Quasi-triangular structures on Hopf algebras with positive bases

Abstract

A basis B of a finite dimensional Hopf algebra H is said to be positive if all the structure constants of H relative to B are non-negative. A quasi triangular structure R ∈ H ⊗ H is said to be positive with respect to B if it has non-negative coefficients in the basis B ⊗ B of H ⊗ H. In our earlier work, we showed that finite dimensional Hopf algebras with positive bases are in one-to-one correspondence with group factorizations G = G+G−. In this paper, we show that positive quasi-triangular structures on such Hopf algebras are given by a pair of homomorphisms ξ, η : G+ → G− satisfying some compatibility conditions. Further properties of such structures are also discussed.