How do you recover a group from the automorphisms of the forgetful functor?

  • Thread starter Jim Kata
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In summary, recovering a group from the automorphisms of the forgetful functor involves using the natural isomorphism between the group and its group of automorphisms, as well as the underlying set of the group. This process allows for the reconstruction of the original group from its automorphisms, providing a useful tool in algebraic structures and category theory.
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Jim Kata
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Ok, I doubt anyone on here will know this, but given a neutral tannakain category there is a bijection between this category and the representations of some group (with some adjectives). I'm not sure how to show that, don't care though. But, to recover the group from the category of representations you look at the automorphisms of the forgetful functor from the category of representations of the the group to the vector space created by the representations of the group. How exactly do you recover the group from the automorphisms of the forgetful functor? Illustrate this with an example.
 
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FAQ: How do you recover a group from the automorphisms of the forgetful functor?

What is Tannaka Krein duality?

Tannaka Krein duality is a mathematical concept that relates the representation theory of a group to the algebraic structure of its associated category. It is a generalization of the well-known Pontryagin duality for locally compact abelian groups.

What is the significance of Tannaka Krein duality?

Tannaka Krein duality has important applications in many areas of mathematics, including algebraic geometry, harmonic analysis, and quantum field theory. It also provides a powerful tool for studying the structure and properties of groups and their representations.

How does Tannaka Krein duality differ from other duality theorems?

Unlike other duality theorems, Tannaka Krein duality involves a category rather than a group or algebraic structure. It also allows for the reconstruction of the group from its category of representations, rather than just its dual.

What is the connection between Tannaka Krein duality and quantum groups?

Tannaka Krein duality plays a crucial role in the study of quantum groups, which are non-commutative generalizations of Lie groups. It provides a framework for understanding the relationship between the representation theory of quantum groups and their underlying algebraic structures.

How is Tannaka Krein duality related to Tannakian categories?

Tannaka Krein duality is closely related to Tannakian categories, which are categories that satisfy certain properties similar to those required for Tannaka Krein duality. In fact, the Tannaka Krein duality theorem can be seen as a generalization of the Tannakian reconstruction theorem for algebraic groups.

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