Semi-simple algebraic group

In mathematics , semi-simple algebraic groups are a term taken from algebraic geometry .

definition

A connected algebraic group over a field is called semi-simple if one of the following equivalent conditions is met: ${\displaystyle G}$ ${\displaystyle k}$

• the maximum continuous resolvable normal subgroup is ${\displaystyle \left\{1\right\}}$
• ${\displaystyle G}$has no nontrivial connected Abelian normal divisor.

Examples

• The special linear group is semi-simple. ${\displaystyle SL_{n}}$
• The projective linear group is semi-simple. ${\displaystyle PGL_{n}}$
• The symplectic group is semi-simple. ${\displaystyle Sp_{2n}}$
• The general linear group , the multiplicative group and the group of invertible upper triangular matrices are not semi-simple .

Semi-simple lie groups

→ Main article : Semi-simple Lie group

For a semi-simple algebraic group over is a semi-simple Lie group. ${\displaystyle G}$ ${\displaystyle k=\mathbb {C} }$ ${\displaystyle G(\mathbb {C} )}$

Not every semi-simple Lie group is a semi-simple algebraic group. An example of this is the universal overlay of . ${\displaystyle SL(2,\mathbb {R} )}$

classification

The classification of semi-simple algebraic groups over an algebraically closed field is analogous to the classification of semi-simple complex Lie groups by Dynkin diagrams .

literature

• JE Humphreys, "Linear algebraic groups", Springer (1975)
• TA Springer, "Linear algebraic groups", Birkhäuser (1981)

Web links

• Semi-simple algebraic group (Encyclopedia of Mathematics)