Questions tagged [coalgebras]
For questions about coalgebras, comultiplication, cocommutativity, counity, comodules, bicomodules, coactions, corepresentations, cotensor product, subcoalgebras, coideals, coradical, cosemisimplicity, ...
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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 $R\subseteq V^{\otimes 2}$, ...
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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 ...
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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 ...
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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 \...
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Coalgebra presentation in terms of generators and relations?
The presentation of associative algebras in terms of generators and relations are very useful as they often give a simple description of a large, potentially infinite dimensional algebra using just a ...
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Coradical filtration and socle series $C_n=\mathrm{Soc}^{n+1}(C)$
I am reading the book Hopf Algebras and Their Actions on Rings by Susan Montgomery. In page 64, she said $C_n=\mathrm{Soc}^{n+1}(C)$, where $C$ is a coalgebra with coradical filtration $\{C_n \}$ and ...
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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, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital coassociative ...
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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:
$$\...
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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, \...
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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 ...
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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 \...
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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 ...
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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' ...
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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 ...
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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-...
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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 \...
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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 ...
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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 ...
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Extend a function as a coderivation
Suppose I have a (linear) function $f:V\to V$ for some vector space $V$. What does it mean to `extend f as a coderivation to $\Lambda V$'?
Extending as a derivation makes sense to me, because I define ...
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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, ...
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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
$$c_{(\sigma(1))} \otimes \dots \...
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In the axioms of a coalgebra, does the *naturalness* of the isomorphisms play any role?
I don't know whether this question makes complete sense, but I'm 90% certain it does.
In the definition of a coalgebra over a field, the fact that $(C \otimes C) \otimes C \cong C \otimes (C \otimes C)...
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Dual of coideals
Let $C$ be a coalgebra with comultiplication $\Delta$ and count $\epsilon$. A coideal $I$ in $C$ is a linear subspace such that
$$\epsilon(I)=0 \qquad \text{and} \qquad \Delta(I)\subset I\otimes C +C\...
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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 ...
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$M^{H^*}= M^{coH}$, $M^H=M^{coH^o}$ 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 ...
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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 ...
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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
$$
...
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Finite support of multiset and distribution functor
It is unclear to me why in the distribution functor as well as in the multiset functor we require finite support. Below, I provided the definition from "Introduction to Coalgebra" by Bart ...
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Subcomodule structure on pure submodule
A number of sources state that the following:$^\ast$ that if $C$ is an $R$-coalgebra ($R$ a commutative ring), with $M$ a right $C$-comodule, and $\iota:K\to M$ a $C$-pure submodule, such that $\rho_M(...
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Example of a quasi-finite Comodule that is not finitely cogenerated
Let $k$ be a field. Let $C$ be a coassociative and counital coalgebra over $k$. Takeuchi defines the notion of quasi-finite comodule as follows: a left $C$-comodule $M$ is quasi-finite if the induced ...
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Dual algebra structure of the divided power coalgebra
Background
Let $K$ be a (unital, associative) commutative ring and consider the dual $K$-module $K[x]^*$ of the polynomial algebra. Then there is a $K$-algebra structure on $K[x]^*$ satisfying
$$x_n \...
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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 ...
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Literature for modal logic and coalgebras (focused on logic and category theory)
I've been looking for literature that focuses on coalgebras and modal logic. But all literature I can find is mostly related to automata theory, or otherwise takes heavy influence from computer ...
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A morphism of coalgebras related to Binomial coefficients
I am looking for a hint to prove that
$$\Delta({t\choose n})= \sum\limits_{i=0}^{n} {t\choose i} \otimes {t\choose n-i} $$
where ${t\choose k}= \frac{t(t-1)...(t-k+1)}{k!}$ is a polynomial in $t$, $\...
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Consequences of coherence in a monoidal category
Let $\mathcal{D}_i = \left( D_i, \otimes_i, \mathbf{1}_i \right)$ be two cocomplete additive monoidal categories and let $\mathcal{F} \colon \mathcal{D}_1 \rightarrow \mathcal{D}_2$ be a strong ...
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(co)algebras of the adjunction between presheaves and bundles
For a topological space $X$ and the lattice of open sets $LX$, there is an extension of the "inclusion" (don't know what to call it) functor $F:LX\to\text{Top}/X$ along the Yoneda embedding $...
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What is a most elementary Coalgebra?
I just read the definition of a coalgebra, defined categorically ( reversing the arrows of the Algebra category ), the given example in the text I am referring to is the Homology on a topological ...
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Is there some practical intuition when working with a cooperad given by cogenerators and corelations?
In the case of algebras and operads, a description by generators and relations is common practice and I have a good understanding of this. A non-symmetric operad $\mathcal{P}$ given by a linear space ...
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Finding dual of matrix algebra
Consider $A=M_n(K)$, the algebra of matrices over the field $K$. Let $\{e_{ij}\}$ be the standard basis for $A$ and $\{X_{ij}\}$ be the basis of $A^*$ dual to $\{e_{ij}\}$.
I need to find the ...
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Supposse $U, V$ and $W$ are subspaces of a coalgebra $C$. Show that $\Delta(U) \subseteq V\otimes W$ implies $U\subseteq V\cap W$.
I'm new to coalgebras and this is a question from section 2.1 of the book "Hopf Algebras" from Davied E Radford.
I tried to pick an element $u \in U$, so $\Delta(u) = u_1\otimes u_2 \in V\...
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Exercise 2.3.24 - Radford's Hopf Algebras
I am having trouble solving the following problem, which is Exercise 2.3.24 of Radford's Hopf Algebras:
Let $C = C_n(k)$, where $n \geq 1$, and let $\{e_{i,j}\}_{1 \leq i,j\leq n}$ be a standard ...
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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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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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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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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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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
$$\text{comultiplication } \Delta: C \to C\otimes_k C \text{ and} $$
$$\text{counit } \epsilon: C \to k$$ ...
- 1,044
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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(...
user839372
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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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