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

Modules over the dual of an infinite dimensional coalgebra

Let $k$ be a field and let $A$ be a finite dimensional (unital, associative, not necessarily commutative) $k$-algebra. The $k$-linear dual of $A$ is a coalgebra, and viceversa, the $k$-linear dual of ...
0
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1answer
41 views

Proving uniqueness of antipodes in Hopf algebras

Let $(H,\mu,\nu,\Delta,\epsilon)$ be a Bialgebra where H is the vector space, $\mu, \nu$ are the product and unit whilst $\Delta, \epsilon$ are the coproduct and counit. Now, for $f,g \in end(H)$ ...
3
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1answer
67 views

Sweedler Notation: $\eta\epsilon$ is the identity element of the convolution product

Proof attempt Let $(A, \mu, \eta) $ be an algebra and $(C, \Delta, \epsilon) $ be a coalgebra, both over the same field $k$. Define the convolution product $$*: \mathrm{Hom}(C,A)\otimes \mathrm{Hom}(C,...
3
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1answer
37 views

Showing the triple $(\hom(C,A),@,\mu \epsilon)$ defines an algebra

Showing the triple $(\hom(C,A),@,\mu \epsilon)$ defines an algebra Let $(C,\Delta,\epsilon)$ be a colalgebra and $(A, \mu, \nu)$ be an algebra where $\Delta, \mu$ are the coproduct and product whilst $...
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1answer
45 views

Showing $\lambda: A \otimes C^* \rightarrow \text{Hom}(C,A)$ is a morphism of algebras

Show that $\lambda: A \otimes C^* \rightarrow \text{Hom}(C,A)$ is a morphism of algebras. Let either $C^*$ or $A$ be finite dimensional, and let $\lambda$ be the isomorphism $\lambda: A \otimes C^* \...
0
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1answer
23 views

Prove that $\gamma$ is a map of $C^*$-modules

Let $C$ be a coalgebra and $M$ a $C^*$-module, where $C^*$ is C's dual. Prove that $$\gamma: M \otimes C \rightarrow \text{Hom}_\mathbb{k}(C^*,M) \\ m \otimes c \mapsto [f \mapsto mf(c)]$$ is a $C^*$-...
2
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1answer
45 views

Showing tensor product of coalgebras is a coalgebra.

Let $(C, \Delta, \epsilon)$ and $(C',\Delta', \epsilon')$ be two coalgebras over the field $k$. I'm trying to show that $C \otimes C'$ is a coalgebra for the comultiplication $$\overline{\Delta}:=(id_{...
7
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2answers
118 views

A monic coalgebra morphism whose underlying $\text{Set}$ morphism is not injective

Let $F:\mathscr A\to \mathscr A$ be a functor. Consider the following category $\mathscr C$. The objects are arrows $A\to F(A)$. If $\alpha:A\to F(A)$ and $\beta:B\to F(B)$ are two objects, then a ...
0
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1answer
33 views

Confusing with definition of (co)algebra

I began to read the book « Hopf algebras » by Sweedler. He gave the definitions of algebra and coalgebra using commutative diagrams. These diagrams just show (co)associativity and (co)unitary ...
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1answer
33 views

A question about infinitesimal bialgebra or Newtonian coalgebra

Recently, I am studing an interesting coalgebra which was called the infinitesimal bialgebra by Joni and Rota. It can be regarded as an algebraic framework for the calculus of divided differences. ...
4
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0answers
74 views

Sweedler notation and String diagrams: coaction on quotient

Let $(C, \Delta, \epsilon)$ be a coalgebra (in the category $Vect$), and $I \subset C$ a right coideal, i.e. $\Delta (I) \subset I \otimes C$. Define a counital coaction $\overline \Delta := (x_{(1)} ...
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0answers
54 views

Coalgebras & Coideals: Why does $ker(\pi \otimes id_C ) = I\otimes C$ hold?

In a proof on comodules and coideals I found the following passage: “Let $C$ be a coalgebra, and $I \subset C$ a vector subspace. Let $\pi$: $C \rightarrow C/I $ be the canonical projection. Consider ...
5
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0answers
94 views

Coideals in the grouplike colagebra are spanned by differences

Let $k$ be a field, and let $S$ be a nonempty set. Let $k[S]$ be the grouplike coalgebra of $S$ over $k$, i.e. the free vector space with basis $S$ equipped with the coproduct $\Delta(s)=s\otimes s$ ...
0
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0answers
27 views

Exterior coalgebra

Let $V$ be a vector space over field. On the exterior algebra $\Lambda(V)$ there is deconcatenation coproduct $\Delta: \Lambda(V) \to \Lambda(V) \otimes \Lambda(V)$ defined on the elements of $V$ as $$...
1
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0answers
21 views

Matrix Coefficients Map a Coalgebra Morphism

Simple question: let $C$ be a coalgebra over a field $k$. Given a finite-dimensional right $C$-comodule $V$ determined by the structure morphism $$a:V\to V\otimes C,$$ we have a natural 'matrix ...
1
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1answer
66 views

Right adjoint of a monad is a comonad

I'm having trouble proving the following statement: If a monad $T$ has a right adjoint $K$, then $K$ is a comonad and the categories of $T$-algebras and $K$-coalgebras are isomorphic. So far I've ...
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0answers
11 views

How do coalgebra eliminators work?

I've been reading about coalgebraic data types, and I've got the impression that they are dual to algebraic data types in that while ADTs are defined in terms of constructors (and terms are built ...
0
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0answers
30 views

The take lemma needs a coinductive proof

In Are coinductive proofs necessary?, the answerer claimed that we cannot prove inductively the take lemma: Two streams that agree on all initial subsequences of given length are the same. I was ...
0
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0answers
11 views

Cocomplete coalgebra and comodule

Let $C$ be a coaugmented coalgebra, i.e., $C= k\oplus \bar{C}$ as coalgebras. We say that $C$ is cocomplete if $$ C\,=\,\bigcup_{n} \mathrm{ker} \big(\bar{\Delta}^{(n)}: C \xrightarrow {\Delta^{(n)}} ...
2
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0answers
50 views

Products In the Categary of Skew-Cocommutative Coalgebras Are Skew Tensor Products

Let $A = \bigoplus^n_{i=0} A_i,B = \bigoplus^b_{i=0} B_i $ be a graded modules over the same commutative ring $R$ . The twisting isomotphism $\tau_{A,B} : A \otimes B \to B \otimes A$ is defined on ...
2
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0answers
27 views

How to get a Hopf algebra from a monoid?

Let $k$ be a field. Let $S$ be a monoid with netural element $e$. Suppose that all $s \in S$ have only finitely many factoriztions $s=ab$, where $a,b \in S$. Then the free k-module $k[S]$ has a ...
2
votes
1answer
66 views

Confusion about dual of comodule

I am confused about something. Please help! :) All objects are vector spaces over a fixed field $k$. Let $C$ be a coalgebra with comultiplication $m:C\to C\otimes C$. Let $M$ be a left comodule ...
2
votes
1answer
50 views

Counterexample to the fundamental theorem of comodules

The Fundamental Theorem of Comodules (aka the Finiteness Theorem for Comodules) states that if $\Bbbk$ is a field, then any element of a comodule over a $\Bbbk$-coalgebra lies in a finite-dimensional ...
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0answers
39 views

Sweedler Notation: limitations and left comodule version.

I have two questions: In the nlab entry about the sweedler notation it reads "One can formalize in fact which manipulations are allowed with such a reduced notation." Has anyone done that/an idea ...
2
votes
1answer
41 views

$\times$ versus $\otimes$ in Definition of $K$-Algebra

While reading about algebras and coalgebras, I keep coming across two definitions of an algebra $A$. One definition uses the Cartesian product $\times$, while another uses the tensor product $\otimes$....
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1answer
41 views

Primitive elements in $k[x]$

I need to find all primitives element of $k[x]$ with the coalgebra structure: $\Delta(x)=1\otimes x+x\otimes 1$ and $\epsilon(x)=0$. If char($k$) is zero, for me it's clear that, within the basis $\{x^...
1
vote
1answer
32 views

Subcoalgebra generated by an element

Let $(A, \Delta, \epsilon)$ be a coalgebra and $f\in A$. What is the subcoalgebra generated by $f$ like? For example, if $A$ is the dual of the quaternions $\mathbb{H}$ (which is $\mathbb{R}$-algebra ...
2
votes
1answer
49 views

Wedge product of coalgebras

I need to prove the following: "Let $U$ and $V$ be subspaces of a coalgebra $(C,\Delta, \epsilon)$. Suppose that $U\subseteq\ker(\epsilon)$. Show that $U\wedge V\subseteq V$ and $V\wedge U\subseteq V$...
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0answers
31 views

Every coalgebra is the quotient of a cosemisimple.

I am trying to show that every coalgebra is the quotient of a cosemisimple coalgebra (I'm actually not sure if it's true or not). Here is my attempted solution: Let $ C $ be a coalgebra. We know ...
1
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1answer
45 views

Is a coalgebra comodule cosemisimple if and only if every subcomodule is a direct summand?

It is well known that if $ R $ is a ring, then every $ R $-module $ M $ is semisimple (that is, $ M $ is the direct sum of simple $ R $-modules) if and only if every submodule of $ M $ is a direct ...
3
votes
0answers
62 views

Explicit formula for the equalizer of coalgebras

The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is ...
2
votes
1answer
105 views

Why is the Chevalley-Eilenberg differential a coderivation?

For every Lie algebra $\mathfrak{g}$ we can consider the Chevalley-Eilenberg complex given by the exterior powers $\bigwedge^n \mathfrak{g}$ together with the differential $d_{\mathrm{CE}} \colon \...
0
votes
1answer
60 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
51 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 ...
0
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0answers
32 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 ...
0
votes
1answer
31 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 ...
5
votes
3answers
189 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 ...
1
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1answer
49 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}, ...
3
votes
1answer
39 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
votes
1answer
157 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 ...
1
vote
1answer
69 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
votes
1answer
56 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
votes
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
votes
1answer
75 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
52 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 ...
6
votes
1answer
284 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
83 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 ...
7
votes
3answers
87 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,...
3
votes
1answer
55 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
105 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 ...