Title: Integrals for Bialgebras Show Chinese title

Title: 双代数的积分

Abstract: A well-known result by Larson and Sweedler shows that integrals on a Hopf algebra can be obtained by applying the Structure Theorem for Hopf modules to the rational part of its linear dual. This fact can be rephrased by saying that taking the space of integrals comes from a right adjoint functor from a category of modules to the category of vector spaces. This observation inspired the categorical approach that we advocate in this work, which yields to a new notion of integrals for bialgebras in the linear setting. Despite the novelty of the construction, it returns the classical definition in the presence of an antipode. We test this new concept on bialgebras that satisfy at least one of the following properties: being coseparable as regular module coalgebras, having a one-sided antipode, being commutative, being cocommutative, or being finite-dimensional. One of the main results we obtain in this process is a dual Maschke-type theorem relating coseparability and total integrals. Remarkably, there are cases in which the space of integrals turns out to be isomorphic to that of the associated Hopf envelope. In particular, this space results to be one-dimensional for finite-dimensional bialgebras, providing an existence and uniqueness theorem for integrals in the finite-dimensional case. Furthermore, explicit computations are given for concrete examples including the polynomial bialgebra with one group-like variable, the quantum plane and the coordinate bialgebra of $n$-by-$n$ matrices. Show Chinese abstract

Abstract: 由Larson和Sweedler的一个著名结果表明，可以通过将霍普夫模的结构定理应用于其线性对偶的有理部分，来获得霍普夫代数上的积分。 这一事实可以重新表述为：取积分空间来自于从模范畴到向量空间范畴的右伴随函子。 这一观察启发了我们在本文中提倡的范畴方法，该方法在线性设置下产生了一个关于双代数积分的新概念。 尽管该构造具有新颖性，但在存在反元素的情况下，它会返回经典的定义。 我们测试这个新概念对于满足以下至少一个性质的双代数：作为常规模余代数是可分离的，具有单侧反元素，是交换的，是余交换的，或者有限维的。 在这个过程中，我们得到的主要结果之一是一个类似于Maschke的对偶定理，将可分离性与全积分联系起来。 值得注意的是，在某些情况下，积分空间被证明与相关霍普夫包络的积分空间同构。 特别是，对于有限维双代数，该空间结果是一维的，从而提供了有限维情况下积分的存在性和唯一性定理。 此外，给出了具体例子的显式计算，包括一个群元变量的多项式双代数、量子平面以及$n$-$n$矩阵的坐标双代数。