给定一个非退化的幂等代数到自身张量积代数的乘子代数上的一个余乘和该乘子代数中的一个幂等元，通过研究余乘和张量积代数上Galois映射的值域与核之间的关系，来定义代数量子群胚，并用其源代数和靶代数来建立该代数量子群胚的余积分与积分理论，从而建立其上的Pontryagin对偶理论，进而利用傅里叶变换理论来证明Radford对极的四次方公式。其次，研究代数量子群胚所余作用的Galois对象的弱K.M.S.函数及对极平方的类似映射的存在性问题。 最后，讨论该代数量子群胚上的模范畴的构造以及Hopf循环上同调，主要通过双李群胚来构造张量范畴和模范畴的正合性，尤其，证明一个带有乘法的循环operad是余循环模，且它的上链复形的上同调是Batalin-Vilkovisky代数且其负循环上同调是一个阶为-2的分次李代数，同时将研究代数量子超群胚和有界型量子群胚上的Nichols代数与Weyl群胚。

Given an algebra A with a non-degenerate product and with a coproduct on A from A to the multiplier algebra of the tensor product of A with itself. We will study the coproduct and the relation between the ranges and the kernels of the Galois maps by an idempotent multiplier in the multiplier algebra of the tensor product of A with itself and give the notion of an algebraic quantum groupoid. Then we will establish the theory of integrals and cointegrals in order to show the Pontryagin duality for these algebraic quantum groupoids. As an application of the duality, we obtain the Radfod's formula for the antipode of an algebraic quantum groupoid by using the Fourier transformation on A.Then we introduce a theory of Galois objects for algebraic quantum groupoids, i.e.,algebras equipped with a Galois coaction by an algebraic quantum groupoid and prove that there are two distinguished weak K.M.S. functions and there is an analogue of the antipode squared. Finally, we will discuss how to construct the new module categories attached to double Lie groupoids, exacts and Hopf cyclic cohomology, in particular, we will show that every cyclic operad with multiplication is a cocyclic module such that the homology of the associated cochain complex is a Batalin-Vilkovisky algebra and its negative cyclic cohomology is a graded Lie algebra of degree -2. Similarly, we will develop our results for an algebraic quantum groupoid to the setting of an algebraic quantum hypergroupoid and a bornological qLet A be an algebra with a non-degenerate product and let \D be a coproduct on A from A to M(A\o A), the multiplier of A\o A. We will study the coproduct \D and the relation between the ranges and the kernels of the Galois maps by an idempotent multiplier E in M(A\o A) and give the notion of an algebraic quantum groupoid. Then we will establish the theory of integrals and cointegrals in order to show the Pontryagin duality for these algebraic quantum groupoids. As an application of the duality, we obtain the Radfod's formula for the antipode of an algebraic quantum groupoid by using the Fourier transformation on A. Then we introduce a theory of Galois objects for algebraic quantum groupoids, i.e., algebras equipped with a Galois coaction by an algebraic quantum groupoid and prove that there are two distinguished weak K.M.S. functions and there is an analogue of the antipode squared. Finally, we will discuss how to construct the new module categories attached to double Lie groupoids, exacts and Hopf cyclic cohomology, in particular, we will show that every cyclic operad with multiplication is a cocyclic module such that the homology of the associated cochain complex is a Batalin-Vilkovisky algebra and its negative cyclic cohomology is a graded Lie algebra of degree -2. We will also develop our the results and Nichols algebras and Weyl groupoids in the setting of an algebraic quantum hypergroupoid and a bornological quantum groupoid.