dual of a coalgebra is an algebra, the

Let R be a commutative ring with unity. Suppose we have a coassociative coalgebra (C,Δ) and an associative algebra A, both over R. Since C and A are both R-modules, it follows that HomR(C,A) is also an R-module. But in fact we can give it the structure of an associative R-algebraPlanetmathPlanetmath. To do this, we use the convolution productPlanetmathPlanetmath. Namely, given morphisms f and g in HomR(C,A), we define their product fg by


where we use the Sweedler notation


for the comultiplication Δ. To see that the convolution product is associative, suppose f, g, and h are in HomR(C,A). By applying the coassociativity of Δ, we may write




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 ε:CR and A is unitary with identity 1:RA, then their compositionMathworldPlanetmathPlanetmath 1ε:CA is the identity for the convolution product.


Let C be a coassociative coalgebra over R. Then R itself is an associative R-algebra. The algebra HomR(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 HomR(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 DualityMirror 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 groupPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is a Hopf algebraPlanetmathPlanetmathPlanetmath.


  • 1 W. Nichols and M. Sweedler, Hopf algebras and combinatoricsDlmfMathworld, in Proceedings of the conference on umbral calculus and Hopf algebras, ed. R. Morris, AMS, 1982.
  • 2 B. Fauser: A treatise on quantum Clifford AlgebrasPlanetmathPlanetmath. 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