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Questions tagged [coalgebras]

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

2
votes
0answers
33 views

Solve $f(x)' + s x f(x) - r f(x) = 0, \quad f(0) =w$ using stream calculus

The ODE $$f(x)' + s x f(x) - r f(x) = 0, \qquad f(0) =w$$ has the solution \begin{equation} f(x) = w\exp\left(r x - \frac{1}{2}s x^2\right). \end{equation} I'm trying to obtain this result using ...
2
votes
1answer
56 views

What is $\exp(rX^2)$ in stream calculus?

In the coinductive calculus of streams (sensu Rutten) $\exp(rX) = 1/(1-rX)$. Is there a similarly nice representation for $\exp(rX^2)$? Edit: I've just received a downvote on this. I'm making a ...
2
votes
1answer
35 views

Isomorphism between $R$-algebra $RG$ and $RG^{\ast}$

Let $R$ be a commutative ring, $G$ be a finite abelian group. Consider a group ring $RG$ as an $R$-coalgebra. Is it true that $RG\simeq RG^{\ast}$ as an $R$-algebra? If the answer is true, please tell ...
5
votes
1answer
75 views

When is a Hopf Algebra isomorphic to a group ring k[G]?

Let $H$ be a Hopf algebra over a field $k$. What are some nice conditions for when $H$ is isomorphic to $k[G]$ for a finite group $G$? The co-multiplication structure on the group algebra $k[G]$ is ...
0
votes
1answer
41 views

I need a hint: how to identify this type of algebra?

Let $C$ be a $k$-coalgebra with basis $\{x_m\}$ where $m \in \{0, 1, ..., n\}$ where $n \geq 0$, with comultiplication defined by $$\Delta(x_m) = \sum_{t=0}^m x_t \otimes x_{m-t}$$ and ...
3
votes
1answer
45 views

Existence of integrals in f.d Hopf algebras

In THE HAAR MEASURE ON FINITE QUANTUM GROUPS, van Daele gives an implausibly short proof of the existence of integrals in a finite-dimensional Hopf algebra. I'm probably overlooking something obvious,...
0
votes
0answers
13 views

Example of coalgebra over $\mathsf{KHaus}$

Let $\mathbb{T}$ be an endofunctor on a category $\mathsf{C}$. A $\mathbb{T}$-coalgebra is a pair $(X, \gamma)$, where $X$ is an object in $\mathsf{C}$ and $\gamma : X \to \mathbb{T}X$ is a morphism ...
2
votes
0answers
35 views

Are “grouplike elements” in quasi-Hopf algebras still invertible?

Suppose $H$ is a quasi-Hopf algebra with non-trivial evaluation $\alpha$. I cannot find any sources about grouplike elements in $H$. What I mean by "grouplike" is $$ \Delta(g) = g\otimes g \quad\text{...
4
votes
1answer
58 views

How to see that endotransformations of fiber functor have a coalgebra structure?

This question is based on section 5.2 in Tensor Categories, by Etingof et al. Note also that the question is pretty much in the title and what follows is just some background along with my fruitless ...
3
votes
1answer
46 views

Is the preimage of a subcoalgebra a subcoalgebra?

The question is in the title. I think yes, and my reasoning is as follows, but something feels fishy and I can't quite put my finger on it. Let $f:C\to D$ be a coalgebra morphism. Suppose that $Y\...
2
votes
1answer
58 views

Whether we can define the finitely generated coideal?

In the question How to understand the coidea of a colagebra?, we had posed the definition of coideals. Let $R$ be a unitary commutative ring and $X$ be a finite subset of $R$. Then the ideal ...
0
votes
1answer
30 views

Kernel of coalgebra homomorphism

If $R$ is a commutative ring, is the kernel of any coalgebra homomorphism $f:C\to D$ a (two sided) coideal of $C$? For $R$ a field this is the case, since we have $(f\otimes f)\circ\Delta_C=\Delta_D\...
2
votes
1answer
38 views

A question on counit of a coalgebra

I am reading an very interesting paper ''The infinitesimal Hopf algebra and the poset of planar forests'' https://arxiv.org/pdf/0802.0442.pdf written by Pro. Foissy. In his paper, on page 4, I don't ...
2
votes
1answer
36 views

Why is the coproduction in the tensor algebra a homomorphism?

Let $V$ be a vector space over a field $K$. And let $T^k V = V \otimes V \otimes\ldots \otimes V $ ($k$-times). Then I am interested in the space $$ T(V) = \bigoplus_{k=0}^\infty T^k V . $$ The ...
2
votes
1answer
80 views

The kernel of a morphism of co-rings is a co-ideal

I would like to show that The Kernel of a coring morphism $\phi:C\rightarrow C'$ between two $R$-corings $(C,\Delta,\epsilon)$ and $(C',\Delta',\epsilon')$ is a coideal. The only point that I can'...
3
votes
1answer
127 views

How to understand the coidea of a colagebra?

Let $C$ be a coalgebra with coproduct $\Delta$ and counit $\epsilon$. Then a subset $I\subseteq C$ is a coideal if $\Delta(I)\subseteq I\otimes C+C\otimes I$ and $\epsilon(I)=0$. My question is why ...
3
votes
1answer
45 views

How the coproduct defines an action on $X\otimes Y$

Given a bialgebra $A$ and two $A$-modules $X$ and $Y$. We can build the tensor product of the underlying vector spaces $X\otimes Y$. What does it mean if one says 'The $A$-module structure on $X\...
9
votes
1answer
168 views

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 ...
2
votes
1answer
51 views

Product in the category of cocommutative coalgebras

Let $(C,\Delta,\epsilon)$ and $(C',\Delta',\epsilon')$ be two coalgebras. Consider their tensor product $C\otimes C'$ and the two coalgebra homomorphisms \begin{align*} \pi:C\otimes C'\to C, \quad c\...
0
votes
1answer
70 views

$n$-fold product is a morphism of coalgebras

Let $(H,\mu,\eta,\Delta,\varepsilon)$ be a bialgebra with antipode $S$ which is cocommutative. On $\text{End}(H)$ we have the product $$f\ast g:=\mu\circ(f\otimes g)\circ\Delta\in\text{End}(H).$$ ...
1
vote
0answers
11 views

Intersection of coideals

"Let $C$ be a coalgebra. If $I$ and $J$ are two coideals of $C$, show that $I\cap J$ is a coideal of $C$." This is an exercise on page 45 of the book "Hopf Algebra" by Sweedler, Moss E(1969). But I ...
2
votes
1answer
33 views

coalgebras are right or left vector spaces

Following the definition of a coalgebra found here https://en.wikipedia.org/wiki/Coalgebra, I was wondering if it is a right or a left vector space or both? Indeed, when we use the Sweedler notation, ...
1
vote
1answer
68 views

How obviously injective is this “graded symmetrizer” map $\operatorname{S}(V) \to \operatorname{T}(V)$?

Starting with a graded vector space $V$, you can construct the tensor algebra $\operatorname{T}(V) := \bigoplus_{n>0} V^{\otimes n}$ and you can construct the symmetric algebra $\operatorname{S}(V) ...
4
votes
2answers
132 views

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 ...
0
votes
0answers
23 views

When is the Lie coalgebra of indecomposables cofree?

Suppose I have a cofree coalgebra $A$ with a grading $A=\bigoplus_{N\geq 0} A_N$. The Lie coalgebra of indecomposables is defined by $L := \frac{A_{>0}}{A_{>0}A_{>0}}$ where $A_{>0}:= \...
2
votes
0answers
42 views

Leibniz rule and Alexander-Whitney coproduct

Is there anything more than a superficial similarity between the following? The Alexander-Whitney coproduct $\Delta$ on the tensor algebra $\bigotimes^\bullet V$ of a vector space $V$ is defined by ...
2
votes
1answer
77 views

What are the primitive elements in a polynomial hopf algebra with primitive indeterminates?

Is there a result that says that in any polynomial Hopf algebra $K[X_1, X_2, ...]$ over a field $K$ with indeterminates primitive, the primitive elements are precisely the linear homogeneous ...
1
vote
0answers
51 views

Is the algebra dual to a graded coalgebra graded?

Given a graded coalgebra $C = \bigoplus_{n\geq 0} C_n $ with coproduct $$\Delta : C_n \to \bigoplus_{i=0}^n C_i\otimes C_{n-i} $$ must we have that the dual $C^* = \bigotimes_{n\geq 0}C_n^*$ is a ...
1
vote
0answers
41 views

The associated graded of a filtered coalgebra

Given a coalgebra $C$ with a filtration $F$ such that $\Delta(F^n C)\subset \sum_{i=0}^n F^i C\otimes F^{n-i} C$, how does the coproduct manifest in the associated graded? Do we get something to the ...
8
votes
1answer
141 views

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 ...
3
votes
1answer
126 views

algebra vs Dual of a coalgebra

Let $(A,m,u, \Delta, \varepsilon)$ be a bialgebra. Taking dual, $(A^\star, \Delta^\star,\varepsilon^\star)$ is a algebra. What is the relationship between the two algebras $(A, m, u)$ and $(A^\star, \...
4
votes
1answer
54 views

Are filtrations given by comodules structures?

Let $R$ be a commutative ring and let $M$ be some $R$-module. Is there a coalgebra $A$ such that $A$-comodule structures on $M$ (i.e. the $R$-linear maps $M \to M \otimes_R A$ satisfying the two usual ...
1
vote
1answer
54 views

Difference between (co)algebras and $F$-(co)algebras

I was reading the page on coalgebras and it made a mention to $F$-coalgebras in the first paragraph as though $F$-coalgebras are just a specific type of coalgebras. However, I am having a difficulty ...
1
vote
1answer
61 views

Question on Sweedler notation and a property of the transpose

So I was trying to prove that if $(H,m, \eta, \Delta, \epsilon)$ is a Hopf algebra with antipode $s$, then $s$ is an antimorphism of co-algebras, that is $\Delta \ s = (s \otimes s) \ \Delta^{op}$ ...
5
votes
1answer
495 views

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 ...
2
votes
1answer
104 views

Question about the Sweedler notation and the proof of a property of the antipode

Let $(H,m, \eta, \Delta, \epsilon)$ be a Hopf algebra with antipode $s$. A basic property states that $s$ is an antimorphism of algebras. That is, $s(xy)= s(y)s(x)$. The usual proof involves showing ...
3
votes
2answers
144 views

If $H$ is a Hopf algebra, do we have $H^{cop}$ is a Hopf algebra?

Let $H=(H, m, u, \Delta, \epsilon, S)$ be a Hopf algebra, see for example the lecture notes, where $m$ is the multiplication, $u$ is the unit, $\Delta$ is the comultiplication, $\epsilon$ is the ...
0
votes
0answers
58 views

To what extent are final coalgebras simply a conceptual tool? (In light of a theory with initial algebras)

In the SEP article on non-well-founded set theories, they make the point that (up to isomorphism at least), working in a non-well-founded set theory such as $AZF$ yields no more expressive power over $...
2
votes
2answers
303 views

Is the dual of a module naturally a comodule?

This question is basically an extension of the following fact: given a finite dimensional, associative, unital $k$-algebra $A$, then the vector dual $A^*$ is a coassociative, counital coalgebra with ...
1
vote
0answers
33 views

Is the set finite words over an alphabet a final coalgebra*?

I am studying what coinduction is. In particular, I am reading that coinductive datatypes can be defined as elements of a final coalgebra for a given polynomial endofunctor on $\tt Set$. I've seen ...
-1
votes
2answers
129 views

notation of counitality using Sweedler notation

($(H, \Delta, \epsilon)$ a $K$-coalgebra) I was wondering how one could justify the following notation of counitality using Sweedler notation $$ \epsilon(a_{(1)})a_{(2)} = a = a_{(1)}\epsilon(a_{(2)}...
1
vote
2answers
149 views

Every coalgebra is the sum of its finite-dimensional subcoalgebras

In the article about coalgebra in Wikipedia, it says that Every coalgebra is the sum of its finite-dimensional subcoalgebras I want to know how to prove this but I have no idea where to start. I ...
1
vote
2answers
118 views

Relation between left coideal and coideal.

Let $H$ be a Hopf algebra. A coideal $C$ is a subset of $H$ such that $\Delta(C) \subset H \otimes C + C \otimes H$. A left coideal $C'$ is a subset of $H$ such that $\Delta(C') \subset H \otimes C'$. ...
1
vote
0answers
31 views

Question on a paper by Shnider and Sternberg

I'm reading the paper 'The cobar resolution and a restricted deformation theory' by S. Shnider and S. Sternberg. I do not understand the definition of the map $\delta_p$ defined in section $3$. The ...
2
votes
1answer
116 views

Why is the cofree coalgebra defined in this way?

According to wikipedia, the cofree coalgebra (not requiring it to be a bialgebra) is defined with the coproduct (over v) defined as $$\Delta: V\to V\boxtimes V$$ $$\Delta: v \mapsto v\boxtimes 1 + 1\...
2
votes
1answer
96 views

Is every algebra a coalgebra?

I was looking to study the topic of coalgebras and few examples were given. In particular, I didn't see anywhere this example: let $A$ be any $\mathbb{K}$-algebra, in particular it is a $\mathbb{K}$ ...
4
votes
2answers
555 views

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 '...
1
vote
0answers
38 views

How can one define a set of formulas inductively, without some base propositions being defined first?

In some contexts (i.e. coalgebraic logic/coalgebraic modal logic) I see people commonly define (coalgebraic) languages (indeed sets of formulas) without considering any atomic propositions (or base ...
3
votes
0answers
70 views

Where can I find coderivations?

I have recently stumbled upon the notion of a coderivation. I had been working with twisted coderivations for a while, without realizing they there was a name for them (and probably some nice ...
0
votes
0answers
60 views

Cosemisimple coalgebras and uniqueness of their cosimple decompositions [duplicate]

A coalgebra $C$ is called cosimple if it has no subcoalgebras. It is called cosemisimple if it is a direct sum of simple coalgebras. Is this direct sum decomposition unique? Explicitly, can there ...