OPPOSITE SKEW COPAIRED HOPF ALGEBRAS

Let A be a Hopf algebra with a linear form σ : k → A⊗A, which is convolution invertible, such that σ 21 (∆⊗id)τ (σ(1)) = σ 32 (id ⊗ ∆)τ (σ(1)). We define Hopf algebras, (A σ , m, u, ∆ σ , ε, S σ ). If B and C are opposite skew copaired Hopf algebras and A = B ⊗ k C then we find Hopf algebras, (A [σ] , m B ⊗ m C , u B ⊗ u C , ∆ [σ] , ε B ⊗ ε C , S [σ] ). Let H be a finite dimensional commutative Hopf algebra with dual basis {h i } and {h ∗ i }, and let A = H op ⊗ H ∗ . We show that if we define σ : k → H op ⊗ H ∗ by σ(1) = P h i ⊗ h ∗ i then (A [σ] , m A , u A , ∆ [σ] , ε A , S [σ] ) is the dual space of Drinfeld double, D(H) ∗ , as Hopf algebra.