Editing Linear algebraic group (section)

==The Lie algebra of an algebraic group==
The [[Lie algebra]] <math>\mathfrak g</math> of an algebraic group ''G'' can be defined in several equivalent ways: as the [[Zariski tangent space|tangent space]] ''T''<sub>1</sub>(''G'') at the identity element 1 ∈ ''G''(''k''), or as the space of left-invariant [[derivation (abstract algebra)|derivation]]s. If ''k'' is algebraically closed, a derivation ''D'': ''O''(''G'') → ''O''(''G'') over ''k'' of the coordinate ring of ''G'' is '''left-invariant''' if
:<math>D \lambda_x = \lambda_x D</math>
for every ''x'' in ''G''(''k''), where λ<sub>''x''</sub>: ''O''(''G'') → ''O''(''G'') is induced by left multiplication by ''x''. For an arbitrary field ''k'', left invariance of a derivation is defined as an analogous equality of two linear maps ''O''(''G'') → ''O''(''G'') ⊗''O''(''G'').<ref>Milne (2017), section 10.e.</ref> The Lie bracket of two derivations is defined by [''D''<sub>1</sub>, ''D''<sub>2</sub>] =''D''<sub>1</sub>''D''<sub>2</sub> − ''D''<sub>2</sub>''D''<sub>1</sub>.

The passage from ''G'' to <math>\mathfrak g</math> is thus a process of [[differentiation (mathematics)|differentiation]]. For an element ''x'' ∈ ''G''(''k''), the derivative at 1 ∈ ''G''(''k'') of the [[conjugation (group theory)|conjugation]] map ''G'' → ''G'', ''g'' ↦ ''xgx''<sup>−1</sup>, is an [[automorphism]] of <math>\mathfrak g</math>, giving the [[adjoint representation]]:
:<math>\operatorname{Ad}\colon G \to \operatorname{Aut}(\mathfrak g).</math>

Over a field of characteristic zero, a connected subgroup ''H'' of a linear algebraic group ''G'' is uniquely determined by its Lie algebra <math>\mathfrak h \subset \mathfrak g</math>.<ref>Borel (1991), section 7.1.</ref> But not every Lie subalgebra of <math>\mathfrak g</math> corresponds to an algebraic subgroup of ''G'', as one sees in the example of the torus ''G'' = (''G''<sub>''m''</sub>)<sup>2</sup> over '''C'''. In positive characteristic, there can be many different connected subgroups of a group ''G'' with the same Lie algebra (again, the torus ''G'' = (''G''<sub>''m''</sub>)<sup>2</sup> provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.