I’ve decided to start this series with a few posts on the Lie algebra of an algebraic group. This seems to me the first real technical aspect of the classical theory that arises in Humphreys’ book. We shall loosely follow this book as a guide, but we shall also deviate and look at more scheme-theoretic treatments.
We will define a functor from the category of algebraic groups over a commutative ring to Lie algebras over . The idea is that Lie algebras are often easier to work with than algebraic groups directly, so the Lie algebra will help us with things (such as classification problems). For now, what we do will be general enough so that we do not need to assume that is a field.
We shall look at several definitions of a Lie algebra of an algebraic group, and prove that they are all equivalent. After this, we shall examine what this functor does to morphisms (the “differential of a morphism”) and then give a few examples of why this process is useful.
Let be a commutative ring, let be our algebraic group and be the representing Hopf algebra of . In the classical language, is just the coordinate algebra of the -points of the algebraic group scheme .
A good example for the reader to keep in mind is , which associates to every -algebra the group of units of . In particular . This is an algebraic group scheme represented by the algebra that has a Hopf algebra structure induced by the multiplication in the group of units. Of course, eventually we will want to consider more interesting algebraic groups, but this will be a good test case for our initial definition of the Lie algebra, which will definitely not be the easiest to work with!
The First Definition
A derivation is a -module homomorphism that satisfies the Leibniz rule
for all . We denote the set of derivations by .
If , then the morphism of varieties given by induces a morphism of -algebras, which we call left translation. The term “translation” refers to translation of functions because is the -algebra of -valued functions on . We call a derivation left-invariant if it satisfies for all .
Now, given any -algebra we can associate to it a -Lie algebra by defining the Lie bracket as for all . It is left as an exercise for the reader to show:
- The bracket of two derivations is a derivation, and hence the -module of derivations is a -Lie algebra.
- If are left-invariant then is also left-invariant.
Thus the Lie subalgebra of all left-invariant derivations in is a Lie algebra via the formula , and we call it the Lie algebra of .
Despite the relative simplicity of the above definition, computing the Lie algebra of is quite tricky. The reader should work out the Lie algebra for the example of , using the following sketch if desired. In this case, the representing Hopf algebra is . It follows from the Leibniz rule that a derivation in this case is determined by and that is in fact the only possibility, which follows from left-invariance.
Thus the Lie algebra is -dimensional, and so the Lie bracket is identically equal to . Notice how the dimension of the Lie algebra associated to is the same as the dimension of as an algebraic variety. This in fact will always turn out to be the case, as we shall see later.
This computation was fairly straightforward, but for more complicated algebraic groups, working out the Lie algebra could be rather tricky. It’s not even clear from this definition that the Lie algebra is in fact finite dimensional in all cases.
What About Morphisms?!
Now, let us see how the above definition is actually functorial. For this, we should explain how an algebraic group morphism gives rise to a corresponding morphism of Lie algebras.
Keeping with the functorial language, a morphism of algebraic groups is a natural transformation that is also a group homomorphism for each -algebra . Via Yoneda, these natural transformations correspond naturally bijectively to Hopf algebra homomorphisms . This is just a generalisation of the classical case where a morphism of algebraic groups is determined by the corresponding Hopf algebra morphism.
Thus we may as well just consider a morphism of Hopf algebras. (Of course, if we’re coming from the group side, we don’t need to verify that it is a Hopf-algebra morphism in the first place!). Now the question I pose to the dear reader is: given a left-invariant derivation , how can we use to get a left-invariant derivation on ? I encourage the reader to try a few things before moving onto the answer, which will be contained in the next post.