Glossary of algebraic groups

There are a number of mathematical notions to study and classify algebraic groups.

In the sequel, G denotes an algebraic group over a field k.

notion	explanation	example	remarks
linear algebraic group	A Zariski closed subgroup of ${\rm GL}_n$ for some n	${\rm SL}_n$	Every affine algebraic group is isomorphic to a linear algebraic group, and vice-versa
affine algebraic group	An algebraic group which is an affine variety	${\rm GL}_n$, non-example: elliptic curve	The notion of affine algebraic group stresses the independence from any embedding in ${\rm GL}_n$
commutative	The underlying (abstract) group is abelian.	${\mathbb G}_a$ (the additive group), ${\mathbb G}_m$ (the multiplicative group),[1] any complete algebraic group (see abelian variety)
diagonalizable group	A closed subgroup of $(\mathbb{G}_m)^n$, the group of diagonal matrices (of size n-by-n)
simple algebraic group	A connected group which has no non-trivial connected normal subgroups	${\rm SL}_n$
semisimple group	An affine algebraic group with trivial radical	${\rm SL}_n$, ${\rm SO}_n$	In characteristic zero, the Lie algebra of a semisimple group is a semisimple Lie algebra
reductive group	An affine algebraic group with trivial unipotent radical	Any finite group, ${\rm GL}_n$	Any semisimple group is reductive
unipotent group	An affine algebraic group such that all elements are unipotent	The group of upper-triangular n-by-n matrices with all diagonal entries equal to 1	Any unipotent group is nilpotent
torus	A group that becomes isomorphic to $(\mathbb{G}_m)^n$ when passing to the algebraic closure of k.	${\rm SO}_2$	G is said to be split by some bigger field k' , if G becomes isomorphic to Gm^n as an algebraic group over k'.
character group X∗(G)	The group of characters, i.e., group homomorphisms $G\rightarrow{\mathbb G}_m$	$X^*(\mathbb{G}_m)\cong\mathbb{Z}$
Lie algebra Lie(G)	The tangent space of G at the unit element.	${\rm Lie}({\rm GL}_n)$) is the space of all n-by-n matrices	Equivalently, the space of all left-invariant derivations.

References

1. These two are the only connected one-dimensional linear groups, Springer 1998, Theorem 3.4.9

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
• Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713