ANALOGS OF q-SERRE RELATIONS IN THE YANG-BAXTER ALGEBRAS

1. Yang-Baxter algebras (YBA), introduced in [1, 2, 3], generalize the wideknown FRT construction [4] in the following sense: to any numerical matrix solution R of the Yang-Baxter equation there is associated a bialgebra containing the FRT one as a sub-bialgebra. Generally, this construction may provide examples of (new) bialgebras and Hopf algebras [5]. In several aspects, there is some similarity of YBA with so-called inhomogeneous quantum groups [6, 7], and also with Majid’s scheme of double bosonization [8]. However, in YBA no extra (dilation) generators appear, whereas additional (analogous to q-Serre) relations are not necessarily quadratic. The main goal of the present note is to refine the concept and definition of these generalized Serre relations, first introduced in [1] and further studied in [5]. Here we obtain Serre-like relators as the elements in the kernel of some bilinear form, and the main result of the present paper is a condition (21) for such relators.