The invariant vectors of the adjoint representation

For a connected affine group over an algebraically closed field of characteristic zero, the invariant vectors of its rational adjoint representation are exactly the center of its tangent Lie algebra. Thus the entire group fixes each central tangent vector after extension to any commutative value algebra, even one with nilpotents.

This is the comparison needed to show that Lie-central tangent vectors give central dual-number points. The action on the center as a subrepresentation is trivial.

References

• J. S. Milne, Algebraic Groups (2017), Chapters 10 and 22.
• J. E. Humphreys, Linear Algebraic Groups, §13.

theorem TauCeti.FiniteTypeCommHopfAlgCat.coact_adjointComodule_eq_tmul_one_iff_mem_center {k : Type u} [Field k] [CharZero k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) [ConnectedSpace (PrimeSpectrum ↑H.obj)] (x : Module.Dual k (Bialgebra.CotangentSpace k ↑H.obj)) : Comodule.coact x = x ⊗ₜ[k] 1 ↔ x ∈ LieAlgebra.center k (Module.Dual k (Bialgebra.CotangentSpace k ↑H.obj))

For a connected group over an algebraically closed field of characteristic zero, a tangent vector is rationally fixed under the adjoint action exactly when it lies in the center of the tangent Lie algebra.

theorem TauCeti.FiniteTypeCommHopfAlgCat.endOfPoint_tmul_of_mem_center {k : Type u} [Field k] [CharZero k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) [ConnectedSpace (PrimeSpectrum ↑H.obj)] (x : Module.Dual k (Bialgebra.CotangentSpace k ↑H.obj)) (hx : x ∈ LieAlgebra.center k (Module.Dual k (Bialgebra.CotangentSpace k ↑H.obj))) {A : Type u_1} [CommSemiring A] [Algebra k A] (g : ↑H.obj →ₐ[k] A) (a : A) : (Comodule.endOfPoint (Module.Dual k (Bialgebra.CotangentSpace k ↑H.obj)) g) (a ⊗ₜ[k] x) = a ⊗ₜ[k] x

Every algebra-valued point fixes the scalar extension of each central tangent vector of a connected affine group over an algebraically closed field of characteristic zero. The value algebra may be nonreduced.