On the center of the small quantum group

Lachowska, Anna

doi:10.1016/s0021-8693(03)00033-4

Cited by 17 publications

(20 citation statements)

References 12 publications

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“…In the latter case the answer was first found in [Ke95]: the dimension of the center of u q (sl 2 ), with q a primitive root of unity of odd degree l ≥ 3, equals 3l−1 2 , which is unexpectedly large (the number of inequivalent irreducible representations of u q (sl 2 ) is l). For higher rank, [La03] contains a description of the largest known central subalgebra, and provides a lower bound for the dimension of the center. In particular, the principal block of the center of u q (g), whose structure is independent of l by [AJS94], contains a subalgebra of dimension 2|W | − 1, where W is the Weyl group associated with g. For g = sl 2 , this subalgebra coincides with the whole center.…”

Section: Motivationmentioning

confidence: 99%

The Center of Small Quantum Groups I: The Principal Block in Type A

Lachowska

2017

International Mathematics Research Notices

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We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The cases of sl 3 and sl 4 are computed explicitly. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum sl m at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group S m .

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Section: Motivationmentioning

confidence: 99%

The Center of Small Quantum Groups I: The Principal Block in Type A

Lachowska

2017

International Mathematics Research Notices

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“…The χ map of the same elements in Ch yields the localized, i.e., Cardy states [23]. 3 The Drinfeld and Radford-map images of only the (traces over) irreducible representations do not span the entire center; this is an essential complication compared with the quantum sℓ(2) in [20] and in [5].…”

Section: Theorem Multiplication In the Gmentioning

confidence: 99%

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

Feigin

Gainutdinov

Semikhatov

et al. 2007

Journal of Mathematical Physics

181

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ABSTRACT. We derive and study a quantum group g p,q that is Kazhdan-Lusztig-dual to the W -algebra W p,q of the logarithmic (p, q) conformal field theory model. The algebra W p,q is generated by two currents W + (z) and W − (z) of dimension (2p−1)(2q−1), and the energy-momentum tensor T (z). The two currents generate a vertex-operator ideal R with the property that the quotient W p,q /R is the vertex-operator algebra of the (p, q) Virasoro minimal model. The number (2pq) of irreducible g p,q representations is the same as the number of irreducible W p,q -representations on which R acts nontrivially. We find the center of g p,q and show that the modular group representation on it is equivalent to the modular group representation on the W p,q characters and "pseudocharacters." The factorization of the g p,q ribbon element leads to a factorization of the modular group representation on the center. We also find the g p,q Grothendieck ring, which is presumably the "logarithmic" fusion of the (p, q) model. CONTENTS

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“…15 The small quantum groups have been the subject of some constant interest, see, e.g., [46,47,48] and the references therein.…”

Section: 21mentioning

confidence: 99%

Factorizable ribbon quantum groups in logarithmic conformal field theories

Semikhatov

2008

Theor Math Phys

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ABSTRACT. We review the properties of quantum groups occurring as Kazhdan-Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.

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On the center of the small quantum group

Cited by 17 publications

References 12 publications

The Center of Small Quantum Groups I: The Principal Block in Type A

The Center of Small Quantum Groups I: The Principal Block in Type A

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

Factorizable ribbon quantum groups in logarithmic conformal field theories

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