# 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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### 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 ...
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### 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 ...
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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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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 $$... • 3,042 1 vote 0 answers 49 views ### Finite support of multiset and distribution functor It is unclear to me why in the distribution functor as well as in the multiset functor we require finite support. Below, I provided the definition from "Introduction to Coalgebra" by Bart ... • 68 1 vote 1 answer 70 views ### Subcomodule structure on pure submodule A number of sources state that the following:^\ast that if C is an R-coalgebra (R a commutative ring), with M a right C-comodule, and \iota:K\to M a C-pure submodule, such that \rho_M(... • 925 2 votes 0 answers 48 views ### Example of a quasi-finite Comodule that is not finitely cogenerated Let k be a field. Let C be a coassociative and counital coalgebra over k. Takeuchi defines the notion of quasi-finite comodule as follows: a left C-comodule M is quasi-finite if the induced ... 3 votes 0 answers 226 views ### Dual algebra structure of the divided power coalgebra Background Let K be a (unital, associative) commutative ring and consider the dual K-module K[x]^* of the polynomial algebra. Then there is a K-algebra structure on K[x]^* satisfying$$x_n \...
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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 ...