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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 algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...s they are a different way of classifying simple Lie algebras. References * Lie algebras {{Algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Vogan
David Alexander Vogan, Jr. (born September 8, 1954) is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups. While studying at the University of Chicago, he became a Putnam Fellow in 1972. He received his Ph.D. from M.I.T. in 1976, under the supervision of Bertram Kostant. In his thesis, he introduced the notion of lowest K type in the course of obtaining an algebraic classification of irreducible Harish Chandra modules. He is currently one of the participants in the Atlas of Lie Groups and Representations. Vogan was elected to the American Academy of Arts and Sciences in 1996. He served as Head of the Department of Mathematics at MIT from 1999 to 2004. In 2012 he became Fellow of the American Mathematical Society. He was president of the AMS in 2013–2014. He was elected to the National Academy of Sciences in 2013. He was the Norbert Wiener Chair of Mathematics at MIT until his retirement in 2020. Publica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' Dynkin diagrams will be explicitly so named. Classification of semisimple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [x,y]. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative property, associative. Lie algebras are closely related to Lie groups, which are group (mathematics), groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected space, connected Lie group unique up to finite coverings (Lie's third theorem). This Lie group–Lie algebra correspondence, correspondence allows one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Satake Diagram
In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. 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 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 reductive 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
