Questions tagged [coalgebras]
For questions about coalgebras, comultiplication, cocommutativity, counity, comodules, bicomodules, coactions, corepresentations, cotensor product, subcoalgebras, coideals, coradical, cosemisimplicity, ...
124 questions
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Coradical filtration for comodules is exhaustive
It is a standard fact in the theory of coalgebras and comodules that, given a coalgebra $C$ and a comodule $M$ over $C$, the coradical filtration
$$M_n := \Delta^{-1}(M\otimes C_0 + M_{n-1}\otimes C)$$...
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What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
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Hopf algebras actions
Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...
CommunityBot
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Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
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On reflexive bialgebras
Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...
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Infinite-dimensional, non-unital Frobenius algebras
A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital ...
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$C^\infty$-coring
We know that there the so called smooth algebras also known as $C^\infty$-rings. They can play an important role in modern treatment of differential geometry. Is there a coring analogue?
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Reference for cocommutative coalgebras
I'm looking for references on cocommutative coalgebras where I can see them as kind of infinitesimal spaces. I'm trying to understand this post Why do Lie algebras pop up, from a categorical point of ...
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Is there a way in which "space" of random variables on $\mathbb{R}$ is canonically a coaugmented coalgebra?
Consider the "space" of random variables with finite expectation on $\mathbb{R}$ in the following sense: we fix the Borel $\sigma$-algebra on $\mathbb{R}$, and put random variables in ...
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Can't parse a statement in an article on coalgebras and umbral calculus
This question is cross-posted from MSE.
I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", ...
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When are topoi of coalgebras atomic?
A geometric morphism is atomic if its inverse image is logical. Now consider a Grothendieck topos $\varepsilon$ and its terminal geometric morphism $\Gamma : \varepsilon \rightarrow Set$, topos is ...
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Has anyone studied factoring as a CO-product?
In factorization, like integer factorization, you start with an integer and end up with a kind-of list of pairs of other elements, namely the factors.
I want to explore the "Co-ness" of this....
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Subcoalgebras of symmetric algebra
Consider the symmetric algebra $S(V)$, with its coalgebra structure: $\Delta(x)=1\otimes x+x\otimes1$ on $V$, extended multiplicatively. What are its subcoalgebras?
In some vague sense, they seem to ...
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Hopf algebra and coideal question
Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put
$$B^+:= B\cap \ker(\epsilon).$$
It can be shown that $B^+$ is a two-...
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Is there a coalgebraic definition of filtered algebras?
If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define
$$...
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Equivalent definitions of pro-unipotent coalgebras
I'm trying to find a reference in the literature for equivalence of the following two definitions of pro-unipotent coalgebras.
Definition Let be $H$ a coagumented coalgebra and let $\Delta \colon H \...
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Limits in subcategories of Powerset-coalgebras
Let $F:Set\to Set$ be a functor. An $F$-coalgebra is a pair $\mathcal{A}=(A,\alpha)$ where $\alpha:A\to F(A)$ is arbitrary map.
Given $F$-coalgebras $\mathcal{A}=(A,\alpha)$ and $\mathcal{B}=(B,\beta)$...
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Lie coalgebra with no finite-dimensional subcoalgebras
In Walter Michaelis' paper Lie Coalgebras, he gives on page 9 an explicit example of a Lie coalgebra which is not the union of its finite-dimensional Lie subcoalgebras. In fact, Michaelis' example has ...
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Coproduct for a Frobenius algebra
The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
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A special class of regular languages: "circular" languages. Is it known?
We can define a subclass of the regular languages. Fix an alphabet $\Sigma$. Define the "circular" languages (actually, the name already exists to denote a different thing it seems, used in the field ...
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Inverse limit of chains of Eilenberg Mac Lane spaces
Let $... \to G_2 \to G_1$ an inverse system of abelian groups with inverse limit $G$, let $n \geq 2$ and $F$ a field. The induced inverse system $$... \to C_*(K(G_2,n);F) \to C_*(K(G_1,n);F) \ (*)$$
...
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Homotopy fixed points vs coalgebras
Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...
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Morphism of conilpotent coalgebras
I have a stupid question about morphisms of two conilpotent coalgebras $\phi:X\to Y$. Is it a morphism of coaugmented coalgebras such that $\phi(F_n(X))\subseteq F_n(Y)$? Here $F_n$ denotes the ...
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Conilpotent coalgebras as pushouts of trivial coalgebras
Let $K$ be a field and $C$ a non-counital conilpotent coassociative coalgebra over $K$
whose underlying $K$-vector space is finite dimensional.
Question: Can one obtain $C$ by iterately taking ...
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179
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Which categories of presheaves are algebraically cocomplete?
We say that a category is algebraically complete when every endofunctor has an initial algebra. Similarly, a category is algebraically cocomplete when every endofunctor has a final coalgebra.
Assuming ...
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How do you get the rational submodule of a $C^*$-module (equivalent to a $C$-comodule)?
Let $C$ be a coalgebra over a field $K$. Let $M$ be a $C^*$-module. I am trying to understand the rational submodule of $M$, which will carry the structure of a $C$-comodule.
It seems to me that the ...
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"Question-answer" bisimulation
I often come across relations that would be defined as a bisimulation, except that the label match can be "inexact", that is, in the bisimulation game, a move labelled with "a" can be replied to with "...
CommunityBot
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360
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Different Bialgebra/Hopf algebra structures on coalgebras
Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
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What is a coalgebra?
A coalgebra is a triple $(A,\Delta,\epsilon)$ consisting of a vector space, a coproduct, and a counit. Now as we all know, just like the unit in an algebra, the counit of a coalgebra is unique, i.e. ...
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Universal property of the cocomplete category of models of a limit sketch
Let $\mathscr{S}$ be a limit sketch in a small category $\mathcal{E}$, i.e. just a collection of cones in $\mathcal{E}$. Then its category $\mathbf{Mod}(\mathscr{S})$ of models (i.e. functors $\...
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Bialgebra maps and Hopf algebra maps
Let $H$ and $H'$ be two Hopf algebras, and let $\phi:H \to H'$ be an bialgebra map. Then is $\phi$ automatically a Hopf algebra map?
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Classifying Hopf algebras that admit a single irreducible comodule
Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k ...
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What is a homotopy between $L_\infty$-algebra morphisms
A $L_\infty$-algebra can be defined in many different ways. One common way, that
gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
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Frobenius algebras associated to posets and coalgebra structures
Let $P$ be a finite poset that we assume for simplicity to be bounded (that is it has a global maximum M and minimum m).
Let k be a field, then the classical incidence algebra $kP$ has $k$-vector ...
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Cotensor products (in monoidal categories) without regularity
In Internal Categories and Quantum Groups, Aguiar defines the cotensor product of two bicomodules as follows. Let
$(\mathcal{V},\otimes_{\mathcal{V}},\mathbf{1}_{\mathcal{V}})$ be a monoidal category;...
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Reconstruction of coalgebras
In the paper Reconstruction of hidden symmetries of Bodo Pareigis in the subsection "3.1 Reconstruction of coalgebras" there is the following proposition (3.3.).
Let $\mathcal{C}$ be a ...
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Do chains send homotopy inverse limits of spaces to homotopy inverse limits of $E_\infty$-coalgebras?
Let $X_\bullet := ... X_2 \to X_1$ be a tower of connected and simple spaces
with the following properties:
The induced tower $H_\ast(X_\bullet; \mathbb{F}_p)$ of graded $\mathbb{F}_p$-vector spaces
...
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Comparing Hochschild (co)homology for algebras and coalgebras
Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes
$$M\otimes A^{\otimes n}$$
and
$$\...
CommunityBot
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Show that a certain element is a linear combination of tensors
I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here.
Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that ...
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Monoidal structures on modules over derived coalgebras
Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can ...
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Contramodule as direct limit of its finitely generated subcontramodules
$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \...
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A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module
For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...
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$M$ comodule if and only if $N$ and $L$ comodules
Let $k$ be a field, $C$ a $k$-coalgebra, and $M$ a left $C$-comodule. Then, for a short exact sequence
$$
0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0
$$
of vector spaces, we have that $N$...
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Definition of subcoalgebra over a commutative ring
Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.
Notes I'm reading give the following definition:
$D$ is called subcoalgebra of $C$ if the ...
- 7,746
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Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
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How to work with co-multiplication?
Let $C$ be a coalgebra and $\Delta: C \to C\otimes C$ a co-multiplication map. Then, due the co-associative property we can consider $\Delta^m$. But how is defined $\Delta^{m}: C \to C^{\otimes m}$?
...
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Injectivity criterion for surjective coalgebra maps: does it hold in full generality?
Let $\mathbf{k}$ be a commutative ring.
Let $C$ be a filtered $\mathbf{k}$-coalgebra. This means a $\mathbf{k}$-coalgebra equipped with an increasing $\mathbf{k}$-module filtration $C^0 \subseteq C^1 \...
CommunityBot
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presentability rank of categories of coalgebras
The following theorem is relatively classical:
Theorem: Given an accessible endofunctor, (co)pointed endofunctor or (co)monad $T$ on a locally presentable category $C$, then the category of $T$-(co)...
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How is a descent datum the same as a comodule structure?
For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong S\...
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Reference for Hopf algebra applications to Feynman diagrams
I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
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