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Algebraic torus

In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by G m {\displaystyle \mathbf {G} _{\mathbf {m} }} , G m {\displaystyle \mathbb {G} _{m}} , T {\displaystyle \mathbb {T} } , or GL ⁡ ( 1 ) {\displaystyle \operatorname {GL} (1)} , is a type of commutative affine algebraic group commonly found in projective algebraic…

Tori in semisimple groups, Overview & Algebraic tori over fields

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Overview

Algebraic tori over fields

Definition

Weights and coweights

Tori in semisimple groups

Tori and geometry

Algebraic tori over an arbitrary base scheme

Arithmetic invariants

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Algebraic torus

Nodes61
Edges60
Triples13
Avg. degree1.97
Density0.032787
Components1

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Algebraic torus

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related to Definition · 8
Algebraic torus → Given, GL1, Gm/L, Gm/S, Gm/U, In, Most, One

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Important terminology

displaystyle torus group tori algebraic mathbf field groups mathbb rank split finite given -rank maximal isomorphic lie real scheme defined

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
symmetric spacesinstance ofas real manifolds.Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them0.80text
buildingsinstance ofas real manifolds.Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them0.80text
the etale topologyinstance ofIf the torus is locally trivializable with respect to a weaker topology0.80text
then the sheaves of groups descend to the same topologiesinstance ofIf the torus is locally trivializable with respect to a weaker topology0.80text
these representations factor through the respective quotient groupoidsinstance ofIf the torus is locally trivializable with respect to a weaker topology0.80text
Algebraic torusrelated to DefinitionGiven0.60section
Algebraic torusrelated to DefinitionGm/S0.60section
Algebraic torusrelated to DefinitionIn0.60section
Algebraic torusrelated to DefinitionGL10.60section
Algebraic torusrelated to DefinitionGm/U0.60section
Algebraic torusrelated to DefinitionOne0.60section
Algebraic torusrelated to DefinitionGm/L0.60section

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