Questions tagged [coalgebras]

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

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Hopf Algebra Structure of fixed ring of group algebra

In Lindsay Childs' "Taming Wild extensions, Hopf algebras and local Galois module theory" , theorem 6.8 [part of Greither and Pareigis' Hopf Galois correspondence theorem] we have a ...
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58 views

For a terminal coalgebra, map is isomorphism

I'm trying to solve Emily Riehl's "category theory in context", question 1.6.vi (1.6.6). The exercise says: A coalgebra for an endofunctor $T: C \rightarrow C$ is an object $c \in C$ ...
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“Basic” coalgebra structures on $R$-modules

Lately I have been reading about coalgebras and wondering about natural ways one can make any $R$-module into an $R$-coalgebra. Two examples of such constructions are given by tensor coalgebras and ...
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1answer
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How to compute the terminal coalgebra (or initial algebra) of a given functor

This is sort of a soft question. I'll start with an example. Fix a set $A$, and consider the functor $F\colon \mathbf{Set} \to \mathbf{Set}$, $F\colon X \mapsto A \times X$. Now, we know that the ...
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1answer
53 views

Koszul dual cooperad of the associative operad

I am trying to compute the $k$-modules of $\mathcal{As}^¡$, the Koszul dual cooperad of the associative operad $\mathcal{As}$. I am using sections 7.1.4 and 7.2.1 of Algebraic Operads to try to do ...
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1answer
34 views

Axioms of a coalgebra restated using Sweedler's notation

I'm struggling with understanding manipulation using Sweedler's notation at a very fundamental level. I don't understand the equivalence of the axioms of coalgebras in the standard notation [Coproduct ...
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65 views

Hopf “algebroid” structure of a groupoid convolution algebra?

To male thinks simple as possible, lets say we have a discrete group $G.$ Then the then the group algebra $\mathbb{C}[G]$ (of finitely supported complex valued functions on $G$) has a convolution and ...
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How to prove the reduced comultiplication of a coaugmented coalgebra is coassociative?

a coalgebra over a field $k$ is a vector space $C$ over $k$ together with $k$-linear maps $$\text{comultiplication } \Delta: C \to C\otimes_k C \text{ and} $$ $$\text{counit } \epsilon: C \to k$$ ...
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Show that a certain element is a linear combination of tensors

Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that the map $$T: A \otimes A \to A \otimes A: a \otimes b \mapsto \Delta(a)(1 \otimes b)$$ is surjective. We write $\Delta(...
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22 views

Isomorphism between two Hopf algebras

Let $k$ be a field, over which we consider algebras and coalgebras. A $k$-coalgebra is a comonoid object in $k$-modules, and a $k$-algebra is a monoid object in $k$-modules. A $k$-bialgebra is ...
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29 views

Is there a first order theory of coalgebras?

Is there a first order theory of coalgebras? The motivation for this question is trying to understand what a coalgebra is without appealing to category theory, or using categorical concepts as little ...
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Showing that $U(\mathfrak{sl}_2)$ is a coalgebra

We know that there is a coalgebra structure on $U(\mathfrak{sl}_2)$ as follows for any $z\in \mathfrak{sl}_2$: $$\Delta(z)=1\otimes z+z\otimes 1, \qquad \epsilon(z)=0.$$ Can someone be so kind to ...
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Finding all subcoalgebras of a coalgebra $C$

Let $(S, \le)$ be a partially ordered finite set. Let $C$ the vector space with basis $\{e_{i,j} | i,j \in S, i \le j\}$, which turns out to be a coalgebra with comultiplication and counit given by: $$...
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1answer
70 views

Dual of an algebra is a coalgebra

Let $A$ be an algebra over the commutative unital ring $k$ that is finitely generated and projective as a $k$-module. Let $A^*= \operatorname{Hom}_k(A,k). $ Then the natural map $$i: A^* \otimes A^* \...
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When is there an “intuitive” functor from F-coalgebras to T-coalgebras?

Suppose $F, T : Set \rightarrow Set$ are two functors on the category of sets. Let $F^{coalg}$, $T^{coalg}$ denote the categories of $F$, respectively $T$ coalgebras. Vaguely, I'm interested in when ...
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Do the regular monomorphisms in the category of coalgebras coincide with the injective ones?

Given a commutative ring $k$ (not necessarily a field), do the regular monomorphisms in the category of (coassociative and counital as usual) $k$-coalgebras (henceforth denoted $k-\mathrm{Coalg}$) ...
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Can we intersect coalgebras?

Let $k$ be a commutative ring, which I am unwilling to assume is a field, and suppose $(C_i)$ is a collection of coassociative $k$-subcoalgebras1 of a coassociative $k$-coalgebra $C$. Is there always ...
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76 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 ...
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67 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 $f,g \in end(H)$ ...
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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,...
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48 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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52 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^* \...
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1answer
25 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 $C^*$-...
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134 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_{...
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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 ...
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41 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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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. ...
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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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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 ...
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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 $\Delta(s)=s\otimes s$ ...
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23 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 ...
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1answer
78 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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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 ...
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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 ...
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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 ...
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78 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 ...
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1answer
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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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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 ...
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1answer
52 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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42 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^...
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49 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 ...
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1answer
50 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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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 ...
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1answer
54 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 ...
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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 ...
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152 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 \...
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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 ...
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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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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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1answer
62 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 ...