Lie correspondence between nilpotent Lie algebras and unipotent algebraic groups

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Statement

General version

Suppose K is a field of characteristic zero. The Lie correspondence is a correspondence:

nilpotent Lie algebras over K \leftrightarrow unipotent algebraic groups over K

This correspondence defines an equivalence of categories over Set. More specifically, for any Lie algebra \mathfrak{g} over K, we can define a unipotent algebraic group over K with the same underlying set, where the group operations are defined in terms of the Lie algebra operations using the Baker-Campbell-Hausdorff formula (and we can go backward from groups to Lie algebras using the inverse Baker-Campbell-Hausdorff formula).

Note that the Lie correspondence does not require any notion of topology or convergence on the field because, even though the Baker-Campbell-Hausdorff formula has infinite length, in practice we only need to use finitely many terms because of the nilpotency assumption.

For matrices: linear Lie algebras and linear algebraic groups

In the case of matrices, the correspondence can be made explicit by means of the exponential map.

Related notions

  • The Lazard correspondence is a generalization to the situation where we are not over a field of characteristic zero, but the Lie ring and group are both uniquely divisible for primes up to the nilpotency class.