Representation theory of semisimple Lie algebras information
Lie groups and Lie algebras
Classical groups
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
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Simple Lie groups
Classical
An
Bn
Cn
Dn
Exceptional
G2
F4
E6
E7
E8
Other Lie groups
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Simple Lie algebra
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Dynkin diagrams
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Root system
Weyl group
Real form
Complexification
Split Lie algebra
Compact Lie algebra
Representation theory
Lie group representation
Lie algebra representation
Representation theory of semisimple Lie algebras
Representations of classical Lie groups
Theorem of the highest weight
Borel–Weil–Bott theorem
Lie groups in physics
Particle physics and representation theory
Lorentz group representations
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Scientists
Sophus Lie
Henri Poincaré
Wilhelm Killing
Élie Cartan
Hermann Weyl
Claude Chevalley
Harish-Chandra
Armand Borel
Glossary
Table of Lie groups
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In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory.[1] The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra (over ); in particular, it gives a way to parametrize (or classify) irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.
There is a natural one-to-one correspondence between the finite-dimensional representations of a simply connected compact Lie group K and the finite-dimensional representations of the complex semisimple Lie algebra that is the complexification of the Lie algebra of K (this fact is essentially a special case of the Lie group–Lie algebra correspondence). Also, finite-dimensional representations of a connected compact Lie group can be studied through finite-dimensional representations of the universal cover of such a group. Hence, the representation theory of semisimple Lie algebras marks the starting point for the general theory of representations of connected compact Lie groups.
The theory is a basis for the later works of Harish-Chandra that concern (infinite-dimensional) representation theory of real reductive groups.
^Knapp, A. W. (2003). "Reviewed work: Matrix Groups: An Introduction to Lie Group Theory, Andrew Baker; Lie Groups: An Introduction through Linear Groups, Wulf Rossmann". The American Mathematical Monthly. 110 (5): 446–455. doi:10.2307/3647845. JSTOR 3647845.
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