These notes are from the February 23, 2016 lecture of 18.757, *Representations of Lie Algebras*, taught by Laura Rider.

Fix a field and let be a finite group. In this post we will show that one can reconstruct the group from the monoidal category of -modules (i.e. its -representations).

## 1. Hopf algebras

We won’t do anything with Hopf algebras *per se*, but it will be convenient to have the language.

Recall that an associative -algebra is a -vector space equipped with a map and (unit), satisfying some certain axioms.

Then a **-coalgebra** is a map

called comultiplication and counit respectively, which satisfy the dual axioms. See \url{https://en.wikipedia.org/wiki/Coalgebra}.

Now a **Hopf algebra** is a bialgebra over plus a so-called **antipode** . We require that the diagram

commutes.

Given a Hopf algebra **group-like** element in is an element of

**Exercise 1**

Show that is a group with multiplication and inversion .

Now the example

**Example 2** **(Group algebra is Hopf algebra)**

The group algebra is a Hopf algebra with

- , as expected.
- the counit is the trivial representation.
- comes form extended linearly.
- takes extended linearly.

**Theorem 3**

The group-like elements are precisely the basis elements .

*Proof:* Assume is grouplike. Then by assumption we should have

Comparing each coefficient, we get that

This can only occur if some is and the remaining coefficients are all zero.

## 2. Monoidal functors

Recall that **monoidal category** (or tensor category) is a category equipped with a functor which has an identity and satisfies some certain coherence conditions. For example, for any we should have a natural isomorphism

The generic example is of course suggested by the notation: vector spaces over , abelian groups, or more generally modules/algebras over a ring .

Now take two monoidal categories and . Then a **monoidal functor** is a functor for which we additionally need to select an isomorphism

We then require that the diagram

commutes, plus some additional compatibility conditions with the identities of the ‘s (see Wikipedia for the list).

We also have a notion of a natural transformation of two functors ; this is just making the squares

commute. Now, suppose is a monoidal functor. Then an **automorphism** of is a natural transformation which is invertible, i.e. a natural isomorphism.

## 3. Application to

With this language, we now reach the main point of the post. Consider the category of modules endowed with the monoidal (which is just the tensor over , with the usual group representation). We want to reconstruct from this category.

Let be the forgetful functor

It’s easy to see this is in fact an monoidal functor. Now let be the set of monoidal automorphisms of .

The key claim is the following:

**Theorem 4** **( is isomorphic to )**

Consider the map

Here, the natural transformation is defined by the components

Then is an isomorphism of groups.

In particular, using only structure this exhibits an isomorphism . Consequently this solves the problem proposed at the beginning of the lecture.

*Proof:* It’s easy to see is a group homomorphism.

To see it’s injective, we show gives isn’t the identity automorphism. i.e. we need to find some representation for which acts nontrivially on . Now just take the regular representation, which is faithful!

The hard part is showing that it’s surjective. For this we want to reduce it to the regular representation.

**Lemma 5**

Any is completely determined by .

*Proof:* Let be a representation of . Then for all , we have a unique morphism of representations

If we apply the forgetful functor to this, we have a diagram

Next, we claim

**Lemma 6**

is a grouplike element of .

*Proof:* Draw the diagram

This implies surjectivity, by our earlier observation that grouplike elements in are exactly the elements of .