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I have to figure out the duals to a couple of graded algebras. This requires a comultiplication (also called a coproduct in Hatcher). Hatcher's book shows what form the comultiplication must take using a cohomology argument for the cohomology of H-spaces. I do not know that these algebras are the cohomology of some H-spaces. Is there a purely algebraic argument to figure out a comultipication for an arbitrary graded algebra or do I need some extra data.

If you were wondering, I need to figure the duals of $\Lambda (y)$ and $\Gamma [ \gamma ] $.

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There is no way to "figure out" a comultiplication for an arbitrary graded algebra. Some algebras are not bialgebras in any way, and those algebras that can be made into bialgebras can usually turned into bialgebras in many different ways.

You will probably have to be more specific about the algebras you have in mind (at the very least, explain the notation you are using!)

The following, earlier text answers another question, the one in the title...

Doesn't the following obvious construction work? If $A$ and $B$ are graded algebras you can present them as quotients of free algebras $T(V)$ and $T(W)$ modulo homogeneous ideals generated by sets of homogeneous elements $R_A\subset T(V)$ and $R_B\subset T(W)$, so that $A=T(V)/\langle R_A\rangle$ and $B=T(W)/\langle R_B\rangle$. Then $A\sqcup B$ is $T(V\oplus W)/\langle R_A\cup R_B\rangle$.

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So how does this give me a graded homomorphism $\Delta : A \to A \otimes A$? This nicely satisfies the categoric definition but I have no idea how this gives me a product structure on the dual. – Sven Apr 13 '11 at 12:47
Hmmm. What you want it to define a comultiplicationon algebras. I answered the question you asked in your title: how does the coproduct of algebras work. You should rewrite your question so that it asks clearly what you want... – Mariano Suárez-Alvarez Apr 13 '11 at 15:40
Thank you for the clarification. The two books I was referencing called the map a coproduct. – Sven Apr 13 '11 at 16:11
@Sven: your first comment to this answer confuses me: are you aware that the two uses of the work coproduct in "categorical coproduct" and "coproduct of a bialgebra/Hopf algebra" are rather unrelated, right? – Mariano Suárez-Alvarez Apr 13 '11 at 16:27

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