dual of a coalgebra is an algebra, the
Let be a commutative ring with unity. Suppose we have a coassociative coalgebra and an associative algebra , both over . Since and are both -modules, it follows that is also an -module. But in fact we can give it the structure of an associative -algebra. To do this, we use the convolution product. Namely, given morphisms and in , we define their product by
where we use the Sweedler notation
for the comultiplication . To see that the convolution product is associative, suppose , , and are in . By applying the coassociativity of , we may write
Since has an associative product, it follows that .
In the foregoing, we have not assumed that is counitary or that is unitary. If is counitary with counit and is unitary with identity , then their composition is the identity for the convolution product.
Let be a coassociative coalgebra over . Then itself is an associative -algebra. The algebra is called the algebra dual to the coalgebra .
We have seen that any coalgebra dualizes to give an algebra. One might expect that a similar construction could be performed on to give a coalgebra dual to . 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.
- 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.
B. Fauser: A treatise on quantum Clifford Algebras. Konstanz,
- 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|
|Date of creation||2013-03-22 16:34:20|
|Last modified on||2013-03-22 16:34:20|
|Last modified by||mps (409)|
|Defines||dualities of algebraic structures|