dual of a coalgebra is an algebra, the

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 $\mathrm{Hom}_{R}({C},{A})$ 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 $\mathrm{Hom}_{R}({C},{A})$ , we define their product $f g$ 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 $\mathrm{Hom}_{R}({C},{A})$ . 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.

Example.

Let $C$ be a coassociative coalgebra over $R$ . Then $R$ itself is an associative $R$ -algebra. The algebra $\mathrm{Hom}_{R}({C},{R})$ is called the algebra dual to the coalgebra $C$ .

We have seen that any coalgebra dualizes to give an algebra. One might expect that a similar construction could be performed on $\mathrm{Hom}_{R}({A},{R})$ 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.

References

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).

Title	dual of a coalgebra is an algebra, the
Canonical name	DualOfACoalgebraIsAnAlgebraThe
Date of creation	2013-03-22 16:34:20
Last modified on	2013-03-22 16:34:20
Owner	mps (409)
Last modified by	mps (409)
Numerical id	8
Author	mps (409)
Entry type	Derivation
Classification	msc 16W30
Related topic	GrassmanHopfAlgebrasAndTheirDualCoAlgebras
Related topic	DualityInMathematics
Related topic	QuantumGroups
Defines	dualities of algebraic structures