A group is , where

There are a bunch of axioms.

Let be the diagonal map.

A group object in a category is …

The category should have

Assume has products and a terminal object. A group object in a category is such that the aboveabove are satisfied.

Let and define the “functor of points”

If is a group object, then is a group (and vice versa). That is, the functor factors through . For example,

A group object is such that factors through .

In , we have

Some facts about :

We have

This is a group under matrix multiplication. (For , we get a group scheme, for a field, we get an algebraic group.)

In Lie groups, group schemes, algebraic groups, we can talk about tangent spaces. (Algebraic groups are smooth.)

In cases where there is a tangent space, the Lie algebra of is

the tangent space at the identity. Often written with .

The identity is .

In vector calculus, we learn that the tangent space is orthogonal to , where the space is given by . Here, our function is

So the tangent space is given by

So tangent spaces are “easy”.

Let . A curve in is given as

Then is a tangent vector. If we put in the condition that , then we have

Notice that this was a completely algebraic computation: we didn’t care about terms or higher. This is really a computation in the ring .

Let be an algebraic group (over a field). Then

where

Alternatively, we are looking at the composition

As a functor,

The Lie algebra

To see this, is fairly easy to compute since is linear in each variable. The entries in are determinants of matrices of the submatrices. At the identity, the entries of are all (for on-diagonal entries) or (for off-diagonal entries). The trace is the sum of the diagonal entries.