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| {{distinguish|Harish-Chandra homomorphism}}
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| In [[mathematics]], the '''Harish-Chandra isomorphism''', introduced by {{harvs|txt|last=Harish-Chandra|year=1951}},
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| is an [[isomorphism]] of commutative rings constructed in the theory of [[Lie algebra]]s. The isomorphism maps the [[center (algebra)|center]] ''Z''(''U''(''g'')) of the [[universal enveloping algebra]] ''U''(''g'') of a [[reductive Lie algebra]] ''g'' to the elements ''S''(''h'')<sup>''W''</sup> of the [[symmetric algebra]] ''S''(''h'') of a [[Cartan subalgebra]] ''h'' that are invariant under the [[Weyl group]] ''W''. | |
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| ==Fundamental invariants==
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| Let ''n'' be the '''rank''' of ''g'', which is the dimension of the Cartan subalgebra ''h''. [[H. S. M. Coxeter]] observed that ''S''(''h'')<sup>''W''</sup> is a [[polynomial ring|polynomial algebra]] in ''n'' variables (see [[Chevalley–Shephard–Todd theorem]] for a more general statement). Therefore, the center of the universal enveloping algebra of a reductive Lie algebra is a polynomial algebra. The degrees of the generators are the degrees of the fundamental invariants given in the following table.
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| {| class="wikitable" style="text-align:center"
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| ! Lie algebra || [[Coxeter number]] ''h'' || [[Dual Coxeter number]] || Degrees of fundamental invariants
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| | '''R'''</sub> || 0 || 0 || 1
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| | A<sub>''n''</sub> || ''n'' + 1 || ''n'' + 1 || 2, 3, 4, ..., ''n'' + 1
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| | B<sub>''n''</sub> || 2''n'' || 2''n'' − 1 || 2, 4, 6, ..., 2''n''
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| | C<sub>''n''</sub> || 2''n'' || ''n'' + 1 || 2, 4, 6, ..., 2''n''
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| | D<sub>''n''</sub> || 2''n'' − 2 || 2''n'' − 2 || ''n''; 2, 4, 6, ..., 2''n'' − 2
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| | E<sub>6</sub> || 12 || 12 || 2, 5, 6, 8, 9, 12
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| | E<sub>7</sub> || 18 || 18 || 2, 6, 8, 10, 12, 14, 18
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| | E<sub>8</sub> || 30 || 30 || 2, 8, 12, 14, 18, 20, 24, 30
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| | F<sub>4</sub> || 12 || 9 || 2, 6, 8, 12
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| | G<sub>2</sub> || 6 || 4 || 2, 6
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| |}
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| For example, the center of the universal enveloping algebra of ''G''<sub>2</sub> is a polynomial algebra on generators of degrees 2 and 6.
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| ==Examples==
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| *If ''g'' is the Lie algebra ''sl''(2, '''R'''), then the center of the universal enveloping algebra is generated by the [[Casimir invariant]] of degree 2, and the Weyl group acts on the Cartan subalgebra, which is isomorphic to '''R''', by negation, so the invariant of the Weyl group is simply the square of the generator of the Cartan subalgebra, which is also of degree 2.
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| == Introduction and setting ==
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| Let ''g'' be a [[semisimple Lie algebra]], ''h'' its [[Cartan subalgebra]] and λ, μ ∈ ''h''* be two elements of the [[weight space]] and assume that a set of [[Positive root#Positive roots and simple roots|positive roots]] Φ<sup>+</sup> have been fixed. Let ''V''<sub>λ</sub>, resp. ''V''<sub>μ</sub> be [[highest weight module]]s with highest weight λ, resp. μ.
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| === Central characters ===
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| The ''g''-modules ''V''<sub>λ</sub> and ''V''<sub>μ</sub> are representations of the [[universal enveloping algebra]] ''U''(''g'') and its [[center (algebra)|center]] acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vector). So, for ''v'' in ''V''<sub>λ</sub> and ''x'' in ''Z''(''U''(''g'')),
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| :<math>x\cdot v:=\chi_\lambda(x)v</math>
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| and similarly for ''V''<sub>μ</sub>.
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| The functions <math>\chi_\lambda, \,\chi_\mu</math> are homomorphims to scalars called ''central characters''.
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| ==Statement of Harish-Chandra theorem==
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| For any λ, μ ∈ ''h''*, the characters <math>\chi_\lambda=\chi_\mu</math> if and only if λ+δ and μ+δ are on the same [[orbit (group theory)|orbit]] of the [[Weyl group]] of ''h''*, where δ is the half-sum of the [[positive root]]s.<ref>Humphreys (1972), p.130</ref>
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| Another closely related formulation is that the [[Harish-Chandra homomorphism]] from the center of the [[universal enveloping algebra]] ''Z''(''U''(''g'')) to ''S''(''h'')<sup>''W''</sup> (the elements of the symmetric algebra of the Cartan subalgebra fixed by the Weyl group) is an [[isomorphism]].
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| == Applications ==
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| The theorem may be used to obtain a simple algebraic proof of [[Weyl's character formula]] for finite dimensional representations.
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| Further, it is a necessary condition for the existence of a nonzero homomorphism of some highest weight modules (a homomorphism of such modules preserves central character). A simple consequence is that for [[Verma module]]s or [[generalized Verma module]]s ''V''<sub>λ</sub> with highest weight λ, there exist only finitely many weights μ such that a nonzero homomorphism ''V''<sub>λ</sub> → ''V''<sub>μ</sub> exists.
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| ==See also==
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| * [[Translation functor]]
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| * [[Universal enveloping algebra]]
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| * [[Infinitesimal character]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| *{{Citation | last1=Harish-Chandra | title=On some applications of the universal enveloping algebra of a semisimple Lie algebra | jstor=1990524 | mr=0044515 | year=1951 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=70 | pages=28–96}}
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| *{{cite isbn|0387900535}}
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| *{{Citation | last1=Humphreys | first1=James E. | author1-link=James Humphreys (mathematician) | title=Representations of semisimple Lie algebras in the BGG category O | publisher=AMS | year=2008 | isbn=978-0-8218-4678-0 | page=26}}
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| *{{Citation | last1=Knapp | first1=Anthony W. | first2=David A. |last2=Vogan| title=Cohomological induction and unitary representations | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series | isbn=978-0-691-03756-1 | mr=1330919 | year=1995 | volume=45}}
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| *Knapp, Anthony, ''Lie groups beyond an introduction'', Second edition, pages 300–303.
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| [[Category:Lie algebras]]
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| [[Category:Representation theory of Lie algebras]]
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| [[Category:Theorems in algebra]]
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