基于galois对象的新bornological量子群的构造-周楠.pdf

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1、Journal of Southeast University (English Edition) Vol.32, No.4, pp.524 526 Dec. 2016 ISSN 1003 7985Construction of new bornological quantum groupsbased on Galois objectsZhou Nan Wang Shuanhong(Department of Mathematics, Southeast University, Nanjing 211189, China)Abstract: Let A be a bornological qu

2、antum group and R abornological algebra. If R is an essential A-module, then thereis a unique extension to M(A)-module with 1x =x. There is aone-to-one corresponding relationship between the actions of Aand the coactions of A. If R is a Galois object for A, thenthere exists a faithful -invariant fun

3、ctional on R. Moreover,the Galois objects also have modular properties such as algebraicquantum groups. By constructing the comultiplication ,counit , antipode S and invariant functional on R R, R R can be considered as a bornological quantum group.Key words: bornological quantum groups; actions and

4、coactions; Galois theory; Galois objectsDOI:10.3969/ j. issn.1003 -7985.2016.04.022Received 2015-09-14.Biographies: Zhou Nan ( 1991 ), male, graduate; Wang Shuanhong(corresponding author), male, doctor, professor, shuanhwang .Citation:Zhou Nan, Wang Shuanhong. Construction of new bornologi-cal quant

5、um groups based on Galois objectsJ. Journal of Southeast U-niversity (English Edition), 2016, 32(4): 524 526. DOI: 10. 3969/ j.issn.1003 -7985.2016.04.022.I n 1994, van Daele first introduced the concept of mul-tiplier Hopf algebra1 and studied algebraic quantumgroups2 . An algebraic quantum group i

6、s a multiplierHopf algebra with invertible antipode equipped with aHaar integral. The basic example of multiplier Hopf alge-bra is the algebra of complex functions with finite supportfor a group. In order to include more examples such assmooth convolution algebras of Lie groups, Voigt3 intro-duced t

7、he concept of a bornological quantum group. Mo-reover, van Daele and Wang4 generalized it to the bo-rnological quantum hypergroups case. Note that borno-logical quantum groups are considered over the bornologi-cal vector spaces. The bornological vector space is veryimportant when studying various pr

8、oblems in noncommu-tative geometry and cyclic homology5 6 .Galois objects play an important role in the operator al-gebra framework and they provide equivalences of certaincategories. Motivated by the theory, de Commer7 devel-oped the theory of the Galois objects for algebraic quan-tum groups. So, i

9、t is natural to consider the Galois ob-jects for bornological quantum groups.As a generalization of the theory in Ref. 7 8. Westudy the ( co) action on bornological quantum groups,and construct the bornological quantum groups throughthe Galois objects. The algebras in this paper are over thefield C

10、of the complex numbers and the Sweelder notionis used for the coproduct. For two completed bornologicalvector spaces V and W, the tensor product is denoted byV W.1 Actions and Coactions of Bornological Quan-tum GroupsDefinition 1 A bornological quantum group is an es-sential bornological algebra A s

11、atisfying the approxima-tion property together with a comultiplication : A M(A A) such that all Galois maps associated to areisomorphisms and a faithful left invariant functional : A C.A morphism between bornological quantum groups Aand B is an essential algebra homomorphism f: A Bsuch that (f f) =

12、f.Definition 2 Let A be a bornological quantum group.An essential A-module is an A-module X such that themodule action induces a bornological isomorphism A AX X.Dually, we have the concept of an essential comodule.Let A be a bornological quantum group. Assume that Ris a bornological algebra over C p

13、robably without a unitbut with non-degenerate product.Proposition 1 Let R be an essential A-module. If xR and ax =0 for all a A, then x =0.Proposition 2 Let R be an essential A-module, thenthere is a unique extension to a left M(A)-module and 1x= x where 1 M(A).Proof It is very natural to define m(a

14、x) = (ma) x forall x R, a R and m M(A). Since R is essential, wehave 1x = x for all x. The action is well-defined. Assumethat aixi = 0, xi R, ai R . Choose e A such thateai = ai for all i. For any m M(A), we have m(aixi) = (mai)xi = (me)(aixi) =(me) aixi = 0Therefore, we can define the action of M(A

15、) by m(ax) =(ma)x.Proposition 3 Let A be a bornological quantumgroup. If we denote M as the category of essential left A-modules and morphisms, then M is a monoidal categorywith unit.万方数据 The unit is C, and the module structure over C is ac =(a)c for a A and c C.Definition 3 Let R be an essential A-

16、module. We saythat R is a left A-module algebra if a(xy) = (a(1) x)(a(2) y) for all a A and x, y R.Proposition 4 Let R be a left A-module algebra. Wedefine a multiplication on R A by (x a)(y b) = x(a(1) y) a(2) b for all x, y R and a, b A. Then R A is an essential bornological algebra.Definition 4 L

17、et A be a bornological quantum groupand R is a bornological algebra. is called the coactionof A on R if there is an essential injective homomorphism: R M(R A) satisfying1) (R)(1 A) R A and (1 A)(R) R A;2) ( id) = (id ). is called reduced if (R 1)(R) R A. If is re-duced, we also have (R)(R 1) R A. In

18、 this case,R is called an A-comodule algebra.Proposition 5 Let (A, ) be a bornological quantumgroup. If R is an A-comodule algebra, then R is an A-module algebra.Proof The action of A on R is defined by b x = ( id )(1 a)(x) for all x R, b = (a ), where aA. Then we need to check ( ab) x = a ( b x) an

19、da (xy) = (a(1) x)(a(2) y) , where a, b A. Therest proof is standard and the essentialness is easy tocheck.Proposition 6 Let (A, ) be a bornological quantumgroup. If R is an A-module algebra, then R is an A-co-module algebra.Proof The coaction here is defined as(r)(1 b) = S-1(a(1) ) r ( a(2) )(1 b)(

20、r) = S-1(a(2) ) r ( a(1) )where b = ( a) and b = ( a) for a, a A. With thiscoaction, R is an A-comodule algebra.Theorem 1 Let A be a bornological quantum groupand R a bornological algebra. R is A-module algebra ifand only if R is an A-comodule algebra.2 Galois Objects and Main ConstructionsDefinitio

21、n 5 Let A be a bornological quantum groupand is a coaction of A on a bornological algebra R. Anelement f M(R) is coinvariant if (f) = f 1. Let RcoAbe the set of all coinvariants in M(R), and RcoA is a unitalsubalgebra.Definition 6 Let A be a bornological quantum group,and R is a bornological algebra

22、. Then, the coaction de-fined above is called Galois coaction if is reduced andthe mapV: R RcoAR R A:x y (x 1)(y)is bijective.Definition 7 Let be a right Galois coaction of a bo-rnological quantum group ( A, ) on R. Then ( R, ) iscalled a right Galois object for A if RcoA is the scalar field.Theorem

23、 2 Let R be a right A-Galois object. Thereexists a faithful -invariant functional R on R such that(id )(r) = R(r)1 for all r R. Moreover, thereexists a non-zero invariant functional R on R.Proof For all r, s R and a A, let x = ( id ) (r). We compute(x)(s)(1 a) = (xs)(1 a) =(id id )( id)(r)(s 1)(1 a

24、1) =(id id )(id )(r)(s(0) s(1) a 1) =r(0) s(0) (r(1) )s(1) a = (x 1)(s)(1 a)Since R is a Galois object, we have ( x) = x 1 =R(r) for some scalar R ( r). So, we have defined abounded linear functional R on R. It is easy to obtain -invariance and faithfulness. Setting rR ( r) = R( r(0) r) (r(1) ), her

25、e r is the element such that rR is non-zero.Proposition 7 Let A be a bornological quantum groupand R is an A-Galois object. R and R are defined inTheorem 2. Then for all r, s R, we have1) There exists a unique invertible element R M(A)such that R(rR) = R(r);2) There exists a unique bounded algebra a

26、utomor-phism of R such that R(r(s) = R( sr). We call the modular automorphism.Now, we construct a new bornological quantum groupdenoted by (X, X) which is spanned by , 1 X for , 1 R. Note that R = R( r) r R and , 1 A(a) =( 1)(a), (a) is the element in M(R R) satis-fying (r 1)(a) = V -1(r a)(a) (1 r)

27、. Since thereis a natural bijection between R A R and X, we mainlyconsider space X.Define the multiplication on X as x , 1 X = x ,1 X for , 1 R and x X. The associativity is straight-forward.The comultiplication X: X M(X X) is defined asX(, 1 X) = (1) , 1(2) (2) , 1(1) for , 1 R.The essential algebr

28、a homomorphism X: X C is de-fined as X(, 1 X) = (1) 1(1).The antipode SX is defined as SX: X X: , 1 X R(1), X for , 1 R. Remember that R ( ) = R, A is the modular element in the dual A and R is themodular automorphism of R.Finally, given a map : R R: ( r) r, the function-al X: X C: , 1 X (1) is a le

29、ft invariant func-tional.Theorem 3 Together with the maps X, X, SX, X, Xis a bornological quantum group.525 Construction of new bornological quantum groups based on Galois objects万方数据References1 van Daele A. Multiplier Hopf algebras J. Trans AmerMath Soc, 1994, 342 ( 2): 917 932. DOI: 10. 1090/s0002

30、-9947-1994-1220906-5.2 van Daele A. An algebraic framework for group dualityJ. Advances in Mathematics, 1998, 140(2):323 366.DOI:10.1006/ aima.1998.1775.3 Voigt C. Bornological quantum groups J. Pacific Jour-nal of Mathematics, 2008, 235(1): 93 135. DOI: 10.2140/ pjm.2008.235.93.4 van Daele A, Wang

31、S H. Pontryagin duality for bornolog-ical quantum hypergroups J. Manuscripta Mathematica,2009, 131 ( 1): 247 263. DOI: 10. 1007/ s00229-009-0318-8.5 Meyer R. Smooth group representations on bornologicalvector spaces J. Bull Sci Math, 2004, 128(2): 127166. DOI:10.1016/ j. bulsci.2003.12.002.6 Voigt C

32、. Equivariant periodic cyclic homology J. Jour-nal of the Institute of Mathematics of Jussieu, 2007, 6(4): 689 763. DOI:10.1017/ s1474748007000102.7 de Commer K. Galois objects for algebraic quantumgroups J. Journal of Algebra, 2009, 321(6): 17461785. DOI:10.1016/ j. jalgebra.2008.11.039.8 Drabant B

33、, van Daele A, Zhang Y H. Actions of multi-plier Hopf algebras J. Comm Algebra, 1999, 27(9):4117 4172. DOI:10.1080/00927879908826688.基于Galois对象的新bornological量子群的构造周楠 王栓宏(东南大学数学系,南京211189)摘要:设A为bornological量子群,R为bornological代数.如果R为essential A模,那么R可以扩张为M(A)模并且满足1x = x. A上的作用与A上的余作用之间有一个一一对应的关系.若R是A上的Galois对象,则R上存在一个忠实的不变泛函,且拥有类似于代数量子群的modular性质.最后,通过构造R R上的余乘、余单位、対极S和不变泛函,使之成为bornological量子群.关键词:bornological量子群;作用和余作用; Galois理论; Galois对象中图分类号:O153.3625 Zhou Nan and Wang Shuanhong万方数据