Fixed vectors and differentiation

For a connected affine group over an algebraically closed field of characteristic zero, a vector in a rational representation is fixed exactly when its differentiated action is zero. More generally, a linear map intertwines rational representations exactly when it intertwines their differentials. Neither representation needs to be finite dimensional.

The morphism criterion follows by applying the Lie-stable subspace criterion to the graph of the linear map. Applying it to a map from the trivial representation gives the fixed vector criterion. In particular this allows infinitesimal invariants, such as the center of the tangent Lie algebra, to be recognized as rational invariants.

References

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

The graph argument uses Submodule.coact_mem_range_of_forall_differential_mem.

theorem LinearMap.map_coact_iff_map_differential {k : Type u} [Field k] [CharZero k] [IsAlgClosed k] {H : TauCeti.FiniteTypeCommHopfAlgCat k} [ConnectedSpace (PrimeSpectrum ↑H.obj)] {M : Type w} [AddCommGroup M] [Module k M] [TauCeti.Comodule k (↑H.obj) M] {N : Type x} [AddCommGroup N] [Module k N] [TauCeti.Comodule k (↑H.obj) N] (f : M →ₗ[k] N) : TensorProduct.map f id ∘ₗ TauCeti.Comodule.coact = TauCeti.Comodule.coact ∘ₗ f ↔ ∀ (d : Derivation k (↑H.obj) (TauCeti.Bialgebra.CounitAlgebra k (↑H.obj) k)) (m : M), f ((TauCeti.Comodule.differential d) m) = (TauCeti.Comodule.differential d) (f m)

For a connected affine group over an algebraically closed field of characteristic zero, a linear map is a rational representation morphism exactly when it intertwines the differentiated actions.

theorem TauCeti.Comodule.coact_eq_tmul_one_iff_forall_differential_eq_zero {k : Type u} [Field k] [CharZero k] [IsAlgClosed k] {H : FiniteTypeCommHopfAlgCat k} [ConnectedSpace (PrimeSpectrum ↑H.obj)] {V : Type w} [AddCommGroup V] [Module k V] [Comodule k (↑H.obj) V] (v : V) : coact v = v ⊗ₜ[k] 1 ↔ ∀ (d : Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k)), (differential d) v = 0

For a connected affine group over an algebraically closed field of characteristic zero, rational invariants are exactly the vectors killed by every tangent vector. No dimension assumption is needed.