# nLab dual bialgebra

Given a field $k$, a $k$-vector space pairing between $k$-bialgebras $H$ and $K$ such that

$\langle h\otimes h', \Delta_K k\rangle = \langle h h', k\rangle$
$\langle \Delta_H h, k\otimes k' \rangle = \langle h, k k'\rangle$

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

The bialgebra pairing which is perfect as $k$-vector space pairing (i. e. if $\langle h, K\rangle = 0$ implies that $h$ is $0$ and $\langle H,k\rangle$ implies that $k$ is $0$) is called the bialgebra duality.

If $H$ and $K$ are Hopf algebras then the compatibility with antipodes is $\langle h, S k \rangle = \langle S h, k\rangle$.

Revised on May 5, 2017 15:29:42 by Mamuka Jibladze (109.172.129.80)