nLab
dual bialgebra

Given a field kk, a kk-vector space pairing between kk-bialgebras HH and KK is a kk-linear map ,:H×Kk\langle, \rangle : H\times K\to k such that

hh,Δ Kk=hh,k \langle h\otimes h', \Delta_K k\rangle = \langle h h', k\rangle
Δ(h),kk=h,kk \langle \Delta (h), k\otimes k' \rangle = \langle h, k\otimes k'\rangle

(where on the left hand side ,\langle,\rangle denotes the map HHKKkH\otimes H\otimes K\otimes K\to k given by hh,kk=h,kh,k\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 kk-vector space pairing (i. e. if h,k=0\langle h, k\rangle = 0 implies that either hh or kk is 00) is called the bialgebra duality.

If HH and KK are Hopf algebras then the compatibility with antipodes is h,Sk=Sh,k\langle h, S k \rangle = \langle S h, k\rangle.

Revised on February 9, 2016 07:25:06 by Tim Porter (95.145.121.139)