe some of the recent progress, particularly in the case of super-groups, where the second fundamental theorem is now quite well understood.

The content of the three lectures will be roughly as follows:
- Lecture 1: Classical theory for GL(C^n)-the first and second fundamental theorems; Schur-Weyl duality. The case of the classical groups O(n) and Sp(2n). Brauer diagrams; the Brauer category; other categories of tangles.
- Lecture 2: Quantum groups and R-matrices; braid group action and strongly multiplicity free modules. Quantum sl2 and the Temperley-Lieb algebra. Higher representations of quantum sl2. Cellular algebras. Quantum versions of the second fundamental theorem.
- Lecture 3: Supersymmetry: super spaces and super Lie algebras. The Grassmann algebra. Adaptation of a geometric idea of Atiyah; the second fundamental theorem for the orthosymplectic super group.