Frobenius manifolds and Frobenius algebra-valued integrable systems
Frobenius manifolds and Frobenius algebra-valued integrable systems
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Frobenius 流形和 Frobenius 代数值可积系统
DOI：
10.1007/s11005-017-0939-x
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发表时间：
2017
影响因子：
1.2
通讯作者：
Zuo Dafeng
中科院分区：
文献类型：
--
作者：
Strachan Ian A. B.;Zuo Dafeng
可积性的概念通常会从具有标量值域的系统扩展到具有代数值域的系统。在这样的扩展中，代数的性质和结构在确保保留可积性方面发挥着核心作用。本文提出了一种新的 Frobenius 代数值可积系统理论。对于源自 Frobenius 流形的系统，这是通过利用此类流形的张量积理论来实现的，由 Kaufmann (Int Math Res Not 19:929–952, 1996)、Kontsevich 和 Manin (Inv Math 124: 313–339, 1996) 开发。通过专门化这种构造，使用固定的弗罗贝尼乌斯代数可以得出这样的理论。更一般地说，我们可以应用相同的思想来构造无值拓扑量子场论。然后研究两类可积演化方程的哈密顿性质：无色散和色散演化方程。讨论了这些想法的应用，并作为一个例子，构建了一个有值修正的卡马萨-霍尔姆方程。
The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper, a new theory of Frobenius algebra-valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann (Int Math Res Not 19:929–952, 1996), Kontsevich and Manin (Inv Math 124: 313–339, 1996). By specializing this construction, using a fixed Frobenius algebraone can arrive at such a theory. More generally, one can apply the same idea to construct an-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an-valued modified Camassa–Holm equation is constructed.