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I have added to primitive element the definition of primitive elements in comodules, and their equivalent characterization in terms of cotensor products. Added also a corresponding remark to cotensor product.
Who is calling it primitive element in the generality of comodules (I guess some topologists) ? For comodules the extensive Hopf algebra literature is equivocal in calling such elements (left) coinvariants! I got a PhD in the subject (introducing related “localized coinvariants” as one of the main objects in my thesis) and spent a decade in the subject and this is the first time I heard that somebody would call the element primitive in such a generality where the good old term coinvariant is generally accepted.
Of course, instead of $1$ in the definition one can have any group like (only if the ground coalgebra is a bialgebra, then of course $1$ is distinguished!). In particular, coalgebras do not know of the difference between $1$, $2$ and $654378$ in a 1-dimensional coalgebra, even if you identify it with the ground ring because of being 1-dimensional or trivial. The trivial $R$-coalgebra is determined by its ground ring only up to an isomorphism of coalgebras, what boils down to an arbitrary scale relative to a bare $R$-module. Otherwise the choice is evil, in $n$Cafe language.
Sure, I have added pointer to “coinvariants”.
For the terminology “primitive” for these elements, see for instance def. 7.19 in this pdf
Maybe you could help me with something:
what’s the precise statement (and what’s a citable source for it) that Hopf algebras with all elements primitive plus maybe some other condition are necessarily tensor/symmetric/exterior algebras?
In symmetric algebra etc. not all elements are primitive, of course, just a specific set of associative algebra generators. These are variants of Milnor-Moore theorem (the original paper of Milnor and Moore, 1956 or so, or in positive characteristic Cartier and then again Quillen) that if the Hopf algebra over a field is generated by the set of ALL its primitive elements then it is the universal enveloping algebra of the Lie algebra consisting of all primitive elements (in positive characteristic I guess the restricted Lie algebra). It has many generalizations like super case and for many other cases of nonassociative algebras (these generalizations were very much studied by Loday and his school). See wikipedia for the (connected) graded case, which is classical to algebraic topologists.
Probably one should look into Cartier’s Primer on Hopf algebras.
There are some more general theorems like when a Hopf algebra is a tensor product of an enveloping algebra and of a group algebra.
P.S. 7.7 of the Rognes notes you cite above.
Thanks. It seems that Cartier and Wikipedia just state the case for characteristic 0. But the original is avaible (here) and May 69 is useful.
I have started a stub for Milnor-Moore theorem.
Correction: for a unital comonoid in a $CMon$-enriched monoidal category, the counit annihilates the primitive elements if the homsets $\mathcal{C}[A,B]$ are cancellative modules over the commutative rig $\mathcal{C}[I,I]$ and if $\mathcal{C}[I,I]$ verifies that $1 \neq 2$ (it’s true “at least” in this case).
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