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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