Hopf Algebra Approaches to Nonlinear Algebraic Equations

Abstract

Let M be a k-vector space and R ∈ Hom(M ^⊗p , M ^⊗q ), we present a general version of the FRT-construction, we provide a method for examining whether an FRT-bialgebra A(R) has a pre-braided structure and whether M can be regarded as an A(R)-dimodule. We show that the FRT-relation plays a fundamental role in determining the algebra structure on the FRT-bialgebra and the compatibility condition of relevant dimodule. As an example, we give a Hopf algebra approach for solving both homogeneous and non-homogeneous nonlinear (algebraic) equations.

Key Words:

• FRT-bialgebra
• Hopf equation
• Yang-Baxter equation

Mathematics Subject Classification (2000):

• 16D90
• 16W30
• 81R50

Acknowledgment

The work was carried out during the visit of the author to the Laboratory of Mathematics for Nonlinear Sciences at Fudan University. The research was supported by the Visiting Funds from the Ministry of Education of China and the National Natural Science Foundation of China. The author thanks the referee for his/her helpful comments.