Last time in this series, we saw the definition of the Lie algebra of a linear algebraic group $ G$ over a arbitrary field $ k$ as the set of differentiations $ f:k[G]\to k$; these are the $ k$-linear maps satisfying $ f(ab) = \epsilon(a)f(b) + f(a)\epsilon(b)$ where $ \epsilon:k[G]\to k$ is the counit morphism corresponding to the identity in $ G$.

In this post we will look at the geometric definition of the tangent space, which is natural when we consider the $ k$-points of $ G$ as a subset of affine space. Furthermore, we shall see an example of the adjoint representation, and how morphisms of algebraic groups correspond to morphisms of Lie algebras in the explicit case of $ G$ embedded into $ \rm{GL}_n$. This will allow us to write down an explicit formula for the adjoint representation.
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In Linear Algebraic Groups 2, we defined the Lie algebra of an algebraic group $ G$ to be the Lie algebra of all left-invariant derivations $ D:k[G]\to k[G]$ where $ k[G]$ is the representing algebra of $ G$. However, we were left trying to figure out exactly how a morphism $ \varphi:G\to H$ determines a morphism $ d\varphi:\mathcal{L}(G)\to\mathcal{L}(H)$. It turns out that the answer is slightly tricky, and thus the Lie algebra in terms of left-invariant derivations $ k[G]\to k[G]$ really can't be the "real" or "most natural" definition!

Now, I could just give the formula for morphisms right away, but I think it would be a bit unmotivated. So before looking at the formula, let us first look at another way of defining the Lie algebra of an algebraic group. In fact, the next defintion is much more natural in that the functoriality of the Lie algebra construction will be clear, whereas in this case it was not.
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Posted by Jason Polak on 14. June 2012 · 1 comment · Categories: algebraic-geometry · Tags: , , ,

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 $ k$ to Lie algebras over $ k$. 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 $ k$ 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.
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