Page "Tannaka–Krein duality" Paragraph 2

Duality theorems of Tannaka and Krein describe the converse passage from the category Π ( G ) back to the group G, allowing one to recover the group from its category of representations.

Moreover, they in effect completely characterize all categories that can arise from a group in this fashion.

Alexander Grothendieck later showed that by a similar process, Tannaka duality can be extended to the case of algebraic groups: see tannakian category.

Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physicists.

A generalization of Tannaka – Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to, quantum supergroups, quantum groupoids and their dual quantum algebroids.