Questions tagged [coalgebras]
For questions about coalgebras, comultiplication, cocommutativity, counity, comodules, bicomodules, coactions, corepresentations, cotensor product, subcoalgebras, coideals, coradical, cosemisimplicity, ...
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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 ...
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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 ...
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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 = ...
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$\mathbb{C}G$-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 ...
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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 ...
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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) ...
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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 ...
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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 $(\mathbb{N},[\text{zero},\text{succ}])$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, \...
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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 ...
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