Heisenberg Double, Pentagon Equation, Structure and Classification of Finite-Dimensional Hopf Algebras

Militaru, Gigel

doi:10.1112/s0024610703004897

Cited by 11 publications

(7 citation statements)

References 15 publications

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“…2 For any locally compact quantum group, a multiplicative unitary can be constructed in terms of the coproduct. Any finite-dimensional Hopf algebra is characterized by an invertible solution of the pentagon equation [83]. The pentagon equation arises as a 3-cocycle condition in Lie group cohomology and also in a category-theoretical framework (see, e.g., [94]).…”

Section: Introductionmentioning

confidence: 99%

Simplex and Polygon Equations

Dimakis

Müller‐Hoissen

2015

SIGMA

103

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Abstract. It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a "mixed order". We describe simplex equations (including the Yang-Baxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of "polygon equations" realizes higher Tamari orders. They generalize the well-known pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1-skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the N -simplex equation to the (N + 1)-gon equation, its dual, and a compatibility equation.

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Section: Introductionmentioning

confidence: 99%

Simplex and Polygon Equations

Dimakis

Müller‐Hoissen

2015

SIGMA

103

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“…There are also some similarities in constructing solutions of both equations, for example the role of the Drinfeld double construction [30] of solutions of the quantum Yang-Baxter equation is replaced by the Heisenberg double [72,76,55,45]. However, in modern theory of quantum groups [4,78,53] (see also [74] for a review, and [58] for discussion of the finite dimensional case) the quantum pentagon equation seems to play more profound role. Remarkably, given a solution of (1.1) satisfying some additional non-degeneracy conditions, it allows to construct all the remaining structure maps of a quantum group and of its Pontrjagin dual simultaneously.…”

Section: Introductionmentioning

confidence: 99%

The pentagon relation and incidence geometry

Doliwa

Sergeev

2014

Journal of Mathematical Physics

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Abstract. We define a map S : D 2 × D 2 D 2 × D 2 , where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev-Petviashvili equation. Finally we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure -the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

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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%

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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Heisenberg Double, Pentagon Equation, Structure and Classification of Finite-Dimensional Hopf Algebras

Cited by 11 publications

References 15 publications

Simplex and Polygon Equations

Simplex and Polygon Equations

The pentagon relation and incidence geometry

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

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