代数、导子与积分
Algebras, Derivations and Integrals
摘要：
在前面的文章中介绍的代数上的积分的背景下，我们得到了几个结果，一类特定的结合代数的单位元。这类代数称为自共轭代数，例如，它们包括$p$阶的Paragrassmann代数、四元数代数和环面代数。我们研究导子和积分之间的关系，证明了黎曼积分的平移不变性的措施和消失的积分的总导数（方便的边界条件）的标准结果的推广。我们还考虑的可能性，给定的积分代数，以定义它的积分在一个子代数，以类似的方式通常的积分流形。在积分测度中投影出子流形。我们证明，这是可能的paragrassmann代数的顺序$p$，一旦我们认为他们作为子代数的代数的$(p+1)\times(p+1)$矩阵。我们还发现子代数上的积分与直接定义的积分是一致的。作为副产品，我们可以定义在一维格拉斯曼代数上的积分为在$2\times 2$矩阵上的迹。
英文摘要：
In the context of the integration over algebras introduced in a previous paper, we obtain several results for a particular class of associative algebras with identity. The algebras of this class are called self-conjugated, and they include, for instance, the paragrassmann algebras of order $p$, the quaternionic algebra and the toroidal algebras. We study the relation between derivations and integration, proving a generalization of the standard result for the Riemann integral about the translational invariance of the measure and the vanishing of the integral of a total derivative (for convenient boundary conditions). We consider also the possibility, given the integration over an algebra, to define from it the integral over a subalgebra, in a way similar to the usual integration over manifolds. That is projecting out the submanifold in the integration measure. We prove that this is possible for paragrassmann algebras of order $p$, once we consider them as subalgebras of the algebra of the $(p+1)\times(p+1)$ matrices. We find also that the integration over the subalgebra coincides with the integral defined in the direct way. As a by-product we can define the integration over a one-dimensional Grassmann algebra as a trace over $2\times 2$ matrices.
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