论文标题
区域的区域产生混洗代数
Areas of areas generate the shuffle algebra
论文作者
论文摘要
我们考虑单词上半剃须刀的反对称，我们称之为“区域”操作员，因为它对应于迭代综合签名的符号符号区域。张量代数是该操作员下的所谓托尔卡拉代数。我们表明，区域操作员的迭代应用足以恢复路径的迭代综合签名。正如已知第二个级别的“信息”添加到第一个级别相当于路径组成部分之间的区域一样，这意味着随后级别添加的所有信息都等同于迭代区域。在进入这个主要结果的路上，我们表征了（均匀）生成的混洗代数集。我们最终讨论了区域操作员与离散集成和随机集成之间的兼容性，并在区域区域的线性跨度上得出结论。
We consider the anti-symmetrization of the half-shuffle on words, which we call the 'area' operator, since it corresponds to taking the signed area of elements of the iterated-integral signature. The tensor algebra is a so-called Tortkara algebra under this operator. We show that the iterated application of the area operator is sufficient to recover the iterated-integral signature of a path. Just as the "information" the second level adds to the first one is known to be equivalent to the area between components of the path, this means that all the information added by subsequent levels is equivalent to iterated areas. On the way to this main result, we characterize (homogeneous) generating sets of the shuffle algebra. We finally discuss compatibility between the area operator and discrete integration and stochastic integration and conclude with some results on the linear span of the areas of areas.
