Research any topic before you write.

Find related topics. | Discover entities. | See connections. | Build a topical map.

Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is semisimple, i.e. a direct sum of irreducible representations. Reductive groups include some of the most important…

Characters, Real reductive groups & Torsors and the Hasse principle

Use the mouse wheel or two fingers (on touchscreens) to zoom in and out of the map.

Research this topic

Explore the main themes, entities and connections around Reductive group. Start with the topic map, then use the sections below for research and deeper semantic analysis.

Explore this topic

Start with a few of the strongest sections from the source topic. These are research directions, not a list of keywords you must use.

Topics to explore

Browse the full topic structure. Each item opens a new analysis centered on that subject.

Overview

Definitions

Examples

Other characterizations of reductive groups

Roots

Parabolic subgroups

Classification of split reductive groups

Reductive group schemes

Real reductive groups

Representations of reductive groups

Non-split reductive groups

Structure of semisimple groups as abstract groups

Lattices and arithmetic groups

The Galois action on the Dynkin diagram

Torsors and the Hasse principle

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

Map overview Semantic statistics

Reductive group

Nodes190
Edges189
Triples263
Avg. degree1.99
Density0.010526
Components1

How this topic connects Entity context

See the strongest relationship patterns around the current topic before diving into the raw triples.

Reductive group

Top relations

related to References · 94
Reductive group → Academic Press, Algebraic, Algebraic Groups, American Mathematical Society, Andrei, Annals, Armand, Astérisque, Automorphic Forms, Autour, Berlin, BF01404653, BFb0059005, Bibcode, Birkhäuser Boston, Borel, Boston, Brian, Cambridge University Press, Canonical Basis
related to Classification of split reductive groups · 22
Reductive group → An, Bn, By, Cartan, Chevalley, Cn, Dickson, Dn, Dynkin, E6, E7, E8, F4, For, G2, In, It, Lie, The, This
related to The Galois action on the Dynkin diagram · 17
Reductive group → Artin, Dynkin, For, Gal, Galois, Galois-invariant, Generalizing, Gksep, In, Likewise, The Tits, This, Tits, Traditionally, Wedderburn, Witt, Witt's
related to Representations of reductive groups · 15
Reductive group → Borel, Chevalley, Define, For, Furthermore, G-equivariant, G/B, In, Rn, Schur, Then, There, Weil, Weyl, Zn
related to Reductive group schemes · 13
Reductive group → Borel, Chevalley, Dedekind, Extending Chevalley's, For, Grothendieck, JK, Michel Demazure, P1, SL2, Spec, Then SL, This
related to Parabolic subgroups · 12
Reductive group → As, Borel, By, Dynkin, Every, Explicitly, For, For GL, G/P, GL, Let, Thus
related to Real reductive groups · 12
Reductive group → Ad, Also, GL, In, Int, It, L0, Lie, Satake, The, These, Zariski
related to Torsors and the Hasse principle · 12
Reductive group → Also, Aut, For, G-bundle, G-torsors, Galois, H1, Namely, PGL, The, These, Torsors
related to Non-split reductive groups · 11
Reductive group → As, By, Every, Here, If, Mn/r, SL, SO, Some, The, Witt
related to Other characterizations of reductive groups · 10
Reductive group → Every, For, G/Go, GL, Go, In, Lie, Masayoshi Nagata, Note, That

Important terminology Word statistics

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

group reductive field groups algebraic simple connected semisimple subgroup displaystyle split example lie characteristic linear dynkin root every finite real

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Reductive groupis atype of linear algebraic group over a field0.90text
Reductive groupis aconnected group G admitting a faithful semisimple representation which remains semisimple over its algebraic closure kal. page 424.Simple reductive groupsA linear algebraic grou…0.90text
Reductive groupis aconnected group G admitting a faithful semisimple representation which remains semisimple over its algebraic closure kal. page 4240.90text
Reductive groupis ageneral linear group GL n0.90text
Reductive groupis aspecial linear group SL0.90text
Reductive groupis achoice of root basis and also a choice of trivialisation of the one-dimensional additive group corresponding to each simple root0.90text
Reductive groupis aLie group G such that there is a linear algebraic group L over R whose identity component0.90text
the real numbers R or a number fieldinstance ofbut for many fields0.80text
the classification is well understoodinstance ofbut for many fields0.80text
number fieldsinstance ofand they are understood for some other fields0.80text
but for arbitrary fields there are many open questions.A reductive group over a field k is called isotropic if it has k-rank greater than 0instance ofand they are understood for some other fields0.80text
Reductive grouprelated to Classification of split reductive groupsChevalley0.60section

Related concept clusters Concept neighborhoods

These clusters group vocabulary that occurs around closely connected concepts in the source material.

    Connections between topic areas Semantic bridges

    Bridges can reveal useful research angles that are easy to miss in a flat list of related terms.

    Min side: 3
    For writers, content strategists, SEOs, marketers and creators — from quick topic research to advanced semantic analysis.