Central subgroups and the tangent Lie algebra

The Lie algebra of a central closed subgroup is contained in the center of the ambient Lie algebra. This holds over commutative rings and with arbitrary coefficient algebras, without smoothness or reducedness assumptions. In particular it applies to infinitesimal central subgroups in positive characteristic.

A universally central point acts trivially on tangent vectors: conjugation fixes their dual-number points. Applied to a tangent point of a central subgroup, the infinitesimal adjoint formula then says that the Lie bracket vanishes.

The dual-number argument uses Derivation.snd_conjugate_tangentPoint and Derivation.adDerivation_dualNumber_apply. The method follows the normal-subgroup argument in TauCeti.Algebra.AlgebraicGroup.HopfIdeal.Normal.Tangent.

References

• J. S. Milne, Algebraic Groups (2017), §10, Lie algebras and the adjoint action.
• J. C. Jantzen, Representations of Algebraic Groups, I.7.

theorem TauCeti.HopfIdeal.IsCentral.lieSubalgebra_le_center {R : Type u} {H : Type v} {B : Type (max v w)} [CommRing R] [CommRing H] [HopfAlgebra R H] [CommRing B] [Algebra R B] {I : HopfIdeal R H} (hI : I.IsCentral) : I.lieSubalgebra ≤ ↑B (Derivation R H (Bialgebra.CounitAlgebra R H B)) (LieAlgebra.center B (Derivation R H (Bialgebra.CounitAlgebra R H B)))

The tangent Lie algebra of a central closed subgroup lies in the center of the ambient Lie algebra, with coefficients in any commutative value algebra.