Characters are trivial on the derived subgroup

Every character of an affine group restricts to the trivial character on its derived closed subgroup. In coordinates, a group-like element minus one belongs to the derived defining ideal. This supplies the diagonal equations when restricting a triangular representation to the derived subgroup.

The proof uses the canonical evaluation morphism from the group algebra of the characters and functoriality of the derived subgroup.

References

• J. S. Milne, Algebraic Groups (2017), §6d.

@[simp]

theorem GroupLike.sub_one_mem_derivedDefiningIdeal {R : Type u_1} {H : Type u_2} [CommRing R] [CommRing H] [HopfAlgebra R H] (x : GroupLike R H) : ↑x - 1 ∈ TauCeti.CommHopfAlgCat.derivedDefiningIdeal H

Every character restricts to one on the derived closed subgroup.