On new rational non-dynamical ABCD matrices for quadratic classical and quantum algebras

Abstract

We consider classical and quantum non-dynamical quadratic abcd Lax algebras with rational classical and quantum gl(n)⊗gl(n)-valued abcd-tensors satisfying a set of quadratic “differential” or “semi-dynamical” [3] Yang-Baxter-type equations generalizing those of Fredel and Maillet [2]. The linearization of the corresponding quadratic structure lead to linear tensor structure with the classical gl(n)⊗gl(n)-valued non-dynamical, non-skew-symmetric rational r-matrix coinciding with a quasi-graded deformation of the standard skew-symmetric rational r-matrix [7]. The classical and quantum abcd structures corresponding to the simplest n=2 case are considered in details.