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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 ...
study of
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
s and
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
s, a Satake diagram is a generalization of a
Dynkin diagram In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebr ...
introduced by whose configurations classify simple Lie algebras over the field of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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. ...
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
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...
reductive algebraic groups. Satake diagrams are not the same as
Vogan diagram In mathematics, a Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagram In the mathematical study of Lie alge ...
s 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

*
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 alge ...
s correspond to the Satake diagram with all vertices blackened. * Split Lie algebras correspond to the Satake diagram with only white (i.e., non blackened) and unpaired vertices. * A table can be found at .


Differences between Satake and Vogan diagrams

Both Satake and
Vogan diagram In mathematics, a Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Satake diagram In the mathematical study of Lie alge ...
s 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 nat ...
, 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 \mathfrak with Cartan involution ''θ'' and associated Cartan pair \mathfrak = \mathfrak \oplus \mathfrak (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 � ...
\mathfrak, that is, one for which \theta(\mathfrak)=\mathfrak and \mathfrak\cap\mathfrak is as small as possible (in the presentation above, \mathfrak 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 \theta(\mathfrak)=\mathfrak and \mathfrak\cap\mathfrak 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

* Relative root system * List of irreducible Tits indices


References

* * * * * * * *{{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 language, English: ''Journal for Pure and Applied Mathematics''). History The journal wa ...
, issn=0075-4102 , volume=1971 , issue=247 , pages=196–220, s2cid=116999784 Lie algebras