A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

Semikhatov, A. M.

doi:10.1007/s11005-010-0373-9

Cited by 5 publications

(12 citation statements)

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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%

“…From HÔB ¦ Õ to H q sℓÔ2Õ. In HÔB ¦ Õ, dually to the two steps just mentioned, we take a subalgebra and then a quotient [4]. In the basis chosen above, the subalgebra (which is also a U q sℓÔ2Õ submodule) is the one generated by z, , and λ .…”

Section: 22mentioning

confidence: 99%

“…(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").…”

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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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Section: ¬ Ae Ae Ae Ae Ae Aementioning

confidence: 99%

Section: 22mentioning

confidence: 99%

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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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“…1) Логарифмические конформные теории поля расширяют соответствие/двойственность Каждана-Люстига [11] между вертекс-операторными алгебрами и квантовыми группами (см. обзор [12], а также новое развитие в работах [13]- [16]).…”

Section: Introductionunclassified

Действие квантовой группы $s\ell(2)$ на квантовой плоскости с разделенными степенями в четных корнях из единицы

Semikhatov¹

2010

ТМФ

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Описан нестандартный вариант квантовой плоскости, где базисом являются разделенные степени в четном корне из единицы q = e iπ/p. Ее можно рассматривать как расширение "почти коммутативной" алгебры C[X, Y ], в которой XY = (−1) p Y X, за счет нильпотентов. Для этой квантовой плоскости построен комплекс де Рама весс-зуминовского типа и найдено его разложение на представления 2p 3-мерной квантовой группы Uqsℓ(2) и ее люстиговского расширения U qsℓ(2); действие квантовой группы определено также на алгебре квантовых дифференциальных операторов на квантовой плоскости. Ключевые слова: квантовая плоскость, разделенные степени, люстиговская квантовая группа, неразложимое представление.

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“…For the left and right regular actions of a Hopf algebra B on B ¦ , we use the respective notation b β Üβ ¾ , bÝ β ½ and β b Üβ ½ , bÝ β ¾ , where β È B ¦ and b È B (and Ü , Ý is 1 The "true," underlying motivation (deriving from [12,13,14,15,16,17]) of our interest in HÔB ¦ Õ is entirely left out here. 2 A slight mockery of the statistical-mechanics meaning of a "Heisenberg chain" may give way to a genuine, and deep, relation in the context of the previous footnote.…”

Section: Introductionmentioning

confidence: 99%

Yetter--Drinfeld structures on Heisenberg doubles and chains

Semikhatov

2009

Preprint

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For a Hopf algebra B with bijective antipode, we show that the Heisenberg double HÔB ¦ Õ is a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double DÔBÕ. The braiding structure allows generalizing HÔB ¦ Õ B ¦cop ³ B to "Heisenberg n-tuples" and "chains" . . . ³B ¦cop ³B³B ¦cop ³B³..., all of which are Yetter-Drinfeld DÔBÕ-module algebras. 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.

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A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

Cited by 5 publications

References 72 publications

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

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

Действие квантовой группы $s\ell(2)$ на квантовой плоскости с разделенными степенями в четных корнях из единицы

Yetter--Drinfeld structures on Heisenberg doubles and chains

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