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Questions tagged [coalgebras]

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

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15 views

The quotient of a connected graded bialgebra

Let $k$ be a field. $H$ is called a connected graded bialgebra, if there are k-submodules $H^{n}$, $n \geq 0$, of $H$ such that: $H^0=k$; $H=\oplus _{n=0} ^{\infty} H^n$; $H^p H^q \subseteq H^{p+q}, ...
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0answers
24 views

The proof of $S(gh)=S(h)S(g)$ of an antipode in Hopf algebras

Let $H=(H, m, \Delta, \mu, \epsilon, S)$ be a Hopf algebra. Then there is a property of antipde $S$: $$S(gh)=S(h)S(g)$$ where $g,h \in H$. I have seen in some materials the proof of this property ...
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1answer
32 views

How is the differential induced by $d_C$ on $\Omega C$ defined for $(C,d_C)$ is a dga coalgebra?

Again I am confused about something regarding the cobarconstruction of a dga coalgebra $(C,d_C)$. The cobar construction of $C$ is the dga algebra $(T(s^{-1}\bar{C}),d_1+d_2)$ where $d_2$ is induced ...
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1answer
29 views

Does the differential of an augmented dga algebra fix the augmentation ideal?

I am reading about the bar/cobar construction in the book Algebraic Operads. The differential on the bar construction of a augmented dga algebra $A$ is a sum of two differentials $d_1+d_2$ where $d_1$ ...
2
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1answer
41 views

How does coassociativity of a coalgebra $C$ imply that the derivation on $\Omega C$ is a differential?

I am trying to show that $d²=0$ where $d$ is the derivation on $T(s^{-1}\bar{C})$ induced by the map $s^{-1}\bar{C}\to T(s^{-1}\bar{C})$ defined by $$s^{-1}x\mapsto -\sum (-1)^{|x_{(1)}|}s^{-1}x_{(1)}\...
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0answers
49 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
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1answer
66 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
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1answer
44 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 ...
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1answer
89 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 ...
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1answer
74 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
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1answer
49 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,...
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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 ...
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0answers
37 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
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1answer
69 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
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1answer
49 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
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1answer
63 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
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1answer
38 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\...
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1answer
41 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
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1answer
38 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
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1answer
82 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
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1answer
134 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
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1answer
51 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
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1answer
181 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
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1answer
57 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
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1answer
71 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 $$f\ast g:=\mu\circ(f\otimes g)\circ\Delta\in\text{End}(H).$$ ...
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0answers
13 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 ...
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1answer
34 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, ...
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1answer
72 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
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2answers
136 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 ...
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0answers
24 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}:= \...
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0answers
44 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
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1answer
79 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 ...
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0answers
55 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 ...
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0answers
42 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
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1answer
142 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
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1answer
137 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
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1answer
55 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
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1answer
55 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
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1answer
63 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
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1answer
557 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
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1answer
107 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
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2answers
151 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
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0answers
60 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
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2answers
317 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 ...
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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 ...
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votes
2answers
137 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)}...
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2answers
161 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 ...
0
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2answers
137 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
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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
127 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\...