Abstract：LetA be a bornological quantum group and R a bornological algebra. If R is an essential A-module, then there is a unique extension to M(A)-module with 1x=x. There is a one-to-one corresponding relationship between the actions of A and the coactions of 
. If R is a Galois object for A, then there exists a faithful δ-invariant functional on R. Moreover, the Galois objects also have modular properties such as algebraic quantum groups. By constructing the comultiplication Δ, counit ε, antipode S and invariant functional φ on 
, 
can be considered as a bornological quantum group.
Key words：bornological quantum groups; actions and coactions; Galois theory; Galois objects
In 1994, van Daele first introduced the concept of multiplier Hopf algebra[1] and studied algebraic quantum groups[2]. An algebraic quantum group is a multiplier Hopf algebra with invertible antipode equipped with a Haar integral. The basic example of multiplier Hopf algebra is the algebra of complex functions with finite support for a group. In order to include more examples such as smooth convolution algebras of Lie groups, Voigt[3] introduced the concept of a bornological quantum group. Moreover, van Daele and Wang[4] generalized it to the bornological quantum hypergroups case. Note that bornological quantum groups are considered over the bornological vector spaces. The bornological vector space is very important when studying various problems in noncommutative geometry and cyclic homology[5-6].
Galois objects play an important role in the operator algebra framework and they provide equivalences of certain categories. Motivated by the theory, de Commer[7] developed the theory of the Galois objects for algebraic quantum groups. So, it is natural to consider the Galois objects for bornological quantum groups.
As a generalization of the theory in Ref.[7-8]. We study the (co)action on bornological quantum groups, and construct the bornological quantum groups through the Galois objects. The algebras in this paper are over the field C of the complex numbers and the Sweelder notion is used for the coproduct. For two completed bornological vector spaces V and W, the tensor product is denoted by 
W.
Definition 1 A bornological quantum group is an essential bornological algebra A satisfying the approximation property together with a comultiplication Δ: 
such that all Galois maps associated to Δ are isomorphisms and a faithful left invariant functional φ: A→C.
A morphism between bornological quantum groups A and B is an essential algebra homomorphism f: A→B such that 
f)Δ=Δf.
Definition 2 Let A be a bornological quantum group. An essential A-module is an A-module X such that the module action induces a bornological isomorphism 
AX≅X.
Dually, we have the concept of an essential comodule.
Let A be a bornological quantum group. Assume that R is a bornological algebra over C probably without a unit but with non-degenerate product.
Proposition 1 Let R be an essential A-module. If x∈R and ax=0 for all a∈A, then x=0.
Proposition 2 Let R be an essential A-module, then there 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(ax)=(ma)x for all x∈R, a∈R and m∈M(A). Since R is essential, we have 1x=x for all x. The action is well-defined. Assume that ∑aixi=0,xi∈R, ai∈R. Choose e∈A such that eai=ai for all i. For any m∈M(A), we have
∑m(aixi) =∑(mai)xi=∑(me)(aixi)=
(me)∑aixi=0
Therefore, we can define the action of M(A) by m(ax)=(ma)x.
Proposition 3 Let A be a bornological quantum group. If we denote M as the category of essential left A-modules and morphisms, then M is a monoidal category with unit.
The unit is C, and the module structure over C is ac=ε(a)c for a∈A and c∈C.
Definition 3 Let R be an essential A-module. We say that 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.
⊗a)(y⊗b)=∑x(a(1)y)⊗a(2)b for all x,y∈R and a,b∈A.
a.
Definition 4 Let A be a bornological quantum group and R is a bornological algebra. Γ is called the coaction of A on R if there is an essential injective homomorphism Γ: 
A) satisfying
1) Γ(R)(1⊗A)⊆
⊗A)Γ(R)⊆
Γ.
Γ is called reduced if (R⊗1)Γ(R)⊆
A. If Γ is reduced, we also have Γ(R)(R⊗1)⊆A. In this case, R is called an A-comodule algebra.
Proposition 5 Let (A,Δ) be a bornological quantum group. If R is an A-comodule algebra, then R is an 
-module algebra.
⊗a)Γ(x)) for all x∈R, b=φ(a·),where a∈A.
k.
Proposition 6 Let (A,Δ) be a bornological quantum group. If R is an A-module algebra, then R is an -comodule algebra.
Proof The coaction here is defined as
Γ(r)(1⊗b)=∑S-1(a(1))·r⊗φ(·a(2))
(1⊗
⊗
where b=φ(·a) and b′=ψ(·a′) for a,a′∈A.
a.
Theorem 1 Let A be a bornological quantum group and R a bornological algebra. R is A-module algebra if and only if R is an -comodule algebra.
Definition 5 Let A be a bornological quantum group and Γ is a coaction of A on a bornological algebra R. An element f∈M(R) is coinvariant if Γ(f)=f⊗1. Let RcoA be the set of all coinvariants in M(R), and RcoA is a unital subalgebra.
Definition 6 Let A be a bornological quantum group, and R is a bornological algebra. Then, the coaction Γ defined above is called Galois coaction if Γ is reduced and the map
⊗y→(x⊗1)Γ(y)
is bijective.
Definition 7 Let Γ be a right Galois coaction of a bornological quantum group (A,Δ) on R. Then (R,Γ) is called a right Galois object for A if RcoA is the scalar field.
Theorem 2 Let R be a right A-Galois object. There exists a faithful δ-invariant functional φR on R such that 
φ)(Γ(r))=φR(r)1 for all r∈R. Moreover, there exists a non-zero invariant functional ψR on R.
Γ(x)(Γ(s)(1⊗a))=Γ(xs)(1⊗a)=
⊗1))(1⊗a⊗1))=
⊗s(1)a⊗1))=
r(0)s(0)⊗φ(r(1))s(1)a=(x⊗1)(Γ(s)(1⊗a))
Since R is a Galois object, we have Γ(x)=x⊗1=φR(r) for some scalar φR(r). So, we have defined a bounded linear functional φR on R. It is easy to obtain δ-invariance and faithfulness.
o.
Proposition 7 Let A be a bornological quantum group and R is an A-Galois object. φR and ψR are defined in Theorem 2. Then for all r,s∈R, we have
1) There exists a unique invertible element δR∈M(A) such that φR(rδR)=ψR(r);
2) There exists a unique bounded algebra automorphism σ of R such that φR(rσ(s))=φR(sr). We call σ the modular automorphism.
{
}
⊗1)β(a)=V-1(r⊗a)β(a)(1⊗r).
X.
X. The associativity is straightforward.
The comultiplication 
X) is defined as
ΔX([ρ,ρ1]X)=[ρ(1),ρ1(2)]⊗[ρ(2),ρ1(1)]
.
The essential algebra homomorphism εX:X→C is defined as εX([ρ,ρ1]X)=ρ(1)ρ1(1).
R.
l.
Theorem 3 Together with the maps ΔX,εX,SX,φX, X is a bornological quantum group.
References：
[1]van Daele A. Multiplier Hopf algebras [J]. Trans Amer Math Soc, 1994, 342(2):917-932. DOI:10.1090/s0002-9947-1994-1220906-5.
[2]van Daele A. An algebraic framework for group duality [J]. Advances in Mathematics, 1998, 140(2):323-366. DOI:10.1006/aima.1998.1775.
[3]Voigt C. Bornological quantum groups [J]. Pacific Journal of Mathematics, 2008, 235(1):93-135. DOI:10.2140/pjm.2008.235.93.
[4]van Daele A, Wang S H. Pontryagin duality for bornological quantum hypergroups [J]. Manuscripta Mathematica, 2009, 131(1):247-263. DOI:10.1007/s00229-009-0318-8.
[5]Meyer R. Smooth group representations on bornological vector spaces [J]. Bull Sci Math, 2004, 128(2):127-166. DOI:10.1016/j.bulsci.2003.12.002.
[6]Voigt C. Equivariant periodic cyclic homology [J]. Journal of the Institute of Mathematics of Jussieu, 2007, 6(4): 689-763. DOI:10.1017/s1474748007000102.
[7]de Commer K. Galois objects for algebraic quantum groups[J]. Journal of Algebra, 2009, 321(6):1746-1785. DOI:10.1016/j.jalgebra.2008.11.039.
[8]Drabant B, van Daele A, Zhang Y H. Actions of multiplier Hopf algebras [J]. Comm Algebra, 1999, 27(9):4117-4172. DOI:10.1080/00927879908826688.
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
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δ-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
上的余乘Δ、余单位ε、対极S和不变泛函φ,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：O153.3
Citation：：Zhou Nan, Wang Shuanhong．Construction of new bornological quantum groups based on Galois objects[J]．Journal of Southeast University (English Edition),2016,32(4):524-526．
DOI：10.3969/j.issn.1003-7985.2016.04.022.
DOI：10.3969/j.issn.1003-7985.2016.04.022
Received 2015-09-14．
Biographies：Zhou Nan(1991—),male,graduate; Wang Shuanhong(corresponding author), male, doctor, professor,shuanhwang@yahoo.com.