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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