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Algebra Structures Arising from Yang–Baxter SystemsBarbu R. Berceanu a b , Florin F. Nichita b & Călin Popescu b

a Abdus Salam School of Mathematical Sciences , GC University , Lahore , Pakistanb Institute of Mathematics of the Romanian Academy , Bucharest , Romania

To cite this article: Barbu R. Berceanu , Florin F. Nichita & Călin Popescu (2013) Algebra Structures Arising from Yang–BaxterSystems, Communications in Algebra, 41:12, 4442-4452, DOI: 10.1080/00927872.2012.703736

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Communications in Algebra®, 41: 4442–4452, 2013Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2012.703736

ALGEBRA STRUCTURES ARISING FROMYANG–BAXTER SYSTEMS

Barbu R. Berceanu1�2, Florin F. Nichita2, and Calin Popescu21Abdus Salam School of Mathematical Sciences, GC University,Lahore, Pakistan2Institute of Mathematics of the Romanian Academy, Bucharest, Romania

Yang–Baxter operators from algebra structures appeared for the first time in [11, 22,23]. Later, Yang–Baxter systems from entwining structures were constructed in [8].In fact, Yang–Baxter systems are equivalent with braid systems. In this paper we showthat braidings and entwinings of various algebraic structures—in particular, algebrafactorisations—can be constructed from a braid system, whence from a Yang–Baxtersystem as well.

Key Words: Algebra factorisation; Braid system; Entwining structure; Yang–Baxter system.

2010 Mathematics Subject Classification: 16T20; 16T25; 17B37; 81R50.

1. INTRODUCTION

The quantum Yang–Baxter equation plays a crucial role in theoretical physics,in the theory of quantum groups, in knot theory, in the theory of braided monoidalcategories, etc. There are several versions of Yang–Baxter equations, and they arerelated to various algebraic structures.

Due to the complexity of various integrable models the need arose forextensions and generalizations of Yang–Baxter equations and related algebraicstructures. One such generalization, termed a Yang–Baxter system, was proposed in[15] in the context of non-ultralocal integrable systems [14]. Motivated by the needfor developing a general theory of non-commutative principal bundles, entwiningstructures were introduced in [7] as generalized symmetries of such bundles. Inrecent years, entwining structures and categories of modules associated to them havebeen used to provide a unification of various types of Hopf modules (cf. [5, 10]),and eventually have led to the revival of the theory of corings (cf. [6, 9]). Thus,Yang–Baxter systems and entwining structures arose in entirely different contexts,were motivated by as far fields as high energy physics and non-commutativegeometry, and have entirely different applications. There was no reason to expectany connection between the two. Yet, in [8], we proved that such a connection, and

Received December 15, 2011; Revised May 31, 2012. Communicated by T. Albu.Address correspondence to Florin F. Nichita, Institute of Mathematics of the Romanian

Academy, P.O. Box 1-764, Bucharest RO-70700, Romania; E-mail: Florin.Nichita@imar.ro

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ALGEBRA STRUCTURES FROM BRAID SYSTEM 4443

a very close one, indeed exists. More precisely, we showed that with any entwiningstructure, one can associate a Yang–Baxter system.

In this paper, we take another approach. We start with a braid system andconstruct braidings and entwinings of various algebraic structures—in particular,algebra factorisations. (The braid systems are equivalent with the Yang–Baxtersystems.) As observed in [7, Proposition 2.7], entwining structures are related toalgebra factorisations via semi-dualisation (see [19, p. 300] for a discussion of algebrafactorisations). While Yang–Baxter operators from algebra structures appeared forthe first time in the context of non-commutative descent theory and quantum groups(see [11, 22, 23]), the framework and the applications of our current constructionscould be in theoretical physics, topology, the theory of Nichols algebras, etc (see [2,12, 16]).

The next section is devoted to a brief review of Yang–Baxter operators,Yang–Baxter systems, entwining structures, algebra factorisations and previousresults on the subject.

Braid systems and their connection to Yang–Baxter systems are consideredin Section 3 which deals with the main results: tensor extensions of braid systemsand braidings and entwinings of various algebraic structures—in particular, algebrafactorisations—arising from braid systems, whence from Yang–Baxter systemsas well.

2. PRELIMINARIES AND REVIEW

Unless otherwise stated, throughout this paper, k is a field and unadornedtensor products are over k. The identity map on a k-vector space V is also denotedby V . All algebras are over k. They are associative and with unit 1. The productin an algebra A is denoted by � � A⊗ A → A, and the unit map is denoted by � �k → A. All coalgebras are over k. They are coassociative and with a counit. Thecoproduct in a coalgebra C is denoted by � � C → C ⊗ C and the counit by � �C → k. We use the standard notation for coalgebras; in particular, for a coalgebraC, we use Sweedler’s notation to denote the coproduct � on elements, i.e., ��c� =∑

c�1� ⊗ c�2�, for all c ∈ C.For any vector spaces V and W , V�W � V ⊗W → W ⊗ V denotes the natural

bijection defined by V�W �v⊗ w� = w⊗ v. Let R � V ⊗ V → V ⊗ V be a k-linear map.Define R12 = R⊗ V , R23 = V ⊗ R, and R13 = �V ⊗ V�V ��R⊗ V��V ⊗ V�V �. Each ofthe Rij is thus a linear endomorphism of V ⊗ V ⊗ V .

An invertible k-linear map R � V ⊗ V → V ⊗ V is called a Yang–Baxteroperator (or simply a YB operator) if it satisfies the equation

R12R23R12 = R23R12R23 (2.1)

Equation (2.1) is usually called the braid equation. It is a well-known fact thatthe operator R satisfies (2.1) if and only if RV�V satisfies the quantum Yang–Baxterequation

R12R13R23 = R23R13R12 (2.2)

For a review of Yang–Baxter operators we refer to [18, 19, 21].

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4444 BERCEANU ET AL.

Example 2.1. The following construction of Yang–Baxter operators is describedin [11]. If A is a k-algebra, then for all nonzero r� s ∈ k, the linear map

RAr�s � A⊗ A → A⊗ A� a⊗ b �→ sab ⊗ 1+ r1⊗ ab − sa⊗ b (2.3)

is a Yang–Baxter operator. The inverse of RAr�s is �RA

r�s�−1�a⊗ b� = 1

rab ⊗ 1+ 1

s1⊗

ab − 1sa⊗ b.

Example 2.2 (From [11]). If C is a coalgebra, then for all nonzero p� t ∈ k, thelinear map

Rp�tC � C ⊗ C → C ⊗ C� c ⊗ d �→ p��c�

∑d�1� ⊗ d�2� + t��d�

∑c�1� ⊗ c�2� − pc ⊗ d

(2.4)

is a Yang–Baxter operator.

Yang–Baxter systems were introduced in [15] as a spectral-parameterindependent generalization of quantum Yang–Baxter equations related to non-ultralocal integrable systems studied previously in [14].

Yang–Baxter systems are conveniently defined in terms of Yang–Baxtercommutators—in Section 3, braid commutators, a close relative of Yang–Baxtercommutators, will be considered and the connection between the two discussed.Given three vector spaces V , V ′, V ′′ and three linear maps R � V ⊗ V ′ → V ⊗V ′, S � V ⊗ V ′′ → V ⊗ V ′′ and T � V ′ ⊗ V ′′ → V ′ ⊗ V ′′, the Yang–Baxter commutator�R� S� T� � V ⊗ V ′ ⊗ V ′′ → V ⊗ V ′ ⊗ V ′′ is defined by

�R� S� T� = R12S13T23 − T23S13R12 (2.5)

In terms of a Yang–Baxter commutator, the quantum Yang–Baxter Eq. (2.2) simplyreads �R� R�R� = 0.

Let V and V ′ be vector spaces. A system of linear maps

W � V ⊗ V → V ⊗ V� Z � V ′ ⊗ V ′ → V ′ ⊗ V ′� X � V ⊗ V ′ → V ⊗ V ′

is called a WXZ-system or a Yang–Baxter system, if the following equations aresatisfied:

�W�W�W� = 0� �W�X�X� = 0� �X�X� Z� = 0 and �Z� Z� Z� = 0

There are several algebraic origins and applications of WXZ-systems (see, forexample, [25, 26], etc.). In fact, Yang–Baxter systems are a special case of braidsystems considered in the next section.

Remark 2.3. Given a WXZ-system, one can construct a Yang–Baxter operator onV ⊕ V ′, provided the map X is invertible. This is a special case of a gluing proceduredescribed in [20, Theorem 2.7] (cf. [20, Example 2.11]). Let R = WV�V , R

′ = ZV ′�V ′ ,U = XV ′�V . Then the linear map

R⊕U R′ � �V ⊕ V ′�⊗ �V ⊕ V ′� → �V ⊕ V ′�⊗ �V ⊕ V ′��

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ALGEBRA STRUCTURES FROM BRAID SYSTEM 4445

given by R⊕U R′�V⊗V = R, R⊕U R′�V ′⊗V ′ = R′, and �R⊕U R′��x′ ⊗ x� = U�x′ ⊗ x�,�R⊕U R′��x⊗ x′� = U−1�x⊗ x′�, x ∈ V , x′ ∈ V ′, is a Yang–Baxter operator.

Entwining structures were introduced in [7] in order to recapture the symmetrystructure of non-commutative (coalgebra) principal bundles or coalgebra-Galoisextensions. For applications and algebraic content of entwining structures, we referto [9, 10].

An algebra A is said to be entwined with a coalgebra C if there exists a linearmap � C ⊗ A → A⊗ C satisfying the following four conditions:

(1) �C ⊗ �� = ��⊗ C��A⊗ �� ⊗ A�;(2) �A⊗ �� = � ⊗ C��C ⊗ ���⊗ A�;(3) �C ⊗ �� = �⊗ C;(4) �A⊗ �� = �⊗ A.

The map is known as an entwining map, and the triple �A�C� is called anentwining structure.

The theorem below [8] shows that entwining structures and Yang–Baxtersystems are intimately related.

Theorem 2.4. Let A be an algebra and let C be a coalgebra. For any s� r� t� p ∈ kdefine linear maps

W � A⊗ A → A⊗ A� a⊗ b �→ sba⊗ 1+ r1⊗ ba− sb ⊗ a�

Z � C ⊗ C → C ⊗ C� c ⊗ d �→ t��c�∑

d�1� ⊗ d�2� + p��d�∑

c�1� ⊗ c�2� − pd⊗ c

Let X � A⊗ C → A⊗ C be a linear map such that X��⊗ C� = �⊗ C and �A⊗ ��X =A⊗ �. Then W�X�Z is a Yang–Baxter system if and only if A is entwined with C by themap �= XC�A.

Remark 2.5. As observed in [7, Proposition 2.7], entwining structures are relatedto algebra factorisations via semi-dualisation (see [19, p. 300] for discussion ofalgebra factorisations). Arguments similar to those in the proof of Theorem 2.4show that, given two algebras A, B, and an algebra factorisation map � � B ⊗ A →A⊗ B, one can construct a Yang–Baxter system with W and Z of the same form asW in Theorem 2.4 and X = �A�B.

3. MAIN RESULTS AND CONSEQUENCES

Our purpose here is to show that braid systems give rise to braidings betweenvarious braided structures in a sense to be made specific in the sequel. In particular,to various entwinings of like or dual algebraic structures. Since Yang–Baxtersystems are equivalent with braid systems, they turn out to be a good source toproduce various entwinings.

Unless otherwise stated, throughout this section modules and tensor productsare over a fixed commutative ring, and operators are not necessarily invertible.

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4446 BERCEANU ET AL.

Given three modules U , V , W , the braid commutator of three morphismsR � U ⊗ V → V ⊗ U , S � U ⊗W → W ⊗ U , T � V ⊗W → W ⊗ V , is the morphism�R� S� T� � U ⊗ V ⊗W → W ⊗ V ⊗ U defined by

�R� S� T� = �T ⊗ U��V ⊗ S��R⊗W�− �W ⊗ R��S ⊗ V��U ⊗ T�

With the usual conventions in braid representation,

Notice that �W�V � W�U � V�U � = 0, deducing fairly easily that the braid andYang–Baxter commutators are related by

�TW�V � SW�U � RV�U � = �R� S� T��U ⊗ W�V ��W�U ⊗ V��W ⊗ V�U � (∗)

The morphisms R, S, T extend naturally to morphisms, also denoted by R, S,T , between the tensor products of the corresponding tensor modules. We describethe extension of R; S and T extend similarly. The components Rm�n � TmU ⊗ TnV →TnV ⊗ TmU of the tensor extension R � TU ⊗ TV → TV ⊗ TU are recursively definedby starting with the obvious identities along the boundary and setting

Rm+1�1 = �Rm�1 ⊗ U��U⊗m ⊗ R� and Rm�n+1 = �V⊗n ⊗ Rm�1��Rm�n ⊗ V�

Pictorially,

Notice that Rm+1�1 = �R⊗ U⊗m��U ⊗ Rm�1� and Rm�n+1 = �V ⊗ Rm�n��Rm�1 ⊗ V⊗n�.Alternatively, but equivalently, we might equally well have set

R1�n+1 = �V ⊗ R1�n��R⊗ V⊗n� and Rm+1�n = �R1�n ⊗ U⊗m��U ⊗ Rm�n�

and noticed that R1�n+1 = �V⊗n ⊗ R��R1�n ⊗ V� and Rm+1�n = �Rm�n ⊗ U��U⊗m ⊗R1�n�.

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ALGEBRA STRUCTURES FROM BRAID SYSTEM 4447

Our results are all consequences of the following lemma, which shows that abraid commutator involving the components of the tensor extensions of R, S, T maybe expressed in terms of braid commutators involving components of lower rank.

Lemma 3.1. The braid commutators of the Rm�n, Sm�p, Tn�p satisfy the recurrencerelations below:

�RST�m�1�1 �Rm�1� Sm�1� T1�1� = �R1�1� S1�1� T1�1�Sm−1�1Rm−1�1

+ R1�1S1�1�Rm−1�1� Sm−1�1� T1�1��

�RST�m�n�1 �Rm�n� Sm�1� Tn�1� = T1�1�Rm�n−1� Sm�1� Tn−1�1�Rm�1

+ Rm�n−1�Rm�1� Sm�1� T1�1�Tn−1�1�

�RST�m�n�p �Rm�n� Sm�p� Tn�p� = Tn�1Sm�1�Rm�n� Sm�p−1� Tn�p−1�

+ �Rm�n� Sm�1� Tn�1�Sm�p−1Tn�p−1

Consequently, if �R� S� T� = 0 on U ⊗ V ⊗W , then �R� S� T� = 0 on TU ⊗ TV ⊗ TW .

Proof. The argument is illustrated in the diagrams below. Thick lines (stripes)represent bundles of strings, and the shaded boxes mark the factors outside the braidcommutators. In each case, the terms in the middle cancel out one another, and theconclusion follows.

We say that the morphisms R, S, T form a braid system on U , V , W if

�R�R�R� = 0� �R� S� S� = 0� �S� S� T� = 0 and �T� T� T� = 0

In view of �∗�, the analogy with Yang–Baxter systems is not accidental at all:Roughly speaking, R, S, T form a braid system if and only if TW�V , SW�U , RV�Uform a ‘Yang–Baxter system.’

The following theorem is a straightforward consequence of the lemma.

Theorem 3.2. The tensor extension of a braid system is again a braid system.

The theorem specialises to several important cases discussed in the sequel. Webegin by considering braided modules.

Recall that a braided module �V� �� is a module V endowed with a braiding�, i.e., a morphism of modules � � V ⊗ V → V ⊗ V satisfying the braid relation

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4448 BERCEANU ET AL.

��� �� �� = 0. By the lemma, the tensor extension of � makes TV into a braidedmodule, the braided tensor module of V . For the moment, we deliberately ignoreother algebraic structures on TV and consider only the natural braiding.

Given two braided modules �V� �� and �V ′� �′�, a morphism of modules� � V ′ ⊗ V → V ⊗ V ′ is a braiding between �V� �� and �V ′� �′� if ��′� �� �� = 0and ����� �� = 0; that is, �′, �, �, in order, form a braid system. Alternatively,but equivalently, W = �V�V , X = �V�V ′ , Z = �′V ′�V ′ form a Yang–Baxter system.Conversely, given a Yang–Baxter system, � = WV�V is a braiding on V , �′ = ZV ′�V ′

is a braiding on V ′, and � = XV ′�V is a braiding between the braided modules �V� ��and �V ′� �′�.

Ignoring the braiding �, the condition ��′� �� �� = 0 suggests that � may belooked upon as a ‘left action’ of the braided module �V ′� �′� on the module V ,making the latter into a left �V ′� �′�-module �V� ��; of course, if �W� � is anotherleft �V ′� �′�-module, a morphism of left �V ′� �′�-modules, f � �V� �� → �W� �, is amodule mophism f � V → W such that �f ⊗ V ′�� = �V ′ ⊗ f�. Similarly, ignoringthe braiding �′, the condition ����� �� = 0 suggests that � may be looked upon asa ‘right action’ of the braided module �V� �� on the module V ′, making the latterinto a right �V� ��-module �V ′� ��; and if �W ′� � is another right �V� ��-module, amorphism of right �V� ��-modules, f ′ � �V ′� �� → �W ′� �, is a module mophism f ′ �V ′ → W ′ such that �V ⊗ f ′�� = �f ′ ⊗ V�.

The theorem below is a consequence of 32.

Theorem 3.3. The tensor extension of a braiding between two braided modules is abraiding between the corresponding tensor modules.

Next, we consider braided algebras [3], also called Yang–Baxter algebras in[13]. A braided algebra �A� �� �� �� is an algebra �A� �� �� such that �A� �� is a braidedmodule whose braiding � is compatible with the algebra structure:

�1� ���⊗ A� = �A⊗ ����⊗ A��A⊗ ��;�2� ��A⊗ �� = ��⊗ A��A⊗ ����⊗ A�;�3� ���⊗ A� = A⊗ �;�4� ��A⊗ �� = �⊗ A.

It is readily checked that A� = �A� ��� �� �� is again a braided algebra, thederived braided algebra of A. The construction may be iterated: ��, ��2, etc., are allbraid products on A. Also, the module A⊗ A acquires a structure of braided algebrawith multiplication ��⊗ ���A⊗ �⊗ A�, unit �⊗ �, and braiding �A⊗ �⊗ A� ��⊗���A⊗ �⊗ A�.

If �V� �� is a braided module, multiplication by concatenation makes theassociated braided tensor module into a braided algebra, the braided tensor algebra�TAV� �� �� ��. Of the four conditions of compatibility of the braiding with thealgebra structure, only the first requires some care; the other are trivial. As to thefirst, it is easily proved by sliding disjoint crossings to rearrange them conveniently.

Since multiplication on TAV is given by concatenation, multiplication on�TAV�� is essentially given by �; it is a braid product on TAV . Conversely, given amorphism of modules � � V ⊗ V → V ⊗ V , consider its tensor module extension � �TAV ⊗ TAV → TAV ⊗ TAV . If �� is a multiplication on TAV , then � is a braiding,

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ALGEBRA STRUCTURES FROM BRAID SYSTEM 4449

and �TAV� �� �� �� is a braided algebra. We deliberately postpone the discussion ofthe universal property of the braided tensor algebra.

The notion of a braided algebra may be extended as follows. An algebra�A� �� �� is entwined with an algebra �A′� �′� �′� if there exists a morphism of modules� � A′ ⊗ A → A⊗ A′ satisfying the following four conditions:

(1) ��A′ ⊗ �� = ��⊗ A′��A⊗ ����⊗ A�;(2) ���′ ⊗ A� = �A⊗ �′���⊗ A′��A′ ⊗ ��;(3) ��A′ ⊗ �� = �⊗ A′;(4) ���′ ⊗ A� = A⊗ �′.

The map � is known as an algebra factorisation map, and the triple �A�A′�� iscalled an algebra factorisation. It is readily checked that ��⊗ �′��A⊗ �⊗ A′� and�⊗ �′ make A⊗ A′ into an algebra, whence the interest in algebra factorisations; inparticular, if � = A′�A, this is precisely the standard tensor product algebra.

If, in addition, A and A′ are both braided algebras with braidings � and �′,respectively, then � is a braiding between A and A′ if it is a braiding between thebraided modules �A� �� and �A′� �′�. Conditions �1�, �3�, along with ����� �� = 0state that �, � and � are morphisms of left �A′� �′�-modules, the left action of �A′� �′�on A⊗ A being given by �A⊗ ����⊗ A�. Similarly, conditions �2�, �4�, along with��′� �� �� = 0 state that the morphisms �′, �′, and �′ are morphisms of right �A� ��-modules.

It is readily checked that � is also a braiding between the derived braidedalgebras A� and A′

�′ and the successive iterates of like order. Also, if � is invertible,then �A⊗ �−1 ⊗ A′� ��⊗ �′��A⊗ �⊗ A′� makes A⊗ A′ into a braided module.

The theorem below is the algebra version of 33; it generalises a lemma from[4, p. 6966].

Theorem 3.4. The tensor extension of a braiding between two braided modules isa braiding between the corresponding braided tensor algebras and all their derivedbraided algebras of like order. In particular, it gives rise to a whole sequence of algebrafactorisations.

Mutatis mutandis, we may go through the preceding and consider braidedcoalgebras and entwinings and braidings between braided coalgebras or a braidedalgebra and a braided coalgebra. Consider the coalgebra counterpart of 34.

Given a braided module �V� ��, the braided tensor coalgebra �TCV��� �� ��is defined dually to the braided tensor algebra. Roughly speaking, � is given bydeconcatenation:

��x1 ⊗ · · · ⊗ xn� =n∑

i=0

�x1 ⊗ · · · ⊗ xi�⊗ �xi+1 ⊗ · · · ⊗ xn�

We are now in a position to state the coalgebra counterpart of 3.4.

Theorem 3.5. The tensor extension of a braiding between two braided modules isa braiding between the corresponding braided tensor coalgebras and all their derivedbraided coalgebras of like order. In particular, it gives rise to a whole sequence ofcoalgebra-coalgebra entwinings.

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4450 BERCEANU ET AL.

The algebra-coalgebra version below is clear.

Theorem 3.6. The tensor extension of a braiding between two braided modules isa braiding between the corresponding braided tensor algebra and the braided tensorcoalgebra and all their derived objects of like order. In particular, it gives rise to awhole sequence of algebra-coalgebra entwinings.

There are similar braidings between like or dual structures on TAs, TCs, a TA,and a TC or vice versa. Such structures are obtained via universal properties.

Let �V� �� be a braided module. The braided tensor algebra TAV has thestandard universal property [1]: Given a braided algebra A, a morphism of braidedmodules f � V → A extends to a unique morphism of braided algebras also denotedf � TAV → A whose components are

fn =(

n−2∏i=0

(�A ⊗ A⊗i

))f⊗n � TAnV → A�

this is also an extension between the derived braided algebras, �TAV�� → A�, andso on. In particular, TAV is made into a coalgebra [1] by extending the morphismof braided modules � � �V� �� → �TAV ⊗ TAV� ��, �x = x⊗ 1+ 1⊗ x, where � =�TAV ⊗ �⊗ TAV���⊗ ���TAV ⊗ �⊗ TAV�, to a morphism of braided algebras � ��TAV� �� �� �� → �TAV ⊗ TAV� �� �⊗ �� ��, where � = ��⊗ ���TAV ⊗ �⊗ TAV�; �is known as the quantum shuffle coproduct on TAV [1]. Each component of �,�n � TAnV → �TAV ⊗ TAV�n =

⊕p+q=n TApV ⊗ TAqV , splits as �n =

∑p+q=n �p�q,

where the �p�q � TAp+qV → TApV ⊗ TAqV are given by the recurrence relation

�p�q = �p�q−1 ⊗ V + �V⊗�p−1� ⊗ �q�1���p−1�q ⊗ V��

with the boundary conditions �p�0 = TApV = TApV ⊗ TA0V and �0�q = TAqV =TA0V ⊗ TAqV . For instance,

�p�1 =p∑

i=0

�V⊗�p−i� ⊗ �1�i� and �1�q =q∑

i=0

��i�1 ⊗ V⊗�q−i���

and the first four one-index �’s are

�0 = TA0V ⊗ TA0V� �1 = TA0V ⊗ TA1V + TA1V ⊗ TA0V�

�2 = TA0V ⊗ TA2V + �TA1V ⊗ TA1V + ��+ TA2V ⊗ TA0V�

�3 = TA0V ⊗ TA3V + �TA1V ⊗ TA2V + �⊗ V + ��⊗ V��V ⊗ ���

+ �TA2V ⊗ TA1V + V ⊗ �+ �V ⊗ ����⊗ V��+ TA3V ⊗ TA0V

The counit is the standard augmentation � on TAV to the ground ring. It turns outthat �TAV��� �� �� is a braided coalgebra [1].

Similarly, the braided tensor coalgebra TCV is cofree in the followingsense [24]: Given a braided conilpotent (coradical) coalgebra C with cokernelC+, a morphism of braided modules f � C+ → V lifts to a unique morphism

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ALGEBRA STRUCTURES FROM BRAID SYSTEM 4451

of braided coaugmented coalgebras C → TCV such that f is the compositeC+ → TC+V

pr−→ V . In particular, consider the braided coalgebra TCV ⊗ TCVwith comultiplication �TCV ⊗ �⊗ TCV���⊗ ��, counit �⊗ � and braiding �TCV ⊗�⊗ TCV���⊗ ���TCV ⊗ �⊗ TCV�. The morphism of braided modules �TCV ⊗TCV�+ → V , x⊗ y �→ ��y�prV x + ��x�prV y, lifts to a unique morphism of braidedcoalgebras � � TCV ⊗ TCV → TCV to produce the quantum shuffle product [24].Each component of the shuffle product, �n � �TCV ⊗ TCV�n =

⊕p+q=n TCpV ⊗

TCqV → TCnV , splits as �n =∑

p+q=n �p�q, where the �p�q � TCpV ⊗ TCqV →TCp+qV are given by the recurrence relation

�p�q = �p�q−1 ⊗ V + ��p−1�q ⊗ V��V⊗�p−1� ⊗ �1�q��

with the boundary conditions �p�0 = TCpV ⊗ TC0V = TCpV and �0�q = TC0V ⊗TCqV = TCqV . For instance,

�p�1 =p∑

i=0

�V⊗�p−i� ⊗ �i�1� and �1�q =q∑

i=0

��1�i ⊗ V⊗�q−i��

The shuffle product makes TCV into an algebra; it is a braided algebra withbraiding �—see [24] for further details and related facts on the topic.

As before, the tensor extension of a braiding between two braided modules isa braiding between the various structures thus obtained.

Remark 3.7. Recall the gluing procedure of R and R′ along U described in23 and consider the corresponding braided tensor algebra TA�V ⊕ V ′�. By 34,TAV ⊗ TAV ′ comes out with a structure of algebra, endowed with a braiding.It seems reasonable to ask what connections might there be between the twoalgebra/braiding structures. Of course, the same question may be asked about thecoalgebra structures on TA’s or TCs or the algebra structures on TCs.

ACKNOWLEDGMENTS

The authors would like to thank the referee for carefully reading themanuscript, making valuable comments, and suggesting a first version ofLemma 3.1.

REFERENCES

[1] Ardizzoni, A., Menini, C., Stefan, D. (2009). Braided bialgebras of Hecke-type.J. Algebra 321:847–865.

[2] Ashraf, R., Berceanu, B. Simple Braids. arXiv:1003.6014v1.[3] Baez, J. C. (1994). Hochschild homology in a braided tensor category. Trans. Amer.

Math. Soc. 334:885–906.[4] Bororwiec, A., Marcinek, W. (2000). On crossed product of algebras. J. Math. Phys.

41:6959–6975.[5] Brzezinski, T. (1999). On modules associated to coalgebra-Galois extensions. J. Algebra

215:290–317.

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 2

2:37

28

Sept

embe

r 20

13

4452 BERCEANU ET AL.

[6] Brzezinski, T. (2002). The structure of corings. Induction functors, Maschke-typetheorem, and Frobenius and Galois-type properties. Algebras Rep. Theory 5:389–410.

[7] Brzezinski, T., Majid, S. (1998). Coalgebra bundles. Comm. Math. Phys. 191:467–492.[8] Brzezinski, T., Nichita, F. F. (2005). Yang–Baxter systems and Entwining Structures.

Comm. Algebra 33:1083–1093.[9] Brzezinski, T., Wisbauer, R. (2003). Corings and Comodules, London Math. Soc.

Lecture Note Series 309. Cambridge: Cambridge University Press.[10] Caenepeel, S., Militaru, G., Zhu, S. (2002). Frobenius and Separable Functors for

Generalized Hopf Modules and Nonlinear Equations. LNM 1787, Berlin: Springer.[11] Dascalescu, S., Nichita, F. (1999). Yang–Baxter operators arising from (co)algebra

structures. Comm. Algebra 27:5833–5845.[12] Grapperon, T., Ogievetsky, O. V. (2012). Braidings of tensor spaces. Lett. Math. Phys.

100:17–28.[13] Hashimoto, M., Hayashi, T. (1992). Quantum multilinear algebra. Tohoku Math. J.

44:471–521.[14] Hlavaty, L., Kundu, A. (1996). Quantum integrability of nonultralocal models through

Baxterization of quantised braided algebra. Int. J. Mod. Phys. A11:2143–2165.[15] Hlavaty, L., Snobl, L. (1999). Solution of the Yang–Baxter system for quantum

doubles. Int. J. Mod. Phys. A14:3029–3058.[16] Isaev, A. P., Ogievetsky, O. V. Braids, Shuffles and Symmetrizers. arXiv:0812.3974v1.[17] Kassel, C. (1995). Quantum Groups. Graduate Texts in Mathematics. Vol. 155. Springer

Verlag.[18] Lambe, L., Radford, D. (1997). Introduction to the Quantum Yang–Baxter Equation and

Quantum Groups: An Algebraic Approach. Mathematics and Its Applications. Vol. 423.Dordrecht: KluwerAcademic Publishers.

[19] Majid, S. (1995). Foundations of Quantum Group Theory. Cambridge: CambridgeUniversity Press.

[20] Majid, S., Markl, M. (1996). Gluing operations for R-matrices, quantum groups andlink-invariants of Hecke type. Math. Proc. Cambridge Philos. Soc. 119:139–166.

[21] Nichita, F. (2009). Non-Linear Equations, Quantum Groups and Duality Theorems. VDMVerlag.

[22] Nichita, F. (1999). Self-inverse Yang–Baxter operators from (co)algebra structures.J. Algebra 218:738–759.

[23] Nuss, P. (1997). Noncommutative descent and non-abelian cohomology. K-Theory12(1):23–74.

[24] Rosso, M. (1998). Quantum groups and quantum shuffles. Invent. Math. 133:399–416.[25] Snobl, L. (2000). Construction of quantum doubles from solutions of Yang–Baxter

system. Czech. J. Phys. 50:187–192.[26] Vladimirov, A. A. (1993). A method for obtaining quantum doubles from the Yang–

Baxter R-matrices. Mod. Phys. Lett. A 8:1315–1321.

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 2

2:37

28

Sept

embe

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