# Questions tagged [coalgebras]

For questions about coalgebras, comultiplication, cocommutativity, counity, comodules, bicomodules, coactions, corepresentations, cotensor product, subcoalgebras, coideals, coradical, cosemisimplicity, ...

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### Do chains send homotopy inverse limits of spaces to homotopy inverse limits of $E_\infty$-coalgebras?

Let $X_\bullet := ... X_2 \to X_1$ be a tower of connected and simple spaces with the following properties: The induced tower $H_\ast(X_\bullet; \mathbb{F}_p)$ of graded $\mathbb{F}_p$-vector spaces ...
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### Show that a certain element is a linear combination of tensors

I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here. Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that ...
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### What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?

Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a ...
245 views

### Meaning of coinvariants of a comodule

Let $M$ be a comodule over the bialgebra $B$, with structure map $\rho:M \to M \otimes B$. The space of coinvariants is defined as $M^{coB}:=\{m \in M~|~\rho(m)=m\otimes 1_B\}$. A book I'm reading ...
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I'm learning "An Introduction to Exterior Differential Systems" by Gregor Weingart. I have some confusions regarding comodules: In section 3, the author defines a comodule: Later, the "space of ...
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### What are regular epimorphisms of coalgebras?

It is known that epimorphisms between coalgebras are surjective maps. My question is What are regular epimorphisms between coalgebras? I am in particular ...
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### Category of bicomodules of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra $H$ is one which is equal to the direct sum of its subcoalgebras. As is well-known, this is equivalent to its category of $H$-comodules being semisimple. Is this also true ...
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### On a dual of Kaplansky's $2^{nd}$ conjecture: admissible algebras?

Kaplansky's second conjecture (on Hopf algebras) deals with "admissible" coalgebras: He calls a coalgebra admissible, if there is an algebra structure making it a Hopf algebra. The conjecture states ...
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### Cocommutativity, comultiplication and coalgebra maps

Given a coalgebra $(C,\Delta,\varepsilon)$, over a field, the following is a well-known property: the comultiplication $\Delta:C\to C\otimes C$ is a coalgebra map if and only if $C$ is ...
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### What are types of coalgebras that are more naturally described by cooperads?

Some background. Let $\mathsf{C}$ be a symmetric monoidal category. An object $X \in \mathsf{C}$ has two operads "naturally" (the two constructions aren't functorial) associated to it: the operad of ...
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### Definition of a cosemisimple Hopf algebra

A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the ...
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### Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...
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### bialgebras on quotient polynomials

Is there a general procedure for constructing a bi-algebra out of a quotient polynomial ring? In particular, how do I construct a bi-algebra corresponding to quotient polynomial ring $<x^2-x-1>$?...
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### An identity satisfied by “Differentiation”

I asked this question in MSE but I did not received any answer. So I repeat it here: Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order ...
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### Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
I have problems finding anything about the cofree Lie coalgebra functor $\mathcal{L}ie^c$ out there. Basically all I found was that it appears in Harrison cohomology and that, given a $\mathbb{Z}$-...