nLab linear algebraic group

Definition

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

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

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

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

Related concepts

• algebraic group

• affine algebraic group

• hyperalgebra

• distribution on an affine group scheme

Literature

Monographs:

• Armand Borel, Linear algebraic groups, Springer (1991) [doi:10.1007/978-1-4612-0941-6]

• Gunter Malle, Donna Testerman: Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge University Press (2012) [doi:10.1017/CBO9780511994777]

See also:

• Gerhard P. 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

category: algebra, algebraic geometry