Lie algebra of an algebraic group
Via the intermediary of a formal group law
Suppose is an algebraic group over a field . The Lie algebra of is a Lie algebra over defined by the following two-step process: we first take the formal group law of the algebraic group to obtain a formal group law over , and then we take the Lie algebra of the formal group law obtained.
- The Lie algebra of an algebraic group is the same as the Lie algebra of its connected component of identity. In particular, the Lie algebra depends only on the connected component of identity.
- The Lie algebra of a quotient of an algebraic group by a discrete closed subgroup is the same as the Lie algebra of the algebraic group.