Constants of coordinate differential calculi defined by Yang–Baxter operators

We investigate in details a first order differential calculus with right partial derivatives set up by a not necessarily invertible Yang–Baxter operator. The optimal algebra for this calculus has a natural structure of a braided Hopf algebra and it is isomorphic to the quantum symmetric algebra. The induced to the optimal algebra and to the free cover algebra calculi are right covariant. They are bicovariant if and only if the related braiding is involutive. By means of the P.M. Cohn theory we show that the subalgebra of constants for the cover free differential algebra is a free algebra and an ad-invariant left coideal. If the given algebra is finitely generated then every differential left ideal is generated by constants, a noncommutative Taylor series decomposition formula is valid, and the category of locally nilpotent modules over the operator algebra is semisimple with the only simple object that is isomorphic to the optimal algebra as a module. We find a necessary and sufficient condition for a 1-form to be a complete differential.