标题： 代数积分理论 显示英文标题

标题： A theory of algebraic integration

摘要： 在本文中，我们将积分的概念扩展到一般的代数。 特别是我们关注一类代数，我们将它们称为自共轭代数，这类代数具有右乘代数和左乘代数等价的性质。 在这种情况下，总是可以定义一个积分，它具有许多通常积分的性质。 例如，如果该代数有一个连续的自同构群，则相应的导子满足通常的分部积分公式。 我们还讨论了如何在子代数上进行积分。 讨论了许多例子，从格拉斯曼代数开始，我们恢复了通常的贝雷辛规则。 也考虑了拟格拉斯曼代数以及矩阵代数。 由于格拉斯曼和拟格拉斯曼代数可以用矩阵表示，我们还展示了它们的积分可以看作是相应矩阵上的迹。 一个有趣的例子是群代数的情况，我们证明了我们的积分定义等价于群的酉不可约表示的求和。 我们还展示了一些在非自共轭代数（玻色子和$q$-玻色子振子）上的积分例子，以及在非结合代数（八元数）上的积分例子。 显示英文摘要

摘要： In this paper we extend the idea of integration to generic algebras. In particular we concentrate over a class of algebras, that we will call self-conjugated, having the property of possessing equivalent right and left multiplication algebras. In this case it is always possible to define an integral sharing many of the properties of the usual integral. For instance, if the algebra has a continuous group of automorphisms, the corresponding derivations are such that the usual formula of integration by parts holds. We discuss also how to integrate over subalgebras. Many examples are discussed, starting with Grassmann algebras, where we recover the usual Berezin's rule. The paraGrassmann algebras are also considered, as well as the algebra of matrices. Since Grassmann and paraGrassmann algebras can be represented by matrices we show also that their integrals can be seen in terms of traces over the corresponding matrices. An interesting application is to the case of group algebras where we show that our definition of integral is equivalent to a sum over the unitary irreducuble representations of the group. We show also some example of integration over non self-conjugated algebras (the bosonic and the $q$-bosonic oscillators), and over non-associative algebras (the octonions).