The QuantumDividedPowerAlgebra is a graded algebra over a ring R[q]. The component in degree n is the free R[q]-module with basis xⁿ. The multiplication is defined on basis elements by x^r.x^s = [r+s,r]_q x^r+s where [r+s,r]_q is the quantum binomial coefficient.

The  DividedPowerAlgebra is a graded algebra over a ring R. The component in degree n is the free R-module with basis xⁿ. The multiplication is defined on basis elements by x^r.x^s = [r+s,r]_q x^r+s where [r+s,r] is the binomial coefficient.

The divided power algebra is a Hopf algebra and is the dual Hopf algebra to R[x]. The coproduct on the divided power Hopf algebra is x^k> x^k x 1 + x^k-1 x x + ... 1 x x^k (where I have used x as an indeterminate and as a tensor product symbol).