Reductive group

This article deals with reductive algebraic groups, for reductive Lie groups see reductive Lie group .

The reductive group is a term in mathematics that is particularly important in representation theory and geometric invariant theory .

definition

A reductive group is an algebraic group over an algebraically closed field that satisfies one of the following equivalent conditions: ${\displaystyle G}$ ${\displaystyle k}$

• The radical of the component of one is an algebraic torus , especially an Abelian group . ${\displaystyle G_{0}}$
• The unipotent radical of is the trivial group. In other words: has no closed, connected and unipotent normal parts. ${\displaystyle G_{0}}$ ${\displaystyle G_{0}}$
• The group is the product of two closed normal divisors and , where semisimple and is an algebraic torus. ${\displaystyle G=ST}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle S}$ ${\displaystyle T}$

In the latter case and is the radical of , the intersection is finite and every semi-simple or unipotent subgroup of is contained in. ${\displaystyle S=\left[G_{0},G_{0}\right]}$ ${\displaystyle T}$ ${\displaystyle G}$ ${\displaystyle S\cap T}$ ${\displaystyle G_{0}}$ ${\displaystyle S}$

In the case is reductive if and only if every representation is completely reducible and this is the case if and only if the adjoint representation is completely reducible. ${\displaystyle char(k)=0}$ ${\displaystyle G}$

In case is reductive if and only if it is the complexification of a connected compact Lie group . ${\displaystyle k=\mathbb {C} }$ ${\displaystyle G}$

Examples

Let be an algebraically closed field. Then the following groups are reductive. ${\displaystyle k}$

• The multiplicative group . ${\displaystyle G_{m}(k)=k^{*}}$
• The general linear group . ${\displaystyle GL(n,k)}$
• The special linear group . ${\displaystyle SL(n,k)}$
• All semi-simple algebraic groups .

literature

• A. Borel, J. Tits: Groupes reductifs. Publ. Math. IHES, 27 (1965) pp. 55-150.
• James E. Humphreys : Linear Algebraic Groups. Springer, New York 1975, ISBN 978-1-4684-9445-7 .
• VL Popov: Hilbert's theorem on invariants. Soviet Math. Dokl., 20: 6 (1979) pp. 1318-1322 Docl. Akad. Nauk SSSR, 249: 3 (1979) pp. 551-555.
• TA Springer: Invariant theory. Lect. notes in math., 585, Springer (1977).

Web links

• Reductive Group (Encyclopedia of Mathematics)
• What is ... a reductive group?
• JSMilne: Reductive Groups