A construction of solutions of an integrable deformation of a commutative Lie algebra of skew hermitian Z×Z-matrices

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 C_as(R)-hierarchy. It consists of a deformation of a commutative Lie algebra C_as(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.