# Questions tagged [coalgebras]

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

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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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### 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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### 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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### 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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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 $\Delta(u) = u_1\otimes u_2 \in V\... 1 vote 1 answer 56 views ### Exercise 2.3.24 - Radford's Hopf Algebras I am having trouble solving the following problem, which is Exercise 2.3.24 of Radford's Hopf Algebras: Let$C = C_n(k)$, where$n \geq 1$, and let$\{e_{i,j}\}_{1 \leq i,j\leq n}$be a standard ... 1 vote 1 answer 139 views ### For a terminal coalgebra, map is isomorphism I'm trying to solve Emily Riehl's "category theory in context", question 1.6.vi (1.6.6). The exercise says: A coalgebra for an endofunctor$T: C \rightarrow C$is an object$c \in C$... 2 votes 1 answer 99 views ### "Basic" coalgebra structures on$R$-modules Lately I have been reading about coalgebras and wondering about natural ways one can make any$R$-module into an$R$-coalgebra. Two examples of such constructions are given by tensor coalgebras and ... 1 vote 1 answer 86 views ### How to compute the terminal coalgebra (or initial algebra) of a given functor This is sort of a soft question. I'll start with an example. Fix a set$A$, and consider the functor$F\colon \mathbf{Set} \to \mathbf{Set}$,$F\colon X \mapsto A \times X$. Now, we know that the ... 3 votes 1 answer 131 views ### Koszul dual cooperad of the associative operad I am trying to compute the$k$-modules of$\mathcal{As}^¡$, the Koszul dual cooperad of the associative operad$\mathcal{As}$. I am using sections 7.1.4 and 7.2.1 of Algebraic Operads to try to do ... 1 vote 1 answer 66 views ### Axioms of a coalgebra restated using Sweedler's notation I'm struggling with understanding manipulation using Sweedler's notation at a very fundamental level. I don't understand the equivalence of the axioms of coalgebras in the standard notation [Coproduct ... 1 vote 0 answers 98 views ### Hopf "algebroid" structure of a groupoid convolution algebra? To male thinks simple as possible, lets say we have a discrete group$G.$Then the then the group algebra$\mathbb{C}[G]$(of finitely supported complex valued functions on$G$) has a convolution and ... 1 vote 1 answer 89 views ### How to prove the reduced comultiplication of a coaugmented coalgebra is coassociative? a coalgebra over a field$k$is a vector space$C$over$k$together with$k$-linear maps $$\text{comultiplication } \Delta: C \to C\otimes_k C \text{ and}$$ $$\text{counit } \epsilon: C \to k$$ ... 3 votes 0 answers 94 views ### 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(... 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 ...