On the Fundamental Series of a Real Semisimple Lie Algebra: Their Irreducibility, Resolutions and Multiplicity Formulae

Enright, Thomas J.

doi:10.2307/1971244

Cited by 67 publications

(44 citation statements)

References 12 publications

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“…We thus begin with some facts regarding Weyl group orbits. The next proposition is proved using the completion functors of T. Enright introduced in [3] and generalized to Kac-Moody algebras in [11]. Observe that, since the morphism of complexes in Proposition 3.8 is unique up to homotopy, the map Wk ó is independent of the choice of the morphism of complexes.…”

Section: Definition Of the Functorsmentioning

confidence: 97%

Local projective resolutions and translation functors for Kac-Moody algebras

Neidhardt¹

1988

Trans. Amer. Math. Soc.

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ABSTRACT. Let g be a Kac-Moody algebra defined by a not necessarily symmetrizable generalized Cartan matrix. We define translation functors and use them to show that the multiplicities (M(wi ■ A) : L(u>2 ■ A)) are independent of the dominant integral weight A, depending only on the elements of the Weyl group. In order to define the translation functors, we introduce the notion of local projective resolutions and use them to develop the machinery of homological algebra in certain categories of g-modules.

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Section: Definition Of the Functorsmentioning

confidence: 97%

Local projective resolutions and translation functors for Kac-Moody algebras

Neidhardt¹

1988

Trans. Amer. Math. Soc.

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“…(w + 1)! This result is given below as Proposition 3.1 and is a mild variation of a result in [5] which concerns only invariant forms. The main result of § 3 is Theorem 3.4 which asserts that if A, Be^(m) and φ e / β (A, B) (resp.…”

Section: T J Enright and R Parthasarathymentioning

confidence: 90%

The transfer of invariant pairings to lattices

Enright¹,

Parthasarathy²

1981

Pacific J. Math.

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“…The composite of the top maps is just the a-module action on Ca(Af) (see [E,Proposition 3.3]). Hence Ca is compatible with the a-module action and we can apply the Lifting Theorem to give us a functor Ca: ¿7g(a) -» ^(a) rendering diagram 4.7.1 commutative.…”

Section: (Continuation Of Example 33)mentioning

confidence: 99%

“…Then by restriction of the action from g to a we obtain an o-module which we denote by the same symbol A. Within representation theory of Lie algebras the following question has arisen naturally in a number of contexts (see [D,E,Jol,Jo2,Jo3,Wa,Z]). What hypothesis on y and A are sufficient to imply that y A admits a g-module structure?…”

Section: Introductionmentioning

confidence: 99%

𝔉-categories and 𝔉-functors in the representation theory of Lie algebras

Cox¹

1994

Trans. Amer. Math. Soc.

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Abstract. The fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring. These include of course Horn, Ext, and Tor as well as the more specialized examples of completion and localization used in the setting of representation theory of a semisimple Lie algebra. In this article we let a be a Lie subalgebra of a Lie algebra g and T be a functor on some category of a-modules. We then consider the following general question: For a g-module E what hypotheses on T and E are sufficient to insure that F(E) admits a canonical structure as a g-module? The article offers an answer through the introduction of the notion of S-categories and ^-functors. The last section of the article treats various examples of this theory.

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On the Fundamental Series of a Real Semisimple Lie Algebra: Their Irreducibility, Resolutions and Multiplicity Formulae

Cited by 67 publications

References 12 publications

Local projective resolutions and translation functors for Kac-Moody algebras

Local projective resolutions and translation functors for Kac-Moody algebras

The transfer of invariant pairings to lattices

𝔉-categories and 𝔉-functors in the representation theory of Lie algebras

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