Lie Algebra Homomorphisms

This lecture explains how to associate a Lie algebra to a linear algebraic group and how to define a Lie algebra homomorphism between the Lie algebras of two algebraic groups. The instructor proves a proposition showing that the differential of a regular homomorphism between algebraic groups preserves the Lie algebra structure. The proof involves considering a homomorphism mapping elements of the Lie algebra of one group to itself, and demonstrating that the differential of this map satisfies certain properties. By establishing a commutative diagram, the instructor shows that the differential of the homomorphism between the Lie algebras of two groups is a Lie algebra homomorphism. The lecture concludes with a summary of the main results and a thank you message to the audience.