nLab linear algebraic group

Definition

A linear algebraic group GG is any (Zariski closed) algebraic subgroup of GL(n,k)GL(n,k) where kk is a field and nn a natural number.

An algebraic group over a field is linear iff it is affine as a kk-scheme.

An algebraic group scheme is affine if the underlying scheme is affine.

The category of affine group schemes over a field kk is the opposite of the category of commutative Hopf algebras over kk.

Literature

Related notions include algebraic group, affine algebraic group, hyperalgebra, distribution on an affine group scheme

  • Armand Borel, Linear algebraic groups, Springer
  • G. Hochschild, Algebraic groups and Hopf algebras, Illinois J. Math. 14:1 (1970), 52-65 euclid
  • Gerhard P. Hochschild, Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics 75, 1981 doi
  • wikipedia: linear algebraic group
  • Akira Masuoka, Hopf algebraic techniques applied to super algebraic groups, arXiv:1311.1261

Last revised on August 22, 2023 at 13:59:12. See the history of this page for a list of all contributions to it.