A differential U-module algebra for U = Ū q sℓ(2) at an even root of unity

Semikhatov, A. M.

doi:10.1007/s11232-009-0035-1

Cited by 4 publications

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References 44 publications

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“…Ön× n¡1 , k 2 ⊲ n q ¡2n n , F ⊲ n ¡q n Ön× n 1 As we have already noted (and as is very clearly seen now), the action restricts to HÔB ¦ Õ and then pushes forward to H q sℓÔ2Õ. There, it restricts to the subalgebra C q Öz, ×, and the isomorphism C q Öz, × Mat p ÔCÕ is actually that of U q sℓÔ2Õ-module algebras [49].…”

Section: The Structure Ofsupporting

confidence: 53%

“…By "truncation," DÔBÕ yields the U q sℓÔ2Õ quantum group that is Kazhdan-Lusztig-dual to the Ôp,1Õ logarithmic models. This quantum group first appeared in [45] and was rediscovered, together with its role in the Kazhdan-Lusztig correspondence, in [7]; its further properties were considered in [29,46,47,48,49] and, notably, recently in [50] (also see [51] for a somewhat larger quantum group). We recall this in 3.1.…”

Section: Remarkmentioning

confidence: 99%

“…where C q Öz, × is the p 2 -dimensional algebra defined by relations (3.11) and (3.12). It is indeed isomorphic to the full matrix algebra Mat p ÔCÕ [49] (also see [52]); hence, H q sℓÔ2Õ Mat p CÖλ ×ßÔλ 2p ¡1Õ¨.…”

Section: The Structure Ofmentioning

confidence: 99%

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A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

Semikhatov

2010

Lett Math Phys

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For a Hopf algebra B, we endow the Heisenberg double HÔB ¦ Õ with the structure of a module algebra over the Drinfeld double DÔBÕ. Based on this property, we propose that HÔB ¦ Õ is to be the counterpart of the algebra of fields on the quantumgroup side of the Kazhdan-Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the U q sℓÔ2Õ quantum group that is Kazhdan-Lusztig-dual to Ôp,1Õ logarithmic conformal models. The corresponding pair ÔDÔBÕ,HÔB ¦ ÕÕ is "truncated" to ÔU q sℓÔ2Õ, H q sℓÔ2ÕÕ, where H q sℓÔ2Õ is a U q sℓÔ2Õ module algebra that turns out to have the form H q sℓÔ2Õ C q Öz, × CÖλ ×ßÔλ 2p ¡ 1Õ, where C q Öz, × is the U q sℓÔ2Õ-module algebra with the relations z p 0, p 0, and z q ¡q ¡1 q ¡2 z .

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Section: The Structure Ofsupporting

confidence: 53%

Section: Remarkmentioning

confidence: 99%

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A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

Semikhatov

2010

Lett Math Phys

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“…The "Heisenberg counterpart" of U q sℓÔ2Õ, its braided commutative Yetter-Drinfeld module algebra H q sℓÔ2Õ, is also likely to play a role in the Kazhdan-Lusztig context [39,4], but this is a subject of future work.…”

Section: ¬ Ae Ae Ae Ae Ae Aementioning

confidence: 99%

“…C q Öz, × in (3.7) -the algebra of "quantum differential operators on a line"-is also a braided commutative Yetter-Drinfeld U q sℓÔ2Õ-module algebra. It is in fact the full matrix algebra [39],…”

Section: Matrix Braided Commutative Yetter-drinfeld Module Algebras mentioning

confidence: 99%

Heisenberg Double ℋ(B*) as a Braided Commutative Yetter–Drinfeld Module Algebra Over the Drinfeld Double

Semikhatov¹

2011

Communications in Algebra

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We study the Yetter-Drinfeld DÔBÕ-module algebra structure on the Heisenberg double HÔB ¦ Õ endowed with a "heterotic" action of the Drinfeld double DÔBÕ. This action can be interpreted in the spirit of Lu's description of HÔB ¦ Õ as a twist of DÔBÕ. In terms of the braiding of Yetter-Drinfeld modules, HÔB ¦ Õ is braided commutative. By the Brzeziński-Militaru theorem, HÔB ¦ Õ DÔBÕ is then a Hopf algebroid over HÔB ¦ Õ. For B a particular Taft Hopf algebra at a 2pth root of unity, the construction is adapted to yield Yetter-Drinfeld module algebras over the 2p 3 -dimensional quantum group U q sℓÔ2Õ. In particular, it follows that Mat p ÔCÕ is a braided commutative Yetter-Drinfeld U q sℓÔ2Õmodule algebra and Mat p ÔU q sℓÔ2ÕÕ is a Hopf algebroid over Mat p ÔCÕ.where, in our case, M È DÔBÕ and A È HÔB ¦ Õ. 1We recall from [1, 2, 3] that for a Hopf algebra H, a left H-module and left H-comodule algebra X is said to be braided commutativeAlso, for any two (left-left) Yetter-Drinfeld H-module algebras X and Y , their braided product X ³Y is defined as the tensor product with the composition(This gives a Yetter-Drinfeld module algebra.) 1.2. Theorem. HÔB ¦ Õ is a braided (DÔBÕ-) commutative algebra. Moreover, HÔB ¦ Õ is the braided product HÔB ¦ Õ B ¦cop ³B, where B ¦cop and B are (braided commutative) Yetter-Drinfeld DÔBÕ module algebras by restriction, i.e., with the DÔBÕ action1.2.1. As a corollary, the Brzeziński-Militaru theorem [2] then "provides one with a rich source of examples of bialgebroids." In particular, for any Hopf algebra B with bijective antipode, the "quadruple" HÔB ¦ Õ DÔBÕ, where the smash product is defined with respect to action (1.2), is a Hopf algebroid over HÔB ¦ Õ. (1.2). The DÔBÕ-action (1.2) first appeared in [4]. To borrow a popular term from string theory [5] (where it was also a borrowing originally), this action may be termed "heterotic" because it is constructed by combining left and right DÔBÕ actions, as we describe in 2.2.2 (and the heterotic string famously combines "left" and "right"). Or because (1.2) "cross-breeds" regular and adjoint actions. A "pseudoadjoint" interpretation ofTrying to quantify how "far" (1.2) is from the adjoint action, we arrive at a useful interpretation of our "heterotic" action by extending Lu's description of the product on HÔB ¦ Õ as a twist of the product on DÔBÕ [6]. The two algebraic structures, DÔBÕ and HÔB ¦ Õ, are defined on the same vector space B ¦ B, and the product (1.1) in HÔB ¦ Õ, temporarily denoted by AE, can be written asfor a certain 2-cocycle η : DÔBÕ DÔBÕ k [6]. In the same vein, the DÔBÕ action on 1 For a Hopf algebra H and a left H-comodule X, we write the coaction δ : X H X as δ ÔxÕ x Ô¡1Õ x Ô0Õ ; then the comodule axioms are Üε,x Ô¡1Õ Ýx Ô0Õ x and x ½

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Quantum sℓ(2) action on a divided-power quantum plane at even roots of unity

Semikhatov

2010

Theor Math Phys

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We describe a nonstandard version of the quantum plane in which the basis is given by divided powers at an even root of unity q = e iπ/p . It can be regarded as an extension of the "nearly commutative" algebra C[X, Y ] with XY = (−1) p Y X by nilpotents. For this quantum plane, we construct a Wess-Zumino-type de Rham complex and find its decomposition into representations of the 2p 3 -dimensional quantum group Uqs (2) and its Lusztig extension U qs (2); we also define the quantum group action on the algebra of quantum differential operators on the quantum plane.

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A differential U-module algebra for U = Ū q sℓ(2) at an even root of unity

Cited by 4 publications

References 44 publications

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

Heisenberg Double ℋ(B*) as a Braided Commutative Yetter–Drinfeld Module Algebra Over the Drinfeld Double

Quantum sℓ(2) action on a divided-power quantum plane at even roots of unity

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