Questions tagged [coalgebras]

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

Filter by
Sorted by
Tagged with
1
vote
0answers
31 views

Sweedler Notation: limtations 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 ...
1
vote
1answer
34 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 $\otimes$....
1
vote
1answer
36 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
25 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
41 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 $V\wedge U\subseteq V$...
1
vote
0answers
21 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
vote
1answer
22 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
51 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
56 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
35 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
votes
0answers
47 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
votes
0answers
31 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
21 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
129 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
vote
0answers
26 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}, ...
2
votes
0answers
28 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
48 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
46 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
51 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
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
50 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
168 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
79 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 ...
6
votes
3answers
77 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,...
2
votes
0answers
45 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
88 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
62 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
86 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
vote
1answer
74 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
63 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
44 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
95 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
167 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
61 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\...
10
votes
1answer
223 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 ...
3
votes
1answer
84 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
80 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
29 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
39 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
86 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
201 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
votes
0answers
68 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
101 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
106 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
59 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
150 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
196 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
56 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
62 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 ...