Frobenius manifolds and Frobenius algebra-valued integrable systems

Strachan, I. A.B. and Zuo, D. (2017) Frobenius manifolds and Frobenius algebra-valued integrable systems. Letters in Mathematical Physics, 107(6), pp. 997-1026. (doi: 10.1007/s11005-017-0939-x)

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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 algebra A,A, one can arrive at such a theory. More generally, one can apply the same idea to construct an AA -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 AA -valued modified Camassa–Holm equation is constructed.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Zuo, Dr Dafeng and Strachan, Professor Ian
Authors: Strachan, I. A.B., and Zuo, D.
College/School:College of Science and Engineering > School of Mathematics and Statistics
College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Letters in Mathematical Physics
ISSN (Online):1573-0530
Published Online:24 January 2017
Copyright Holders:Copyright © 2016 Springer
First Published:First published in Letters in Mathematical Physics 107(6):997-1026
Publisher Policy:Reproduced under a Creative Commons License
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