Let $R$ be a commutative ring with unity. Suppose we have a coassociative coalgebra $(C,\Delta)$ and an associative algebra $A$ , both over $R$ . Since $C$ and $A$ are both $R$ -modules, it follows that $\Hom[R]{C}{A}$ is also an $R$ -module. But in fact we can give it the structure of an associative $R$ -algebra. To do this, we use the convolution product. Namely, given morphisms $f$ and $g$ in $\Hom[R]{C}{A}$ , we define their product $fg$ by $$ (fg)(x) = \sum_x f(x_{(1)})\cdot g(x_{(2)}), $$ where we use the Sweedler notation $$ \Delta(x) = \sum_x x_{(1)}\otimes x_{(2)} $$ for the comultiplication $\Delta$ . To see that the convolution product is associative, suppose $f$ , $g$ , and $h$ are in $\Hom[R]{C}{A}$ . By applying the coassociativity of $\Delta$ , we may write $$ ((fg)h)(x) = \sum_x (f(x_{(1)})\cdot g(x_{(2)}))\cdot h(x_{(3)}) $$ and $$ (f(gh))(x) = \sum_x f(x_{(1)})\cdot (g(x_{(2)}))\cdot h(x_{(3)}). $$ Since $A$ has an associative product, it follows that $(fg)h=f(gh)$ .

In the foregoing, we have not assumed that $C$ is counitary or that $A$ is unitary. If $C$ is counitary with counit $\varepsilon\colon C\to R$ and $A$ is unitary with identity $1\colon R\to A$ , then their composition $1\circ\varepsilon\colon C\to A$ is the identity for the convolution product.

We have seen that any coalgebra dualizes to give an algebra. One might expect that a similar construction could be performed on $\Hom[R]{A}{R}$ to give a coalgebra dual to $A$ . However, this is not the case. Thus coalgebras (based on ``factoring'') are more fundamental than algebras (based on ``multiplying'').

(The proof will be provided at a later stage).

Remark on Al/gebraic Duality-Mirror or tangled `duality' of algebras and `gebras':
An interesting twist to duality was provided in Fauser's publications on al/gebras where mirror or tangled `duality' has been defined for Grassman-Hopf al/gebras. Thus, an algebra not only has the usual reversed arrow dual coalgebra but a mirror (or tangled) gebra which is quite distinct from the coalgebra.

Note: The dual of a quantum group is a Hopf algebra.

Bibliography

1

W. Nichols and M. Sweedler, Hopf algebras and combinatorics, in Proceedings of the conference on umbral calculus and Hopf algebras, ed. R. Morris, AMS, 1982.

2

B. Fauser: A treatise on quantum Clifford Algebras. Konstanz, Habilitationsschrift.
arXiv.math.QA/0202059 (2002).

3

B. Fauser: Grade Free product Formulae from Grassman-Hopf Gebras. Ch. 18 in R. Ablamowicz, Ed., Clifford Algebras: Applications to Mathematics, Physics and Engineering, Birkhäuser: Boston, Basel and Berlin, (2004).

4

J. M. G. Fell.: The Dual Spaces of C*-Algebras., Transactions of the American Mathematical Society, 94: 365-403 (1960).