Lie-stable subspaces in characteristic zero

Let G be a connected affine group of finite type over an algebraically closed field k of characteristic zero, and M a representation of G, that is, a comodule over its coordinate Hopf algebra H. A subspace W ≤ M is a subrepresentation exactly when it is stable under the differentiated action of the Lie algebra Lie(G).

One direction holds over any base ring: subcomodules are stable under the differentiated action. For the converse, the Lie algebra of the stabilizer G_W of W consists of the tangent vectors preserving W, so Lie(G_W) = Lie(G). In characteristic zero both groups are smooth, by Cartier's theorem, so G_W = G because G is connected.

Both hypotheses are needed. In characteristic p the line spanned by xᵖ in the regular representation of 𝔾ₐ is killed by Lie(𝔾ₐ), but its coaction xᵖ ⊗ 1 + 1 ⊗ xᵖ does not lie in it. A nontrivial finite constant group has zero Lie algebra, so every subspace of its regular representation is Lie-stable.

This is the bridge that lets complete reducibility of Lie(G)-representations be transported to representations of G in characteristic zero: a representation of G is completely reducible exactly when every Lie(G)-stable subspace has a Lie(G)-stable complement.

Main declarations

• Submodule.coact_mem_range_of_forall_differential_mem: in characteristic zero, the coaction of a vector of a Lie(G)-stable subspace W lies in W ⊗ H.
• Submodule.exists_subcomodule_iff_forall_differential_mem: in characteristic zero, a subspace is a subcomodule exactly when it is Lie(G)-stable.
• TauCeti.Comodule.isCompletelyReducible_iff_forall_differential_mem: in characteristic zero, a representation is completely reducible exactly when its Lie(G)-stable subspaces have Lie(G)-stable complements.

References

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

theorem Submodule.coact_mem_range_of_forall_differential_mem {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] (W : Submodule k M) (hW : ∀ (d : Derivation k (↑H.obj) (TauCeti.Bialgebra.CounitAlgebra k (↑H.obj) k)), ∀ w ∈ W, (TauCeti.Comodule.differential d) w ∈ W) {w : M} (hw : w ∈ W) : TauCeti.Comodule.coact w ∈ (TensorProduct.map W.subtype LinearMap.id).range

In characteristic zero, the coaction of a vector of a subspace stable under the differentiated action of the Lie algebra of a connected group lies in the tensor product of the subspace with the coordinate algebra.

theorem Submodule.exists_subcomodule_iff_forall_differential_mem {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] (W : Submodule k M) : (∃ (N : TauCeti.Subcomodule k (↑H.obj) M), N.toSubmodule = W) ↔ ∀ (d : Derivation k (↑H.obj) (TauCeti.Bialgebra.CounitAlgebra k (↑H.obj) k)), ∀ w ∈ W, (TauCeti.Comodule.differential d) w ∈ W

In characteristic zero, a subspace of a representation of a connected group over an algebraically closed field is a subrepresentation exactly when it is stable under the differentiated action of the Lie algebra.

theorem TauCeti.Comodule.isCompletelyReducible_iff_forall_differential_mem {k : Type u} [Field k] [CharZero k] [IsAlgClosed k] {H : FiniteTypeCommHopfAlgCat k} [ConnectedSpace (PrimeSpectrum ↑H.obj)] {M : Type w} [AddCommGroup M] [Module k M] [Comodule k (↑H.obj) M] : IsCompletelyReducible k (↑H.obj) M ↔ ∀ (W : Submodule k M), (∀ (d : Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k)), ∀ w ∈ W, (differential d) w ∈ W) → ∃ (W' : Submodule k M), (∀ (d : Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k)), ∀ w ∈ W', (differential d) w ∈ W') ∧ IsCompl W W'

In characteristic zero, a representation of a connected group over an algebraically closed field is completely reducible exactly when every subspace stable under the differentiated action of the Lie algebra has a complement that is again stable under it.