dual bialgebra

Given a field k, a k-vector space pairing between k-bialgebras H and K is a k-linear map ,:H×Kk such that

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

(where on the left hand side , denotes the map HHKKk given by hh,kk=h,kh,k), 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.

Created on August 25, 2011 00:37:07 by Zoran Škoda (