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), 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 Gmn 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

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, vol. 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, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713