Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them

Panaite, Florin

doi:10.1090/conm/441/08501

Cited by 21 publications

(11 citation statements)

References 45 publications

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“…We also note that the proper definition of A would require the quasi-Hopf algebra framework [26]. Defining the Heisenberg double within the quasi-Hopf algebra setting has been proposed, for example, in [27,28] and would be worth investigating further.…”

Section: Appendix a Heisenberg Double Constructionmentioning

confidence: 99%

Heisenberg Doubles for Snyder-Type Models

Meljanac

Pachoł

2021

Symmetry

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A Snyder model generated by the noncommutative coordinates and Lorentz generators closes a Lie algebra. The application of the Heisenberg double construction is investigated for the Snyder coordinates and momenta generators. This leads to the phase space of the Snyder model. Further, the extended Snyder algebra is constructed by using the Lorentz algebra, in one dimension higher. The dual pair of extended Snyder algebra and extended Snyder group is then formulated. Two Heisenberg doubles are considered, one with the conjugate tensorial momenta and another with the Lorentz matrices. Explicit formulae for all Heisenberg doubles are given.

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Section: Appendix a Heisenberg Double Constructionmentioning

confidence: 99%

Heisenberg Doubles for Snyder-Type Models

Meljanac

Pachoł

2021

Symmetry

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“…[8,9,10,11,12,1]. Heisenberg doubles [13,14,15,6], among various smash products, have attracted some attention, notably in relation to Hopf algebroid constructions [16,17,2] (the basic observation being that HÔB ¦ Õ is a Hopf algebroid over B ¦ [16]) and also from various other standpoints and for different purposes [18,7,19,20]. (A relatively recent paper where Yetter-Drinfeld-like structures are studied in relation to "smash" products is [21].)…”

Section: 3mentioning

confidence: 99%

“…This is known, e.g., from [17], where both these actions are discussed and references to the previous works are given. The DÔBÕ action on B ¦ is obtained by restricting the left regular action of DÔBÕ on DÔBÕ ¦ B B ¦ [6],…”

Section: 22mentioning

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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“…For this, we combine two well-known DÔBÕ actions, which can be taken from different sources, among which we prefer the beautiful paper [40].…”

Section: Introductionmentioning

confidence: 99%

“…An algebra A carrying an action of a quasitriangular Hopf algebra H is called R-commutative, or quantum commutative[43,40] if ab ÔR Ô2Õ .bÕÔR Ô1Õ .aÕ for all a, b È A, where the dot denotes the action and R R Ô1Õ R Ô2Õ È H H is the universal R-matrix 3. The coproduct in (2.4) refers to DÔBÕ, and hence, in accordance with the Drinfeld double construction, Ôµ mÕ ½ Ôµ mÕ ¾ Ôµ ¾ m ½ Õ Ôµ ½ m ¾ Õ, with the coproducts of B ¦ and B in the right-hand side.…”

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confidence: 99%

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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Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them

Cited by 21 publications

References 45 publications

Heisenberg Doubles for Snyder-Type Models

Heisenberg Doubles for Snyder-Type Models

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

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

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