mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
introduced by whose configurations classify
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
Lie algebras over the
field
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Expanses of open ground
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of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. The Satake diagrams associated to a Dynkin diagram classify real forms of the complex Lie algebra corresponding to the Dynkin diagram.
More generally, the Tits index or Satake–Tits diagram of a reductive
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
over a field is a generalization of the Satake diagram to arbitrary fields, introduced by , that reduces the classification of reductive algebraic groups to that of anisotropicreductive algebraic groups.
Satake diagrams are not the same as Vogan diagrams of a Lie group, although they look similar.
Definition
A Satake diagram is obtained from a Dynkin diagram by blackening some vertices, and connecting other vertices in pairs by arrows, according to certain rules.
Suppose that ''G'' is an algebraic group defined over a field ''k'', such as the reals. We let ''S'' be a maximal split torus in ''G'', and take ''T'' to be a maximal torus containing ''S'' defined over the separable algebraic closure ''K'' of ''k''. Then ''G''(''K'') has a Dynkin diagram with respect to some choice of positive roots of ''T''. This Dynkin diagram has a natural action of the Galois group of ''K''/''k''. Also some of the simple roots vanish on ''S''. The Satake–Tits diagram is given by the Dynkin diagram ''D'', together with the action of the Galois group, with the simple roots vanishing on ''S'' colored black. In the case when ''k'' is the field of real numbers, the absolute Galois group has order 2, and its action on ''D'' is represented by drawing conjugate points of the Dynkin diagram near each other, and the Satake–Tits diagram is called a Satake diagram.
Examples
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Compact Lie algebra
In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebr ...
s correspond to the Satake diagram with all vertices blackened.
*
Split Lie algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak, \mathfrak) where \mathfrak is a Lie algebra and \mathfrak < \mathfrak is a splitting
Vogan diagrams are used to classify semisimple Lie groups or algebras (or algebraic groups) over the reals and both consist of Dynkin diagrams enriched by blackening a subset of the nodes and connecting some pairs of vertices by arrows. Satake diagrams, however, can be generalized to any field (see above) and fall under the general paradigm of
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...
, whereas Vogan diagrams are defined specifically over the reals. Generally speaking, the structure of a real semisimple Lie algebra is encoded in a more transparent way in its Satake diagram, but Vogan diagrams are simpler to classify.
The essential difference is that the Satake diagram of a real semisimple Lie algebra with
Cartan involution
In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decom ...
''θ'' and associated Cartan pair (the +1 and −1 eigenspaces of ''θ'') is defined by starting from a maximally noncompact ''θ''-stable
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
, that is, one for which and is as small as possible (in the presentation above, appears as the Lie algebra of the maximal split torus ''S''), whereas Vogan diagrams are defined starting from a maximally compact ''θ''-stable Cartan subalgebra, that is, one for which and is as large as possible.
The unadorned Dynkin diagram (i.e., that with only white nodes and no arrows), when interpreted as a Satake diagram, represents the split real form of the Lie algebra, whereas it represents the compact form when interpreted as a Vogan diagram.
See also
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Relative root system
In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a s ...
*
List of irreducible Tits indices
In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field ''k'', not assumed to be algebraically closed. The possible irreducible indices wer ...
References
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*{{Citation , last1=Tits , first1=Jacques , title=Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque , url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002185725 , doi= 10.1515/crll.1971.247.196 , mr=0277536 , year=1971 , journal=
Journal für die reine und angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...