For an <affine algebraic group> $G$, its <Lie algebra> is the <tangent space> at the identity,
$$
\mathfrak g=T_eG.
$$
Equivalently, it is the space of left-invariant derivations of the <coordinate ring> $\mathbb C[G]$. The <Lie bracket> is the commutator of derivations,
$$
[X,Y]=XY-YX.
$$
It is antisymmetric because $[Y,X]=-[X,Y]$. Associativity of composition gives
$$
[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0
$$
after all six triple products cancel in pairs, proving the <Jacobi identity>. This construction is the <Lie algebra of an affine algebraic group>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 102 / 1 / i / Solution

The Lie algebra of an affine algebraic group $G$ is the tangent space $T_eG$ at its identity. Its bracket is the commutator of the corresponding left-invariant derivations of the coordinate ring.


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