Subgroups and subalgebras

This lecture focuses on the relationship between subgroups of linear algebraic groups and sub-Lie algebras of the corresponding Lie algebras. The instructor presents a proposition showing that homomorphisms of linear algebraic groups are uniquely determined by their differentials. Additionally, the lecture covers the intersection of closed subgroups and their Lie algebras, emphasizing the connection between Lie algebras and connected components. The instructor also discusses the image and pre-image of subgroups under homomorphisms, highlighting the correspondence between closed connected subgroups and Lie subalgebras. The lecture concludes with a corollary regarding the injective correspondence between closed connected subgroups and Lie subalgebras, emphasizing the compatibility with inclusions and intersections.