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Questions tagged [coalgebras]

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

234 questions
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Good Introduction to Hopf Algebras with Examples

I want to learn more about hopf algebras but I am having trouble finding a down to earth introduction to the subject with lots of motivation and examples. My algebra knowledge ranges from Dummit and Foote, Atiyah-Macdonald commutative algebra, and…
54321user
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2 answers

What is a most elementary Coalgebra?

I just read the definition of a coalgebra, defined categorically ( reversing the arrows of the Algebra category ), the given example in the text I am referring to is the Homology on a topological space. I am not familiar yet with algebraic topology,…
NotaChoice
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13
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Meaning of the antipode in Hopf algebras?

What I understand so far is that Hopf algebra is a vector space which is both algebra and coalgebra. In addition to this, there is a linear operation $S$, which for each element gives a so-called 'anitpode'. Can anyone give an intuitive explanation…
mavzolej
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13
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1 answer

cofree coalgebra: explicit description from its universal property

Let $k$ be a commutative ring. There is a forgetful functor $$U : \mathsf{Coalg}_k \to \mathsf{Mod}_k$$ from $k$-coalgebras to $k$-modules. This has a right adjoint $R : \mathsf{Mod}_k \to \mathsf{Coalg}_k$, called the cofree coalgebra on a…
Martin Brandenburg
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1 answer

Is there a "Coalgebra - Cogeometry" duality? Good opposite of a category of coalgebras?

So the category of affine schemes is dual to the category of commutative rings, Stone spaces are dual to Boolean algebras, localizable measurable spaces are dual to commutative Von Neumann algebras, and I'm sure there are many more examples. In…
David Myers
  • 1,620
12
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1 answer

Example of $V^* \otimes V^*$ not isomorphic to $(V \otimes V)^*$

There is always an injection between $V^* \otimes V^*$ and $(V \otimes V)^*$ given by $$ f(v^* \otimes w^*)(x \otimes y)=v^*(x)w^*(y), $$ where $x,y \in V$. I've been given to understand that in infinite dimension it is not surjective. Does anybody…
Dac0
  • 9,504
12
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2 answers

Exercises to help a student become accustomed to Sweedler notation

For a coassociative coalgebra $A$, we have a comultiplication map $\Delta \colon A \to A \otimes A$. An element $c \in A$ is sent to a sum of simple tensors, which can be a mess of indices, so we can use Sweedler's notation to try to write down the…
Mike Pierce
  • 19,406
11
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3 answers

Geometric interpretation of the fundamental theorem for coalgebras?

Given an element $m$ in a coalgebra $C$, there always exists a finite-dimensional subcoalgebra $D \subset C$ containing $m$; this is the fundamental theorem for coalgebras. This obviously isn't the case for algebras. However, to make a proper…
Michael Parsons
  • 333
9
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Alexander-Whitney map gives a coalgebra?

Let $R$ be a unital ring with multiplication $\mu\colon R\otimes R \rightarrow R$. Consider the category $\mathcal{Ch}(R-\text{mod})$ of chain complexes of $R$-modules. This category becomes a monoidal category with the tensor product of chain…
Margaret
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Question about comultiplication

I have a question about comultiplication for coalgebras: Suppose $C$ is a coalgebra over the field $k$. How does one show that the comultiplication map $\Delta:C\to C\otimes C$ is a coalgebra map if and only if $C$ is cocommutative? The main problem…
yoshi
  • 1,061
9
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1 answer

Why doesn't the functor $\bar{\mathcal{P}}\bar{\mathcal{P}}$ preserve pullbacks?

I've tried finding examples on my own but the sizes of the sets is a bit hard to manage. In the litterature I've seen this fact referenced in a few places but they all point to Rutten: Universal coalgebra: a theory of systems which mentions, in the…
Strange Brew
  • 455
9
votes
1 answer

Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will this decomposition obey Krull-Schmidt, by which…
Dyke Acland
  • 343
9
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1 answer

Quotient coalgebra by coideal

I want to show, that if $I$ is a coideal in a coalgebra $A$, then $A/I$ is a coalgebra. To be a coideal means $\Delta(I)\subset A\otimes I+I\otimes A$ and $\varepsilon(I)=0$. We can use the induced maps from $A$, which…
Egbert53
  • 123
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Under what conditions is the homology of a dg coalgebra a graded coalgebra?

I'm trying to get a feel for some differential graded (dg) structures. Suppose $C$ is a differential graded coalgebra over a commutative ring $k$, i.e. a graded $k$-module equipped with a coproduct $\Delta : C \to C \otimes C$ and a counit…
user50948
  • 1,439
8
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1 answer

Representations of abelian groups

A classical result is Theorem: Let $G$ be a abelian group and $(V, \rho)$ be a irreducible representation of $G$ over a algebraically closed field $k$. If $V$ is finite dimensional (more generally, if $\mathrm{dim}_k V < |k|)$ then $\mathrm{dim}_k V…
espacodual
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