Tannaka Duality and the FRT-Construction

Abstract

We apply Tannaka duality machinery to a one-generator category, which gives a general version of the FRT-construction for what we have called the homogeneous and non-homogeneous categories. In the case of n-homogeneous category, we present a Hopf algebra approach for solving the mixed Yang-Baxter type equations.One of the functions of Tannaka duality is to provide a key for entry to quantum groups (or Hopf algebras) [3]. The goal of this paper is to describe precisely how it works on a one-generator category N (see below), which gives a more general version of the FRT-construction.The FRT-construction is known for being a Hopf algebra approach of solving some non-linear equations. Faddeev, Reshetikhin and Takhtajan [2] initiated the method for the Yang-Baxter operators. Recently, Militaru [7, 8] adapted the FRT-construction to what he has called the Long equation, the D-equation and the Hopf equation. Many related solutions are obtained by using the Hopf algebra approach. 5717 ½Z; f nþ1 ¼ e I nþ1 J nþ1 +Xð f Þ À Xð f Þ+e I nþ1 J nþ1¼ ðe I n J n e i nþ1 j nþ1 Þ+ðX ðgÞ 1Þ À ðX ðgÞ 1Þ+ðe I n J n e i nþ1 j nþ1 Þ ¼ ½e I n J n ; g n ðe i nþ1 j nþ1 1Þ þ ðX ðgÞ e I n J n Þ ½e i nþ1 j nþ1 ; 1 1 ¼ ½e I n J n ; g n ðe i nþ1 j nþ1 1Þ: LU Downloaded by [Simon Fraser University] at 22:26 16 November 2014Now suppose that f 2 A k ðk > nÞ, set f ¼ g h for g 2 A s , h 2 A t ðs þ t ¼ kÞ, and Z ¼ e I

Hopf Algebra Approaches to Nonlinear Algebraic Equations

“…& Remark. One can find a Tannaka version proof of the universal property in Lu (2001). Apart from the comodule structure, additional relations are required to make M an A(R)-module.…”

“…Let M be an m-dimensional vector space, for any R, F 2 End(M M), we call (R, F ) a mixed Yang-Baxter type pair (Lu, 2001) if R and F satisfy…”

“…In Lu (2001), we discussed another Hopf algebra approach for solving nonlinear equations using Tannaka duality machinery. We examined when an FRT-bialgebra has an antipode, in the homogeneous cases, based on what we have defined as the pre-braided structure on the FRT-bialgebra.…”

“…Definition (Lu, 2001). A pre-braided bialgebra is a bialgebra H equipped with a k-bilinear skew-pairing s on H; that is, Hopf Algebra Approaches 1193…”