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1、Acta Ma$hematica Sinica，English Seres Apr，2014，Vo130，No4，PP567590 Published online：March 15，2014 DOh 101007s101 14014-3240-2 Http：wwwActaMathcorn Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg& The Editorial Ofnce of AMS 2014 YetterDrinfeld Modules over Weak Braided Hopf M
2、onoids and Center Categories JosNicanor AL0NS0 AIJVAREZ Departamento de Matemdticas，Universidad de Vigo，Campus Universitario LagoasMarcosende， E一36280 Vigo，Spain E-mail：jnalonsouvi9oes Ram6n GONZALEZ RODRIGUEZ Departamento de Matemdtica Aplicada II，Universidad de Vigo。 Campus Universitario LagoasMar
3、cosende，E-36310 Vigo，Spain E-mail：rgondmauvigoes Carlos SONEIRA CALVo Departamento de Pedagoxla e Diddctica，Universidade da 0 n，Campus Universitario de Elviria， E-15007 A Corura，Spain E-maif：carl08soneiraudces Abstract In this paper，we introduce several centralizer constructions in a monoidal contex
4、t and establish a monoidal equivalence with the category of Yetter-Drinfeld modules over a weak braided Hopf monoidWe apply the general result to the calculus of the center in module categories Keywords Weak(braided)Hopf monoid，YetterDrinfeld module，center MR(2010)Subject Classification 57T05，18D10，
5、16T05，16T25，81R50 1 lntroduction Weak Hopf algebras were introduced by BShm et a18as a new generalization of Hopf algebras and groupoid algebrasThese structures were firstly defined in the category of vector spaces， but it is possible to extend the definition to the general context of monoidal categ
6、ories(see 1，2，17)where algebras are replaced by monoids，coalgebras by comonoids and bialgebras by bimonoids Conditions defining Yetter-Drinfeld modules were established by Radford to study projec tions of Hopf algebras，and widely applied by Yetter21to explain the relation between many mathematical a
7、nd paysical theoriesIn1we find the extension of RadfordS theory for projec tions of weak Hopf monoids in a strict symmetric monoidal category C where every idempotent morphism splitsThe main result of1j extended to the braided setting in41 establishes a categorical equivalence between the category o
8、f projections associated to a weak Hopf monoid H and the category of Hopf monoids in the category of leftleft Yetter-Drinfeld modules over Received April 25，2013，accepted July 18，2013 Supported by Ministerio de Ciencia e Innovaci6n，project MTM2010-15634，and by FEDER Alonso Alvarez JN，et 01 日T0 show
9、this result the authors introduced the notions of weak Yang Baxter oDerat0r and weak braided Hopf monoid in f1，2_Roughly speaking，a weak braided Hopf monoid in a strict monoidal category is a monoidcomonoid with a weak Yang-Baxter operator，satisfying some compatibility conditionsThe first nontrivial
10、 example of such structure can be obtained by modifying the algebraic structure of a Hopf monoid D in the nonstrict braided monOidal category；yv On the other hand，the center construction was introduced independently by Drinfeld(un- published)，Joyal and Street【12and Majid15，and it associates to a mon
11、oidal category C a braided monoidal category z(c)There is a closely relation with YetterDrinfeld modul a classical result states that for a Hopf monoid，the center of the category of left H modules is equivalent to备 14Moreover，using this fact，if the Hopf monoid is finite，it resuits that 嚣 is equivale
12、nt to the category of modules over the Drinfeld double of日f15AfterwDxds this results were extended to the case of weak Hopf monoids in【11，16 This being the situation，this work has two main objectivesFirstly，if D is a weak braided Hopf monoid in a strict monoidal category C with split idempotents，we
13、study if there is a suitable analogue to the equivalence备 z(nM)cited in the previous paragraphNow， although is monoidal when AD is bijective，we do not know if it is braidedso we are not expected to obtain an equivalence with the center of any categoryUsing the concept of centralizer(see19)，we introd
14、uce the special centralizer as the centralizer of a suitable monoidal subcategory A of o3,4 and we construct a categorical equivalence between and SZ(A，D M)The other objective of this work is to use the relation between and SZ(A，D M)in order to study the category z(HM)defined in I12，showing that to
15、0btain the center in a range of a widely handled module categories the problem can be reduced to the calculus of a special centralizer The work is organized as followsAfter the introduction，the first section consists on a review of weak braided Hopf monoids and weak operators confining to those prop
16、erties that will be explicitly used in the sequel(a general exposition can be found in2，51 1In the second one we recall the monoidal category g from5and introduce the category DM of left Dmodules over a weak braided Hopf monoid DThe first question that arises in the general context is whether DA4 is monoidal，and an affirmative answer is given in Theorem 311making explicit the monoidal structure The
