Algebraic group

The mathematical term of the algebraic group represents the synthesis of group theory and algebraic geometry . A central example is the group of invertible n × n matrices .

definition

An algebraic group is a group object in the category of algebraic varieties over a solid body , d. H. an affine algebraic variety over a field together with ${\displaystyle G}$ ${\displaystyle k}$

• a morphism (multiplication) ${\displaystyle m\colon G\times G\to G}$
• a morphism (inverse element) ${\displaystyle i\colon G\to G}$
• and a distinguished point (neutral element), ${\displaystyle e\in G(k)}$

so that the following conditions are met:

• Associative law : ; ${\displaystyle m\circ (m\times \mathrm {id} _{G})=m\circ (\mathrm {id} _{G}\times m)}$
• neutral element: ; ${\displaystyle m\circ (\mathrm {id} _{G}\times e)=\mathrm {id} _{G}=m\circ (e\times \mathrm {id} _{G})}$
• inverse element: ; here is the inclusion of the diagonal ( ) and the structural morphism. ${\displaystyle m\circ (i\times \mathrm {id} _{G})\circ \Delta _{G}=e\circ \xi =m\circ (\mathrm {id} _{G}\times i)\circ \Delta _{G}}$ ${\displaystyle \Delta _{G}\colon G\to G\times G}$ ${\displaystyle g\mapsto (g,g)}$ ${\displaystyle \xi \colon G\to Spec(k)}$

These conditions are equivalent to the requirement that for every - scheme on the set of - valued points define the structure of an (ordinary) group. ${\displaystyle (m,i,e)}$ ${\displaystyle k}$ ${\displaystyle T}$ ${\displaystyle G(T)}$ ${\displaystyle T}$

Examples

• The additive group : with addition as a group structure. In particular for is the affine line with addition. ${\displaystyle \mathbb {G} _{\mathrm {a} }}$ ${\displaystyle \mathbb {G} _{\mathrm {a} }(T)=\Gamma (T,{\mathcal {O}}_{T})}$ ${\displaystyle T=k}$ ${\displaystyle \mathbb {G} _{\mathrm {a} }=(k,+)}$ ${\displaystyle \mathrm {A} ^{1}}$
• The multiplicative group : with multiplication as a group structure. In particular for is the open subset with multiplication. ${\displaystyle \mathbb {G} _{\mathrm {m} }}$ ${\displaystyle \mathbb {G} _{\mathrm {m} }(T)=\Gamma (T,{\mathcal {O}}_{T}^{\times })}$ ${\displaystyle T=k}$ ${\displaystyle \mathbb {G} _{\mathrm {m} }=(k^{\times },\times )}$ ${\displaystyle \mathrm {A} ^{1}-\left\{0\right\}}$
• The general linear group : ; The right side denotes the group of invertible matrices with entries in the ring . can be identified with. ${\displaystyle \mathrm {GL} _{n}}$ ${\displaystyle \mathrm {GL} _{n}(T)=\mathrm {GL} _{n}(\Gamma (T,{\mathcal {O}}_{T}))}$ ${\displaystyle n\times n}$ ${\displaystyle \Gamma (T,{\mathcal {O}}_{T})}$ ${\displaystyle \mathrm {GL} _{1}}$ ${\displaystyle \mathbb {G} _{\mathrm {m} }}$
• The core of a morphism of algebraic groups is again an algebraic group. For example is an algebraic group. ${\displaystyle f:G\rightarrow H}$ ${\displaystyle \mathrm {SL} _{n}(T)=ker(det-1)}$
• Elliptic curves or, more generally, Abelian varieties .
• Zariski-closed subgroups of algebraic groups are again algebraic groups. Zariski-closed subgroups of are called linear algebraic groups . If an algebraic group is an affine variety, then it is a linear algebraic group. ${\displaystyle \mathrm {GL} _{n}}$
• Unipotent algebraic groups .

Chevalley's Theorem

Every algebraic group over a field of characteristic 0 is (in a unique way) an extension of an Abelian variety by a linear algebraic group. That means that for every algebraic group there is a maximal linear algebraic subgroup , this is normal and the quotient is an Abelian variety : ${\displaystyle G}$ ${\displaystyle G_{aff}}$ ${\displaystyle A(G):=G/G_{aff}}$

${\displaystyle 0\rightarrow G_{aff}\rightarrow G\rightarrow A(G)\rightarrow 0}$.

The picture is the Albanese picture . ${\displaystyle G\rightarrow A(G)}$

Individual evidence

1. ↑ Conrad: Theorem by Chevalley (PDF file; 233 kB)

literature

• James E. Humphreys : Linear Algebraic Groups. Springer, New York 1975, ISBN 3-540-90108-6 .
• Armand Borel : Linear Algebraic Groups. 2nd edition, Springer, New York 1991, ISBN 3-540-97370-2 .
• Tonny A. Springer : Linear Algebraic Groups. 2nd edition, Birkhäuser, Boston 1998, ISBN 3-7643-4021-5 .
• Ina Kersten : Linear Algebraic Groups. Universitätsverlag Göttingen, 2007, ( PDF; 1.4 MB ).

Web links

Algebraic Groups by James S. Milne