Lie Algebra of an Algebraic Group

Let \(G\) be a linear algebraic group over a field \(k\). Its Lie algebra is the tangent space at the identity: \[ \operatorname{Lie}(G)=T_eG. \] Equivalently, if \(A=k[G]\) and \(\mathfrak m_e\subset A\) is the maximal ideal of the identity point, then \[ \operatorname{Lie}(G)\simeq \operatorname{Hom}_k(\mathfrak m_e/\mathfrak m_e^2,k). \]

The group law identifies this tangent space with the space of left-invariant derivations of \(k[G]\). The commutator of derivations gives \(\operatorname{Lie}(G)\) the structure of a Lie algebra over \(k\).

Every homomorphism \(f:G\to H\) induces a Lie algebra homomorphism \[ df_e:\operatorname{Lie}(G)\to \operatorname{Lie}(H), \] and the assignment \(G\mapsto \operatorname{Lie}(G)\) is functorial.