nLab
dual bialgebra

Given a field $k$, a $k$-vector space pairing between $k$-bialgebras $H$ and $K$ is a $k$-linear map $⟨,⟩:H\times K\to k$ such that

$$⟨h\otimes h\prime ,{\Delta }_{K}k⟩=⟨hh\prime ,k⟩$$\langle h\otimes h', \Delta_K k\rangle = \langle h h', k\rangle

$$⟨\Delta (h),k,\otimes k\prime ⟩=⟨h,k\otimes k\prime ⟩$$\langle \Delta (h), k,\otimes k' \rangle = \langle h, k\otimes k'\rangle

(where on the left hand side $⟨,⟩$ denotes the map $H\otimes H\otimes K\otimes K\to k$ given by $⟨h\otimes h,k\otimes k\prime ⟩=⟨h,k⟩⟨h\prime ,k\prime ⟩$), is called the bialgebra pairing.

The bialgebra pairing which is perfect as $k$-vector space pairing (i. e. if $⟨h,k⟩=0$ implies that either $h$ or $k$ is $0$) is called the bialgebra duality.

If $H$ and $K$ are Hopf algebras then the compatibility with antipodes is $⟨h,Sk⟩=⟨Sh,k⟩$.