Commutative groups of semisimple elements are diagonalizable

Let H be the coordinate Hopf algebra of a reduced commutative affine group of finite type over an algebraically closed field k. If every k-point of the group is semisimple, then the group is diagonalizable: the group-like elements span H. If the group is moreover connected, it is a torus.

On a finite-dimensional representation the points act by pairwise-commuting semisimple operators, so the representation is the sum of the joint eigenspaces of these operators. Each nonzero joint eigenvector spans a line stable under every point, and point separation on the reduced coordinate ring makes that line a subcomodule; hence the vector is a weight vector. Applying this to the finite-dimensional subcomodules of the regular representation shows that group-like elements span H.

Conversely, every point of a diagonalizable group is semisimple, and its coordinate Hopf algebra is cocommutative. Combined with the reconstruction theorem TauCeti.DiagonalizableGroup.essImage_coordinateRingFunctor, this characterizes diagonalizable groups among reduced affine groups of finite type. Connectedness is not required for this characterization, so it also applies to smooth finite diagonalizable groups.

Reducedness cannot be dropped: in characteristic p, the group αₚ has only the identity point, which is semisimple, but it is not diagonalizable.

Main declarations

• TauCeti.HopfAlgebra.groupLikeSetSpan_eq_top_of_isSemisimplePoint: a reduced commutative affine group whose points are semisimple is diagonalizable.
• TauCeti.torusCommHopfAlgProperty_of_isSemisimplePoint: if it is moreover connected, it is a torus.
• TauCeti.DiagonalizableGroup.groupLikeSpannedProperty_iff_isCocomm_and_forall_isSemisimplePoint: a reduced affine group is diagonalizable exactly when it is commutative and its points are semisimple.

References

• J. S. Milne, Algebraic Groups (2017), Theorem 12.12.
• T. A. Springer, Linear Algebraic Groups, §§2.4 and 3.2.
• W. C. Waterhouse, Introduction to Affine Group Schemes, §3.2.

theorem TauCeti.HopfAlgebra.groupLikeSetSpan_eq_top_of_isSemisimplePoint (k H : Type u) [Field k] [CommRing H] [HopfAlgebra k H] [IsAlgClosed k] [Algebra.FiniteType k H] [IsReduced H] [Coalgebra.IsCocomm k H] (hss : ∀ (g : WithConv (H →ₐ[k] k)), IsSemisimplePoint g) : Subcoalgebra.groupLikeSetSpan Set.univ = ⊤

A reduced commutative affine group whose points are all semisimple is diagonalizable. Over an algebraically closed field, the coordinate Hopf algebra of such a group of finite type is spanned by its group-like elements.

theorem TauCeti.torusCommHopfAlgProperty_of_isSemisimplePoint {k : Type u} [Field k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) [IsReduced ↑H.obj] [Coalgebra.IsCocomm k ↑H.obj] [ConnectedSpace (PrimeSpectrum ↑H.obj)] (hss : ∀ (g : WithConv (↑H.obj →ₐ[k] k)), HopfAlgebra.IsSemisimplePoint g) : torusCommHopfAlgProperty k H

A reduced connected commutative affine group whose points are all semisimple is a torus. This holds over an algebraically closed field, for a group of finite type.

theorem TauCeti.DiagonalizableGroup.groupLikeSpannedProperty_iff_isCocomm_and_forall_isSemisimplePoint {k : Type u} [Field k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) [IsReduced ↑H.obj] : groupLikeSpannedProperty k H ↔ Coalgebra.IsCocomm k ↑H.obj ∧ ∀ (g : WithConv (↑H.obj →ₐ[k] k)), HopfAlgebra.IsSemisimplePoint g

Over an algebraically closed field, a reduced affine group of finite type is diagonalizable exactly when it is commutative and every rational point is semisimple.