Extensions of the discrete KP hierarchy and its strict version
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EXTENSIONS OF THE DISCRETE KP HIERARCHY AND ITS STRICT VERSION G. F. Helminck,∗ V. A. Poberezhny,† and S. V. Polenkova‡
We show that both the dKP hierarchy and its strict version can be extended to a wider class of deformations satisfying a larger set of Lax equations. We prove that both extended hierarchies have appropriate linearizations allowing a geometric construction of their solutions.
Keywords: pseudodifference operator, hierarchy, linearization, oscillating matrix, wave matrix, geometric construction of solutions DOI: 10.1134/S0040577920090044
1. Introduction Let P sΔ be the algebra of Z×Z matrices with coefficients in a commutative algebra R over k = R or k = C with only a finite number of nonzero diagonals above the central diagonal. We consider two integrable hierarchies in P sΔ: the discrete KP hierarchy and its strict version. The first was introduced in [1] and is a deformation of the commutative algebra k[Λ] (Λ denotes the shift matrix in P sΔ) satisfying a system of compatible Lax equations. The strict version was presented in [2] and is a wider deformation of k[Λ] satisfying a system of Lax equations based on a different decomposition of P sΔ. A functional analytic construction of its solutions was given in [3]. Here, we show that each of the two deformations embeds into a more extensive deformation satisfying a larger set of Lax equations, and we correspondingly call them the extended dKP hierarchy and the extended strict dKP hierarchy. We present linearizations for both of the new systems and give a geometric construction of solutions of both extensions. This paper has the following structure. In Sec. 2, we recall various notions in the algebra P sΔ and the algebra P sΔT of transposed matrices and important properties of both these algebras. In Sec. 3, we describe the initial hierarchies and their extensions. In Sec. 4, we introduce two appropriate modules, one for P sΔ and the other for P sΔT . Further, in the product of these modules, we present a set of equations, called ∗
Korteweg–de Vries Institute, University of Amsterdam, Amsterdam, The Netherlands, e-mail: [email protected] (corresponding author). †
National Research University “Higher School of Economics,” Moscow, Russia; Alikhanov Institute for Theoretical and Experimental Physics, Moscow, Russia; Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia, e-mail: [email protected]. ‡
University of Twente, Enschede, The Netherlands, e-mail: [email protected].
The research of V. A. Poberezhny was performed at the Center for Advanced Studies, Skolkovo Institute of Science and Technology, and was supported by a grant from the Russian Science Foundation (Project No. 19-1100275). Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 3, pp. 367–382, September, 2020. Received March 1, 2020. Revised March 1, 2020. Accepted May 19, 2020. 1140
c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2043-1140
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