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

Abstract

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

Heisenberg double as braided commutative Yetter–Drinfel’d module algebra over Drinfel’d double in multiplier Hopf algebra case

“…Recall from [10] that for a finite dimensional Hopf algebra B and its dual B * , there is an action of Drinfel'd double B * ⊲⊳ B on the Heisenberg double…”

“…As shown in [10], for a finite dimensional Hopf algebra H, there is a Yetter-Drinfel'd D(H)-module algebra structure on the Heisenberg double H(H * ) endowed with a heterotic action of the Drinfel'd double D(H), moreover H(H * ) is braided commutative in terms of the braiding of Yetter-Drinfel'd module.…”

“…In section 4, we apply the results as above to the usual Hopf algebras and derive some interesting results, including the main results in paper [10] as a corollary.…”

Fusion in the entwined category of Yetter-Drinfeld modules of a rank-1 Nichols algebra

“…(1.1) xpr 1 q ν 1 xpr 2 q ν 2 " p´1´|r 1`r2´p | ÿ s"|r 1´r2 |`1 step"2 xpsq ν 1`ν2`p ÿ s"2p´r 1´r2`1 step"2 ppsq ν 1`ν2 , with pprq ν " # 2 xprq ν`2 xpp´rq ν`1 , r ă p, xppq ν , r " p. This is the FHST fusion algebra [45] (also see [12]), which makes part of what we know from [11] (also see [46]) to be an equivalence of representation categories-of the triplet algebra W ppq in the pp, 1q logarithmic conformal models [47,48,49,50,45] and of a small quantum sℓ 2 at the 2pth root of unity, proposed in this capacity in [7,8] and then used and studied, in particular, in [51,52,53,54,55] (this quantum group had appeared before in [56,57,58]).…”

Фьюжн В Обвивающей Категории Модулей Йеттера - Дринфельда Над Алгеброй Николса Ранга 1

“…также [46]), является эквивалентностью категорийкатегории представлений триплетной алгебры W ppq в pp, 1q-логарифмических конформных моделях [45], [47]- [50] и категории представлений малой квантовой группы sℓ 2 в p2pq-м корне из единицы. Данная квантовая группа была предложена в этом качестве в работах [7], [8], а затем использовалась и изучалась, в частности, в работах [51]- [55] (она появлялась ранее в работах [56]- [58]). Тот факт, что фьюжн-алгебра возникает в сплетенном подходе, пропагандируемом в работе [28], вместе с рядом других наблюдений свидетельствует в пользу того, что в отношении логарифмического варианта соответствия Каждана-Люстига (соответствия между вертекс-операторными алгебрами и квантовыми группами) алгебры Николса оказываются во всяком случае ничуть не хуже предложенных ранее квантовых групп [7]- [10], [ (1.3) Разложения (1.2) были высказаны в качестве гипотезы в работе [28], и мы доказываем их в настоящей работе.…”