Integrable deformations in the algebra of pseudodifferential operators from a Lie algebraic perspective

Abstract

Inside the algebra LT Z (R) of Z × Z-matrices with coefficients from a commutative C-algebra R that have only a finite number of nonzero diagonals above the central diagonal, we consider a deformation of a commutative Lie algebra C sh (C) of finite band skew hermitian matrices that is different from the Lie subalgebras that were deformed at the discrete KP hierarchy and its strict version. The evolution equations that the deformed generators of C sh (C) have to satisfy are determined by the decomposition of LT Z (R) in the direct sum of an algebra of lower triangular matrices and the finite band skew hermitian matrices. This yields then the C sh (C)-hierarchy. We show that the projections of a solution satisfy zero curvature relations and that it suffices to solve an associated Cauchy problem. Solutions of this type can be obtained by finding appropriate vectors in the LT Z (R)-module of oscillating matrices, the so-called wave matrices, that satisfy a set of equations in the oscillating matrices, called the linearization of the C sh (C)-hierarchy. Finally, a Hilbert Lie group will be introduced from which wave matrices for the C sh (C)-hierarchy are constructed. There is a real analogue of the C sh (C)hierarchy called the Cas(R)-hierarchy. It consists of a deformation of a commutative Lie algebra Cas(R) of anti-symmetric matrices. We will properly introduce it here too on the way and mention everywhere the corresponding result for this hierarchy, but we leave its proofs mostly to the reader.

Properties of the algebra Psd related to integrable hierarchies

“…A further step [7] is to consider deformations M of ∂ by dressing it with the wider class of invertible operators K from D(0) and by requiring that M should satisfy a similar set of Lax equations as in (1.10), but this time A s should be replaced by the strict differential part of M s . This deformation is called the strict KP hierarchy.…”

“…This deformation is called the strict KP hierarchy. Besides its Lax form, the strict KP hierarchy possesses still two other descriptions: the zero curvature form [7] and the bilinear form [8]. Both the KP hierarchy and its strict version have natural Cauchy problems associated with them and their solvability is discussed in [9].…”

Scaling invariance of the strict KP hierarchy

“…In this section we shortly recall the results needed from [2] about the strict KP-hierarchy in the pseudo differential operators Psd. The algebra Psd is built up as follows: one starts with a commutative algebra R over a field k of characteristic zero and a privileged klinear derivation ∂ : R → R. Given R and ∂, one forms the algebra R[∂] of differential operators in ∂ with coefficients from R. It consists of k -linear endomorphisms of R of the form n i=0 a i ∂ i , a i ∈ R. For simplicity, we assume that the powers of ∂ are R -linear independent, otherwise one has to pass to a cover of R[∂] [3].…”

“…To do that we need another form of the strict KP hierarchy. It was shown in [2] that the strict differential operators {C r } in Psd corresponding to a solution M of the strict KP-hierarchy, satisfy…”

Integrable hierarchies in the N×N-matrices related to powers of the shift operator

“…Integrable hierarchies both for KdV-type equations and Toda-type equations often occur, see [2,3,6,[10][11][12] and [5], at splitting an algebra as a vector space in the direct sum of two Lie algebras, taking a set of commuting elements in one component and deforming these directions with suitable elements from the group corresponding to the other component.…”