Lie algebra of an algebraic group

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Definition

Via the intermediary of a formal group law

Suppose G is an algebraic group over a field k. The Lie algebra of G is a Lie algebra over k defined by the following two-step process: we first take the formal group law of the algebraic group G to obtain a formal group law over k, and then we take the Lie algebra of the formal group law obtained.

Facts

  • 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.