...ar combinations of elements of the form [w1|···|wn] withwi∈H such that |v|+1 = n. Therefore, V must be bigraded. To describe linear maps on C(H), we have to recall the definition of relative signs in =-=[1]-=-. For bigraded symbols x1,x2,...,xn, if (i1,i2,...,in) is a re-ordering of (1, 2,...,n), then the sign of (xi1,xi2,...,xin)relativeto(x1,x2,...,xn), or the sign of the permutation ( ) xi1 , xi2 , ...,...

...ogy operation can also be generalized to bigraded Hopf algebras. Let W be the free resolution of the group of integers modular 2, and θ: W ⊗ C(H) ⊗ C(H) → C(H) be the group module morphism defined in =-=[2]-=-. Then, we have θ(e1 ⊗ u ⊗ v) = (−1) |u| S(u; v), θ(e2 ⊗ u ⊗ v) =(−1) |u| S2(u; v), S ′ (u; v) =(−1) 〈u,v〉 S(v; u). So 1 holds. We only prove 4 of the theorem and omit the proof of 2 and 3, since it i...