Department of Mathematics,

University of Wales Swansea,

Singleton Park, Swansea SA2 8PP,

UK

E-mail: D.Parashar@swansea.ac.uk

**Some new solutions to the one and two parameter forms
of the quantum Yang-Baxter equation**

**Abstract:**

The quantum Yang-Baxter equation (QYBE) has three fundamental forms:
the constant, the one parameter, and the two parameter. Most of the known
solutions to these come from examples of quasitriangular Hopf algebras.
In a somewhat new approach, certain operators called Yang-Baxter (YB) operators
arising from purely algebra structures turn out to be solutions of the
constant QYBE. In this work, we investigate spectral-parameter dependent
YB operators arising from algebra structures that solve the one and two
parameter forms of the QYBE. We construct two families of such operators
which are also connected to bialgebras through the FRT construction. Further,
we also establish a link with YB systems.