The center of a semisimple group is finite #
The scheme-theoretic center of a semisimple affine group is finite over its ground field, including in positive characteristic, where it can be nonreduced. This is the finiteness input for the central isogeny from a semisimple group to its adjoint form.
Over the algebraic closure, the reduced center is smooth. Its identity component is a smooth connected central subgroup of the ambient semisimple group, hence trivial. Finiteness of the component group and of the nilpotent thickening then gives finiteness of the full center; finiteness descends to the ground field.
References #
- J. S. Milne, Algebraic Groups (2017), §21.
- T. A. Springer, Linear Algebraic Groups, §8.1.
theorem
TauCeti.semisimpleCommHopfAlgProperty.moduleFinite_centerCoordinate
{k : Type u}
[Field k]
{H : FiniteTypeCommHopfAlgCat k}
(hH : semisimpleCommHopfAlgProperty k H)
:
The scheme-theoretic center of a semisimple affine group is finite over its ground field. No restriction on the characteristic or reducedness of the center is needed.