Algebraic group
notion explanation example remarks
linear algebraic group A Zariski closed subgroup of G L n { \displaystyle { \rm { GL } } _ { n } } for some n S L n { \displaystyle { \rm { SL } } _ { n } } Every affine algebraic group is isomorphic to a linear algebraic group , and vice versa
affine algebraic group An algebraic group that is an affine variety G L n { \displaystyle { \rm { GL } } _ { n } } , non-example : elliptic curve The notion of affine algebraic group stresses the independence from any embedding in G L n { \displaystyle { \rm { GL } } _ { n } }
commutative The underlying ( abstract ) group is abelian G a { \displaystyle { \mathbb { G } } _ { a } } ( the additive group ) , G m { \displaystyle { \mathbb { G } } _ { m } } ( the multiplicative group ) , any complete algebraic group ( see abelian variety )
diagonalizable group A closed subgroup of ( G m ) n { \displaystyle ( \mathbb { G } _ { m } ) ^ { n } } , the group of diagonal matrices ( of size n -by- n )
simple algebraic group A connected group that has no non-trivial connected normal subgroups S L n { \displaystyle { \rm { SL } } _ { n } }
semisimple group An affine algebraic group with trivial radical S L n { \displaystyle { \rm { SL } } _ { n } } , S O n { \displaystyle { \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 , G L n { \displaystyle { \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 ( G m ) n { \displaystyle ( \mathbb { G } _ { m } ) ^ { n } } when passing to the algebraic closure of k S O 2 { \displaystyle { \rm { SO } } _ { 2 } } G is said to be split by some bigger field k ' , if G becomes isomorphic to G m as an algebraic group over k '
character group X ( G ) The group of characters , i.e. , group homomorphisms G → G m { \displaystyle G\rightarrow { \mathbb { G } } _ { m } } X ∗ ( G m ) ≅ Z { \displaystyle X^ { * } ( \mathbb { G } _ { m } ) \cong \mathbb { Z } }
Lie algebra Lie ( G ) The tangent space of G at the unit element L i e ( G L n ) { \displaystyle { \rm { Lie } } ( { \rm { GL } } _ { n } ) } is the space of all n -by- n matrices Equivalently , the space of all left-invariant derivations
Glossary of algebraic groups
There are a number of mathematical notions to study and classify algebraic groups .
In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety. In terms of category theory, an algebraic group is a group object in the category of algebraic varieties.