Super Duality and Homology of Unitarizable Modules of Lie Algebras

Huang, Po Yi; Lam, Ngau; To, Tze Ming

doi:10.2977/prims/60

Cited by 7 publications

(3 citation statements)

References 5 publications

(13 reference statements)

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“…Corollary 5.8 provides an alternative description involving an infinite Weyl group. Ngau Lam has informed us that both forms of the formula are equivalent[19].…”

mentioning

confidence: 98%

Kostant homology formulas for oscillator modules of Lie superalgebras

Cheng

Kwon

Wang

2010

Advances in Mathematics

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We provide a systematic approach to obtain formulas for characters and Kostant ${\mathfrak u}$-homology groups of the oscillator modules of the finite dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned ${\mathfrak u}$-homology computation are worked out

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“…Corollary 5.8 provides an alternative description involving an infinite Weyl group. Ngau Lam has informed us that both forms of the formula are equivalent[19].…”

mentioning

confidence: 98%

Kostant homology formulas for oscillator modules of Lie superalgebras

Cheng

Kwon

Wang

2010

Advances in Mathematics

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“…in Proposition 3.1 is a monomorphism of Lie superalgebras with * -structures, and we have shown that the map ϕ given in Lemma 2.3 is an isomorphism of Lie superalgebras with * -structures from (g given in [EHW,Sections 8 and 9] (see also [HLT,Theorem 2.5]).…”

Section: Unitarizable Modules Overmentioning

confidence: 83%

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Cheong¹,

Lam²

2023

Preprint

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Let L be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show that the Gaudin Hamiltonians associated to G are diagonalizable with simple spectrum on the space spanned by singular vectors of any fixed weight in L. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the space spanned by singular vectors of any fixed weight in the tensor product of infinitedimensional unitarizable irreducible highest weight modules.

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“…The Weyl group W x n of g x n is defined to be the subgroup of Aut((h x n ) * ) generated by all σ α (cf. [HLT,Section 2.3]).…”

Section: Lie Algebras Of Infinite and Finite Ranksmentioning

confidence: 99%

Projective modules over classical Lie algebras of infinite rank in the parabolic category

Chen

Lam

2020

Journal of Pure and Applied Algebra

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We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a, b, c, d. Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O over infinite rank Lie superalgebra of types a, b, c, d through the super duality is also a Koszul category.Notations. Throughout the paper the symbols Z, N, and Z + stand for the sets of all, positive and non-negative integers, respectively. All vector spaces, algebras, tensor products, et cetera, are over the field of complex numbers C.

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Super Duality and Homology of Unitarizable Modules of Lie Algebras

Cited by 7 publications

References 5 publications

Kostant homology formulas for oscillator modules of Lie superalgebras

Kostant homology formulas for oscillator modules of Lie superalgebras

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Projective modules over classical Lie algebras of infinite rank in the parabolic category

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