On a new class of non-dynamical ABCD algebras for classical and quantum integrable systems

We consider classical and quantum non-dynamical quadratic $abcd$ Lax algebras with classical and quantum $gl\left(n\right)\otimes gl\left(n\right)$-valued $abcd$-tensors satisfying a set of quadratic non-dynamical Yang-Baxter-type equations generalizing those of Fredel and Maillet [1]. We establish a relation of some of these equations with the so-called “semi-dynamical” Yang-Baxter equations of [2]. We show that the linearization of the corresponding quadratic structures lead to linear tensor structures with the classical $gl\left(n\right)\otimes gl\left(n\right)$-valued r-matrices satisfying usual “permuted” classical Yang-Baxter equations [1], [3], [4], [5]. We consider example of our construction associated with the deformed $Z_n$-graded r-matrix of [9], [10], [11] and explicitly construct the corresponding $abcd$-tensors — both classical and quantum. Small n examples are also considered in some details.