Mathematics Colloquia and Seminars

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Description

The theory of Kac algebras provides a unified framework for both group algebras and their duals. In finite dimension this notion coincides with that of $C^*$-Hopf algebras.

Those algebras play an important role in the theory of inclusion of factors; indeed, any irreducible, finite index depth 2 inclusion of factors is obtained as fixed point set under the action of some finite dimensional Kac algebra. There furthermore is a Galois-like correspondance between the lattice of intermediate factors and the lattice of coideal subalgebras of the Kac algebra (the analogue of the lattice of subgroups of a group).

In 1998 Leonid Vainerman constructed the two first infinite families of non trivial (that are neither group algebras nor dual thereof) finite dimensional Kac algebras by deformation of the group algebras of the dihedral groups $D_{2n}$ (resp. the quaternion groups $Q_{2n}$).

In this talk, we present a detailed study of the structure of those two families. In particular, we describe the full lattice of coidealgebras in small dimension, and parts of it for all $n$. We derive the principal graphs of certain inclusions of factors, and reciprocally we use classification results on inclusions in some proofs. We finally give Kac algebra isomorphism and autoduality results and conjectures.

As the first interesting examples are of dimension 12 or more, calculations are quickly impractical by hand. Computer exploration therefore turned out to be an essential guide in this research. Our main purpose being to show how examples of Kac algebras can be studied in practice, most results are illustrated by actual computations that led to their conjecture, hinted to the proofs, or in some cases were used in the proofs.

The talk is self-contained and should not require any particular prerequisite.