nLab dual bialgebra

See also dual gebra for a more general entry.

Definition

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

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

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

$$\langle h, 1_K\rangle = \epsilon_H(h), \,\,\,\,\, \langle 1_H, k\rangle = \epsilon_K(k).$$

(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 (nondegenerate in each variable) as a $k$-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.

Finite duals

As stated in dual gebra, for an algebra $A = (A,m)$, a finite (or restricted or Hopf) dual is the subspace

$$A^\circ = \{f\in A^* | \exists ideal I\subset Ker(f)\subset A, dim(A^*/I)\lt\infty \}\subset A^*.$$

Then $f\in A^\circ$ iff $m^*(f)\subset A^*\otimes A^*$ which is iff $m^*(f)\subset A^\circ\otimes A^\circ$, see Sweedler1969. This subspace is consequently a coalgebra, via the corestriction of $m^*:A^*\to (A\otimes A)^*$. If $A$ is a bialgebra (Hopf algebra) then $A^\circ$ is such as well. We can say this differently: If $H$ is a bialgebra (resp. Hopf algebra) then its finite dual $H^\circ$ has a unique structure of a bialgebra (resp. Hopf algebra) such that the evaluation is a bialgebra pairing.

A general pairing $\langle -,-\rangle : H\otimes K\to K$ of vector spaces underlying Hopf algebras $H$ and $K$ is a bialgebra pairing iff the induced map $H\to K^\circ$, $h\mapsto \langle h, -\rangle$ is a bialgebra map.

Hopf algebra case

If $H$ and $K$ are Hopf algebras then the bialgebra pairing is a Hopf pairing if $\langle h, S k \rangle = \langle S h, k\rangle$. This is however, automatic.

Literature

• Moss E. Sweedler, $()^\circ$, Chapter VI in Hopf algebras, Benjamin, N.Y. 1969

• MO: is-a-bialgebra-pairing-of-hopf-algebras-automatically-a-hopf-pairing, dual-of-a-hopf-algebra, hopf-dual-of-the-hopf-dual, on-reflexive-bialgebras

• Stephen Donkin, On the Hopf algebra dual of an enveloping algebra, Math. Proc. Camb. Phil. Soc. 91 (1982) 215-224

• R. G. Heyneman, David E. Radford, Reflexivity and coalgebras of finite type,J. Alg. 28 (1974) 215-246 pdf

category: algebra