Center and Representations of Infinitesimal Hecke Algebras of 𝔰𝔩<sub>2</sub>

Tikaradze, Akaki; Khare, Apoorva

doi:10.1080/00927870903448740

Cited by 10 publications

(8 citation statements)

References 7 publications

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“…One change from earlier theories of the Category O, is in our allowing "non-strict" RTAs in our setup; this is needed if we want to include infinitesimal Hecke algebras (not over sl 2 ). See [Kh2,KT] for more details.…”

Section: Setups -The General and Hopf Casesmentioning

confidence: 99%

Functoriality of the BGG Category 𝒪††

Khare

2009

Communications in Algebra

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This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group Γ acting on a regular triangular algebra A. We develop Clifford theory for A ⋊ Γ, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when A satisfies one of the "Conditions (S)"; the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module.Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories O; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup -and which yield information about O.

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Section: Setups -The General and Hopf Casesmentioning

confidence: 99%

Functoriality of the BGG Category 𝒪††

Khare

2009

Communications in Algebra

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“…The deformed smash product algebras H λ,κ = H λ,κ (A, V ) encompass a very large family of deformations considered in the literature, including universal enveloping algebras, skew group algebras, Drinfeld orbifold algebras, Drinfeld Hecke algebras, symplectic reflection algebras, rational Cherednik algebras, degenerate affine Hecke algebras and graded Hecke algebras, Weyl algebras, infinitesimal Hecke algebras, and many others. This is an area of research that is the focus of tremendous recent activity; see [11][12][13][14]24,31,32,[40][41][42], and subsequent follow-up works in the literature.…”

Section: 1mentioning

confidence: 99%

Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property

Khare¹

2017

Groups, Rings, Group Rings, and Hopf Algebras

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Abstract. Poincaré-Birkhoff-Witt (PBW) Theorems have attracted significant attention since the work of Drinfeld (1986), Lusztig (1989), and EtingofGinzburg (2002) on deformations of skew group algebras H ⋉ Sym(V ), as well as for other cocommutative Hopf algebras H. In this paper we show that such PBW theorems do not require the full Hopf algebra structure, by working in the more general setting of a "cocommutative algebra", which involves a coproduct but not a counit or antipode. Special cases include infinitesimal Hecke algebras, as well as symplectic reflection algebras, rational Cherednik algebras, and more generally, Drinfeld orbifold algebras. In this generality we identify precise conditions that are equivalent to the PBW property, including a Yetter-Drinfeld type compatibility condition and a Jacobi identity. We also characterize the graded deformations that possess the PBW property. In turn, the PBW property helps identify an analogue of symplectic reflections in general cocommutative bialgebras.Next, we introduce a family of cocommutative algebras outside the traditionally studied settings: generalized nil-Coxeter algebras. These are necessarily not Hopf algebras, in fact, not even (weak) bialgebras. For the corresponding family of deformed smash product algebras, we compute the center as well as abelianization, and classify all simple modules.

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“…,1 . The full central reduction of (♣) provides an isomorphism of [T3,Theorem 3.1]. 11 In Appendix C, we establish explicitly suitably modified versions of (*) and (♠) for the cases m = −1, 0, which do not follow from the above arguments.…”

Section: Completionsmentioning

confidence: 99%