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Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie…

History, Explicit construction of the irreducible root systems & Definitions and examples

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Overview

Definitions and examples

History

Positive roots and simple roots

Dual root system, coroots, and integral elements

Classification of root systems by Dynkin diagrams

Root systems and Lie theory

Properties of the irreducible root systems

Explicit construction of the irreducible root systems

The root poset

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Root system

Nodes91
Edges90
Triples176
Avg. degree1.98
Density0.021978
Components1

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Root system

Top relations

related to References · 51
Root system → Adams, An, Andrew Pressley, Ann, Archived, BF01207837, BF01211904, BF01444109, BF01446792, Birkhäuser, Bourbaki, Brian, Cambridge University Press, Chapters, Chicago Press, Coleman, Coxeter Groups, Die Zusammensetzung, Elements, French
related to history · 13
Root system → Cartan, F4, German, He, Here, Killing, Killing's, Lie, The, Then, This, Wilhelm Killing, Wurzelsystem
related to Properties of the irreducible root systems · 12
Root system → An, Any, Bn, Cn, Dn, Dynkin, If, In, Irreducible, The, There, Weyl
related to F4 · 9
Root system → B3, F4, For F4, Hurwitz, One, R4, The F4, There, This
related to Root systems arising from semisimple Lie algebras · 8
Root system → Cartan, If, Lie, One, Root, See, The, We
related to Weyl chambers and the Weyl group · 8
Root system → Delta, If, Phi, Recall, Since, The, Thus, Weyl
related to Classifying root systems · 7
Root system → Although, Consequently, Conversely, Dynkin, The, Thus, Weyl
related to Constructing the Dynkin diagram · 7
Root system → An, Dynkin, Edges, Given, No, Note, The
related to Definition · 7
Root system → Euclidean, For, If, Integrality, Let, Phi, The
related to Definitions and examples · 7
Root system → A2, As, Euclidean, If, Moreover, R2, These

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Important terminology

root roots system displaystyle simple systems set lattice lie alpha one integer two vectors group phi weyl dynkin algebras perpendicular

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Root systemis aconfiguration of vectors in a Euclidean space satisfying certain geometrical properties0.90text
algebraic groupsinstance ofand some analogues0.80text
Root systemrelated to Classification of root systems by Dynkin diagramsPhi0.60section
Root systemrelated to Classification of root systems by Dynkin diagramsIrreducible0.60section
Root systemrelated to Classification of root systems by Dynkin diagramsDynkin0.60section
Root systemrelated to Classification of root systems by Dynkin diagramsEugene Dynkin0.60section
Root systemrelated to Classification of root systems by Dynkin diagramsThe0.60section
Root systemrelated to Classifying root systemsAlthough0.60section
Root systemrelated to Classifying root systemsWeyl0.60section
Root systemrelated to Classifying root systemsConsequently0.60section
Root systemrelated to Classifying root systemsDynkin0.60section
Root systemrelated to Classifying root systemsConversely0.60section

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