Harish-Chandra class

Harish-Chandra class

In mathematics, Harish-Chandra's class is a class of Lie groups used in representation theory. Harish-Chandra's class contains all semisimple connected linear Lie groups and is closed under natural operations, most importantly, the passage to Levi subgroups. This closure property is crucial for many inductive arguments in representation theory of Lie groups, whereas the classes of semisimple or connected semisimple Lie groups are not closed in this sense.

Definition

A Lie group "G" with the Lie algebra "g" is said to be in Harish-Chandra's class if it satisfies the following conditions:
*"g" is a reductive Lie algebra (the product of a semisimple and abelian Lie algebra).
*The Lie group "G" has only a finite number of connected components.
*The adjoint action of any element of "G" on "g" is given by an action of an element of the connected component of the Lie group of the complexification "g"⊗C.
*The subgroup "G"ss of "G" generated by the image of the semisimple part "g"ss= ["g","g"] of the Lie algebra "g" under the exponential map has finite center.

References

*A. W. Knapp, "Structure theory of Lie groups", in ISBN 0-8218-0609-2


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Harish-Chandra — for|the character in Hindu mythology|HarishchandraHarish Chandra (11 October 1923 16 October 1983) was an Indian born American mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups …   Wikipedia

  • Harish-Chandra Research Institute — Established 1966 Type Research Institution Director Dr. Amitava Raychaudhuri Postgraduates 40 graduate students and 20 post doctoral fellows …   Wikipedia

  • Harish-Chandra character — In mathematics, the Harish Chandra character, named after Harish Chandra, of a representation of a semisimple Lie group G on a Hilbert space H is a distribution on the group G that is analogous to the character of a finite dimensional… …   Wikipedia

  • Charu Chandra Bhattacharya — Born June 19, 1883(1883 06 19) South 24 Parganas, Bengal Province, British India Died August 26, 1961(1961 08 26) (aged 78) Resting place Calcutta, Shantiniketan …   Wikipedia

  • List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

  • Langlands classification — In mathematics, the Langlands classification is a classification of irreducible representations of a reductive Lie group G , suggested by Robert Langlands (1973). More precisely, it classifies the irreducible admissible ( g , K ) modules,for g a… …   Wikipedia

  • Plancherel theorem for spherical functions — In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish Chandra. It is a natural generalisation in non commutative harmonic… …   Wikipedia

  • Zonal spherical function — In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K invariant vector in an… …   Wikipedia

  • Tempered representation — In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the L p space : L 2+ epsilon;( G ) for any epsilon; gt; 0. FormulationThis condition, as just given,… …   Wikipedia

  • Séminaire Nicolas Bourbaki (1950–1959) — Continuation of the Séminaire Nicolas Bourbaki programme, for the 1950s. 1950/51 series *33 Armand Borel, Sous groupes compacts maximaux des groupes de Lie, d après Cartan, Iwasawa et Mostow (maximal compact subgroups) *34 Henri Cartan, Espaces… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”