Skip to main content

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
1 answer
17 views

Bialgebra structure on self dual associative algebra

Given a finite-dimensional associative $\mathbb{k}$-algebra $A$, one can define its dual $A^*$ as the $\mathbb{k}$-vector-space $\operatorname{Hom}_{\mathbb{k}}(A, \mathbb{k})$ with $A$-multiplication ...
Jannik Pitt's user avatar
  • 2,085
0 votes
0 answers
17 views

Example of Coalgebras Without Any Nonzero Cocommutative Elements?

It is known that every coalgebra in characteristic $0$ has a nonzero cocommutative element: find a finite dimensional subcoalgebra, express it as a quotient of some comatrix coalgebra $C_n$, and take ...
SalutaFungo's user avatar
0 votes
0 answers
44 views

Composing stream homomorphisms "apply f" based on dynamical systems (A, f)

I'm trying to do Exercise 99 of Rutten's The method of coalgebra: exercises in coinduction. It says For all functions $f: A \to A$ and $g: A \to A$, prove that $$\text{apply}_g \circ \text{apply}_f = ...
msb15's user avatar
  • 138
1 vote
0 answers
29 views

$\mathbb{C}G$-modules and $\mathbb{C}^{G}$-comodules

I know that representations of a group $G$ are essentially $\mathbb{C}G$-modules. How is it that every $\mathbb{C}G-$module is also equivalent to a $\mathbb{C}^{G}-$comodule. I have not found this ...
NoetherNerd's user avatar
1 vote
0 answers
87 views

The notion of fundamental cogroup

I know there is a functor $\pi_1$ from the category of pointed topological spaces to the category of groups, sending each pointed topological space to its first fundamental group. I know that a group ...
toby flenderson's user avatar
1 vote
1 answer
33 views

Inverse and Composition of Bisimulations

Exercise 63 of Rutten's The Method of Coalgebra: exercises in coinduction asks us to prove that "the collection of all bisimulation relations between two given stream systems is closed under (i) ...
msb15's user avatar
  • 138
1 vote
0 answers
32 views

Determining initial algebras and final coalgebras for a given functor without using limits/colimits

I'm trying to find the final coalgebra for a certain functor but I have no idea how to do that in general, so I was hoping to go through the process with some simpler examples. In section 4.1, The ...
msb15's user avatar
  • 138
0 votes
1 answer
27 views

Characterizing congruences on the algebra of natural numbers

I'm trying to do Exercise 31 in Jan Rutten's book on coalgebras. The goal is to show that, given a characterization of congruences on the initial $N$-algebra $(\mathbb{N},[\text{zero},\text{succ}])$, ...
msb15's user avatar
  • 138
1 vote
2 answers
71 views

Confusion Over Distributive Property in Tensor and External Tensor Products

I've been delving into the properties of tensor ($\otimes$) and external tensor products ($\boxtimes$) within the context of coalgebra, particularly examining how the coproduct $\Delta$ applies to ...
Martin Geller's user avatar
1 vote
1 answer
55 views

Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
Brendan Murphy's user avatar
3 votes
0 answers
41 views

Can't parse a statement in an article on coalgebras and umbral calculus

I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", page 344). The article reads: We ...
Daigaku no Baku's user avatar
1 vote
1 answer
54 views

What does it mean for a map to factor through another map?

In Darij Grinberg's "Hopf algebras in combinatorics", there is a statement about existence of quotient coalgebras: "Indeed, $J ⊗ C + C ⊗ J$ is contained in the kernel of the canonical ...
Daigaku no Baku's user avatar
1 vote
1 answer
110 views

Antipode of a Hopf algebra being an antihomomorphism: unable to follow the proof

A PhD thesis contains the following proof that antipode of a Hopf algebra is algebra antihomomorphism (page 22): Here $\nu = \eta \circ \varepsilon$, where $\eta$ is the unit map and $\varepsilon$ is ...
Daigaku no Baku's user avatar
1 vote
1 answer
48 views

How do I define the coalgebra structure on a field?

A PhD thesis I'm reading contains the following statement (page 21): Consider an algebra $(A, \nabla, \eta)$, where $\nabla$ is multiplication, $\eta$ is unit, which is also a coalgebra $(A, \Delta, \...
Daigaku no Baku's user avatar
1 vote
2 answers
122 views

Why coproduct in the algebra of linear operators is actually a coproduct?

Coproduct in a coalgebra $V$, where $V$ is also a vector space over a field $\mathbb{K}$, is defined as a $\mathbb{K}-$linear map $\Delta : V \rightarrow V\otimes V$ satisfying two diagrams obtained ...
Daigaku no Baku's user avatar
1 vote
0 answers
53 views

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....
Ben Sprott's user avatar
  • 1,281
4 votes
0 answers
73 views

Is this hint for proving the fundamental theorem of coalgebras wrong?

In the book Tensor Categories, Exercise 1.9.4 is to prove the fundamental theorem of coalgebras. They give the following hint: This hint doesn't make sense to me. You can always add and subtract a ...
Chris's user avatar
  • 384
1 vote
0 answers
29 views

Non unital Hopf relation

The following problem is an exercise in Loday-Vallette's Algebraic Operads. I hope I am understanding this correctly. Any suggestions or hints would be appreciated. Show that the restriction of the ...
Chanel Rose's user avatar
2 votes
1 answer
68 views

The double dual of a coalgebra

Let $(C,\Delta)$ be a coalgebra over a field $K$. Denote by $()^{\ast}=Hom(-,K)$. Then we know that $(C^{\ast}, \Delta^{\ast} \circ \rho)$ is an algebra, where $\rho: C^\ast \otimes C^\ast \rightarrow ...
Xiaosong Peng's user avatar
1 vote
1 answer
70 views

Haar integral of a finite dimensional Hopf algebra: an explicit expression

Let $\mathcal{H}$ be a finite dimensional Hopf algebra. A nonzero element $\Omega\in \mathcal{H}$ is called an integral in $\mathcal{H}$ if $$x~\Omega=\epsilon(x)\Omega,~~\forall x\in \mathcal{H}.\tag{...
Lagrenge's user avatar
  • 836
0 votes
1 answer
90 views

Intuition for Coproduct of Grouplike vs Primitive elements in a Coalgebra?

I'm trying to understand Hopf Algebras as a physicist with a limited background in abstract algebra, and I might be in a little over my head. In particular I'm trying to wrap my head around the fact ...
Alex Kritchevsky's user avatar
3 votes
0 answers
122 views

Equivalent definitions of Hopf algebras

Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two ...
ferolimen's user avatar
  • 630
2 votes
0 answers
129 views

On the quadratic coalgebras

It is well known that a quadratic algebra $A(V,R)$ is the quotient of the free associative algebra $T(V)$ over a vector space $V$ by the two-sided ideal $(R)$ generated by $R\subseteq V^{\otimes 2}$, ...
Butters Stotch's user avatar
9 votes
0 answers
138 views

Alexander-Whitney map gives a coalgebra?

Let $R$ be a unital ring with multiplication $\mu\colon R\otimes R \rightarrow R$. Consider the category $\mathcal{Ch}(R-\text{mod})$ of chain complexes of $R$-modules. This category becomes a ...
Margaret's user avatar
  • 1,769
0 votes
0 answers
143 views

Coderivations of cofree algebra

I have troubles with understanding some points in Loday-Vallette book (Algebraic operads). (1) Coderivations Let $T(V)$ be the tensor algebra over a vector space $V$. It is well known that it is a ...
Butters Stotch's user avatar
1 vote
1 answer
147 views

Coideal Definition

In a coalgebra $A$ with comultiplication $\Delta$ and counit $\epsilon$, a two-sided coideal is defined to be a linear subspace satisfying $\Delta (I) \subseteq A\otimes I + I \otimes A$ and $I \...
Aaron's user avatar
  • 195
1 vote
0 answers
81 views

Coalgebra structure on $T(V)=k\oplus V \oplus V\otimes V \oplus V^{\otimes 3}\oplus \cdots $

In the wiki page:https://en.wikipedia.org/wiki/Cofree_coalgebra They discuss two coalgebra structures on $T(V)$ I dropped the tensor between $v_1\otimes \cdots\otimes v_n$, the two coproducts: $$\...
IrbidMath's user avatar
  • 3,185
0 votes
0 answers
46 views

Why are the maps $\eta, \mu, \Delta, \varepsilon$ linear?

Here is the question I am trying to solve: (Divided powers) Consider the vector space $C = k[t]$ of polynomials in one variable. Prove that there exists a unique coalgebra structure $(C, \Delta, \...
user avatar
0 votes
1 answer
16 views

Show that $HK^+$ is a coideal where $K$ is a subHopfalgebra of $H$, $K^+=\mathrm{ker}(\epsilon)\cap K$.

Show that $HK^+$ is a coideal where $K$ is a subHopfalgebra of $H$, $K^+=\mathrm{ker}(\epsilon)\cap K$. Because $\epsilon(ha)=\epsilon(h)\epsilon(a)=0$ for $a\in K^+$, $\forall h \in H$. What we need ...
Z.B. Zuo's user avatar
  • 510
0 votes
1 answer
22 views

Intersection of a chain of coideals in k-coalgebras

In my previous post Why can't you consider coideal generated by sets, where consequently i've shown why intersection of coideals need not to be a coideal, i said that ... the nonempty family $\{ I \...
Rafael H.'s user avatar
  • 124
0 votes
1 answer
83 views

A morphism of bialgebras between two Hopf algebras is necessarily a morphism of Hopf Algebras.

Here is the question I am trying to solve: Use the previous exercise to show that a morphism of bialgebras between two Hopf algebras is necessarily a morphism of Hopf algebras. Here is the previous ...
user avatar
3 votes
1 answer
117 views

Proving the uniqueness of a map

Here is the question I am trying to solve: (Tensor product of coalgebras) Let $(C, \Delta, \varepsilon)$ and $(C', \Delta ', \varepsilon ')$ be coalgebras. Show that the linear maps $\pi: C \otimes C' ...
user avatar
2 votes
0 answers
39 views

Direct product of tensor powers is a coalgebra

$\newcommand{\ot}{\otimes}\newcommand{\op}{\oplus}$Exercise from Kassel's Quantum Groups: Show that the canonical isomorphisms $V^{\ot (n+m)}\cong V^{\ot n}\ot V^{\ot m}$ induce a coalgebra structure ...
Dmitry's user avatar
  • 165
2 votes
1 answer
51 views

Why can't you consider coideal generated by sets.

The therm "coideal generated by a set" don't exist in literature but didn't found anything explaining why, so i formulated an example of a 6-dimensional coalgebra in wich there's a 1-...
Rafael H.'s user avatar
  • 124
0 votes
0 answers
45 views

Change of scalars for comodules as adjunctions?

Let $k$ be a commutative ring and $f: C \to C'$ be a homomorphism of $k$-coalgebras (for simplicity, we can suppose that it is surjective so that $f(C) = C'$). There is a functor: $$f_*: {}^lC-comod \...
Dat Minh Ha's user avatar
1 vote
2 answers
48 views

Coalgebraic description of converse of a binary relation on a set

A binary relation on a set is a coalgebra for the powerset endofunctor on the category of sets. In this coalgebraic setting, how do you construct or characterize the coalgebra which represents the ...
John Stell's user avatar
1 vote
1 answer
56 views

Frobenius algebra structure over complex polynomials modulo $x^2$.

I was trying to define a Frobenius Algebra structure over complex polynomials modulo $x^2$, but I am really struggling to do so. The algebra structure is rather evident, but I've tried many possible ...
Tomás Guija Valiente's user avatar
4 votes
0 answers
98 views

Cotensor product of Hopf algebroids constructed out of Brown-Peterson Spectrum

I am reading Ravenel's green book(Complex Cobordism and Stable Homotopy Groups of Spheres), there is an example in its 306 page: Let $(A, \Gamma) := (\pi_* BP, BP_* BP) \cong (\mathbb{Z}_{(p)}[v_1, ...
Cloudifold's user avatar
2 votes
1 answer
54 views

If $\Delta(c) = \Delta^{\mathrm{op}}(c)$, then under which permutations is $c_{(1)} \otimes \dotsb \otimes c_{(n)}$ invariant?

Let $(C, \Delta)$ be a coalgebra and $c\in C$ an element with $\Delta(c) = \Delta^{\mathrm{op}}(c)$. For certain permutations $\sigma \in S_n$, we will have that $$c_{(\sigma(1))} \otimes \dots \...
Andromeda's user avatar
  • 840
2 votes
1 answer
66 views

In the axioms of a coalgebra, does the *naturalness* of the isomorphisms play any role?

I don't know whether this question makes complete sense, but I'm 90% certain it does. In the definition of a coalgebra over a field, the fact that $(C \otimes C) \otimes C \cong C \otimes (C \otimes C)...
wlad's user avatar
  • 8,215
1 vote
0 answers
27 views

Dual of coideals

Let $C$ be a coalgebra with comultiplication $\Delta$ and count $\epsilon$. A coideal $I$ in $C$ is a linear subspace such that $$\epsilon(I)=0 \qquad \text{and} \qquad \Delta(I)\subset I\otimes C +C\...
Yining Zhang's user avatar
8 votes
1 answer
912 views

Representations of abelian groups

A classical result is Theorem: Let $G$ be a abelian group and $(V, \rho)$ be a irreducible representation of $G$ over a algebraically closed field $k$. If $V$ is finite dimensional (more generally, if ...
espacodual's user avatar
0 votes
1 answer
53 views

$M^{H^*}= M^{coH}$, $M^H=M^{coH^o}$ for a Hopf algebra $H$

Susan Montgomery gave the Lemma 1.7.2 without proof in her book 'Hopf algebras and their actions on rings' which states that $M^{H^*}= M^{coH}$ for a right $H$-comodule $M$ with left $H^*$-module ...
Z.B. Zuo's user avatar
  • 510
1 vote
1 answer
75 views

Quantum plane is a bialgebra

I am reading ‘Hopf algebras and their actions on rings’. Susan wrote the quantum plane as an example at 1.3.9 Example. He said $B = k \langle x,y \mid xy = qyx \rangle$, $0 \neq q \in k$ with ...
Z.B. Zuo's user avatar
  • 510
1 vote
0 answers
69 views

Conceptual proof that Hochschild boundary is a derivation for the shuffle product

Let $k$ be a commutative ring with $1$ and $A$ a commutative unital $k$-algebra ($k$ and $A$ are assumed to be associative). Denote by $(C_\bullet(A),b)$ the Hochschild chain complex of $A$ and let $$ ...
Albert's user avatar
  • 3,042
1 vote
0 answers
49 views

Finite support of multiset and distribution functor

It is unclear to me why in the distribution functor as well as in the multiset functor we require finite support. Below, I provided the definition from "Introduction to Coalgebra" by Bart ...
Alex's user avatar
  • 68
1 vote
1 answer
70 views

Subcomodule structure on pure submodule

A number of sources state that the following:$^\ast$ that if $C$ is an $R$-coalgebra ($R$ a commutative ring), with $M$ a right $C$-comodule, and $\iota:K\to M$ a $C$-pure submodule, such that $\rho_M(...
Blunka's user avatar
  • 925
2 votes
0 answers
48 views

Example of a quasi-finite Comodule that is not finitely cogenerated

Let $k$ be a field. Let $C$ be a coassociative and counital coalgebra over $k$. Takeuchi defines the notion of quasi-finite comodule as follows: a left $C$-comodule $M$ is quasi-finite if the induced ...
Maximilien Péroux's user avatar
3 votes
0 answers
226 views

Dual algebra structure of the divided power coalgebra

Background Let $K$ be a (unital, associative) commutative ring and consider the dual $K$-module $K[x]^*$ of the polynomial algebra. Then there is a $K$-algebra structure on $K[x]^*$ satisfying $$x_n \...
Mop1989's user avatar
  • 116
4 votes
0 answers
130 views

Coalgebraic logic in a general category?

Are there any works on coalgebraic logic (which is, in a way, generalization of modal logic, where we have an object $W$ in a category and a $T$-coalgebra $(W, \gamma)$; modal logic is a spacial case ...
tses's user avatar
  • 255

1
2 3 4 5