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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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1answer
25 views

if $C$ is a filtered coalgebra, does Gr($B\Omega C)\backsimeq B\Omega ($Gr $C)$ hold?

I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects $X\rightarrow$ Gr$(X)$ commutes with direct sums (this seems to be ...
4
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0answers
45 views

final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $F_A$(X) = ${\mathscr{P}_{<κ}}$(A×X) to define games coalgebraically. This is a functor from ...
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0answers
28 views

Reference about the proof of this proposition?

``Let $M$ be a finite dimensional Hopf algebra with antipode $S$ and let $M^*$ be the dual bialgebra. Recalls that $x^* \in M^*$ is a primitive in $M^*$, then $x^*$ is a derivation of $M$. '' I am ...
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0answers
11 views

Convolution in Bialgebras

On the wikipedia page on convolution, there is a section on convolution in bialgebras. It's completely mysterious to me. If it has something to do with the regular concept of convolution, can some one ...
4
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2answers
98 views

Is the category of finite-dimensional $k[x]$-modules a comodule category?

Fix a field $k$, denote by $k[x]$ the polynomial algebra. The category of finite-dimensional modules over $k[x]$ is precisely the category $\mathcal{C}$ consisting of pairs $(V, T_V: V \to V)$ of ...
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0answers
20 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
27 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 ...
3
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1answer
41 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
41 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
46 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)}\...
3
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0answers
51 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
70 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
49 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
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1answer
135 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
76 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 ...
5
votes
3answers
70 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
17 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
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0answers
41 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
81 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
57 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
79 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 ...
1
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1answer
59 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
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1answer
54 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
43 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
90 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'...
4
votes
1answer
155 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
54 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
206 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
72 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
77 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
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0answers
27 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
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1answer
36 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
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1answer
78 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
171 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 ...
2
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0answers
58 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
93 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
88 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
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0answers
50 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
149 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 ...
2
votes
1answer
183 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
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
vote
1answer
60 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
71 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 $\Delta \ s = (s \otimes s) \ \Delta^{op}$ ...
6
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1answer
774 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
114 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
180 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
62 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
376 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
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0answers
35 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
146 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)}...