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 for some 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 | , non-example: elliptic curve | The notion of affine algebraic group stresses the independence from any embedding in |
commutative | The underlying (abstract) group is abelian. | (the additive group), (the multiplicative group),[1] any complete algebraic group (see abelian variety) | |
diagonalizable group | A closed subgroup of , the group of diagonal matrices (of size n-by-n) | ||
simple algebraic group | A connected group which has no non-trivial connected normal subgroups | ||
semisimple group | An affine algebraic group with trivial radical | , | 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, | 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 when passing to the algebraic closure of k. | 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 | ||
Lie algebra Lie(G) | The tangent space of G at the unit element. | ) is the space of all n-by-n matrices | Equivalently, the space of all left-invariant derivations. |
Notes
References
- 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
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