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Adjoint representation

In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L ( n , R ) {\displaystyle \mathrm {GL} (n,\mathbb {R} )} , the Lie group of real n-by-n invertible matrices, then the…

Definition, Properties & Variants and analogues

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Overview

Definition

Adjoint representation of a Lie algebra

Structure constants

Examples

Properties

Roots of a semisimple Lie group

Variants and analogues

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Adjoint representation

Nodes73
Edges72
Triples39
Avg. degree1.97
Density0.027397
Components1

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Adjoint representation

Top relations

related to Roots of a semisimple Lie group · 11
Adjoint representation → Conjugation, If, In, Lie, SL, SLn, The, This, Thus, To, We
related to Properties · 8
Adjoint representation → Ad, By, G0, If, Lie, More, The, Therefore
related to Examples · 7
Adjoint representation → Adg, GL, If, In, Lie, SL, The
related to Variants and analogues · 6
Adjoint representation → According, Alexandre Kirillov, Kirillov, Lie, The, This
is a · 4
Adjoint representation → contragredient representation of the adjoint representation, group homomorphism that sends an invertible n-by-n matrix g, isotropy representation associated to the conjugation action of G around the identity element of G.Derivative of AdOne may always pass from a representation of a Lie group G to…, symplectic manifold
related to Structure constants · 3
Adjoint representation → That, The, Then

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Important terminology

lie displaystyle representation algebra adjoint group mathfrak ad linear derivative given operatorname matrices identity map connected element vector action space

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Adjoint representationis agroup homomorphism that sends an invertible n-by-n matrix g0.90text
Adjoint representationis aisotropy representation associated to the conjugation action of G around the identity element of G.Derivative of AdOne may always pass from a representation of a Lie group G to…0.90text
Adjoint representationis acontragredient representation of the adjoint representation0.90text
Adjoint representationis asymplectic manifold0.90text
Adjoint representationrelated to ExamplesIf0.60section
Adjoint representationrelated to ExamplesLie0.60section
Adjoint representationrelated to ExamplesGL0.60section
Adjoint representationrelated to ExamplesIn0.60section
Adjoint representationrelated to ExamplesAdg0.60section
Adjoint representationrelated to ExamplesSL0.60section
Adjoint representationrelated to ExamplesThe0.60section
Adjoint representationrelated to PropertiesThe0.60section

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