nLab
dual bialgebra

Given a field kk, a kk-vector space pairing between kk-bialgebras HH and KK such that

hh,Δ Kk=hh,k \langle h\otimes h', \Delta_K k\rangle = \langle h h', k\rangle
Δ Hh,kk=h,kk \langle \Delta_H h, k\otimes k' \rangle = \langle h, k 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 hh is 00 and H,k\langle H,k\rangle implies that 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 May 5, 2017 15:29:42 by Mamuka Jibladze (109.172.129.80)