Questions tagged [coalgebras]

Let $k$ be a commutative ring, which I am unwilling to assume is a field, and suppose $(C_i)$ is a collection of coassociative $k$-subcoalgebras1 of a coassociative $k$-coalgebra $C$. Is there always a coalgebra structure on the $k$-submodule…

Let $(S, \le)$ be a partially ordered finite set. Let $C$ the vector space with basis $\{e_{i,j} | i,j \in S, i \le j\}$, which turns out to be a coalgebra with comultiplication and counit given by: $$ \Delta (e_{i,j})= \sum_{k=i}^j e_{i,k} \otimes…

If $R$ is a commutative ring, is the kernel of any coalgebra homomorphism $f:C\to D$ a (two sided) coideal of $C$? For $R$ a field this is the case, since we have $(f\otimes f)\circ\Delta_C=\Delta_D\circ f=0$ so that…

Let $C$ be a $k$-coalgebra with basis $\{x_m\}$ where $m \in \{0, 1, ..., n\}$ where $n \geq 0$, with comultiplication defined by $$\Delta(x_m) = \sum_{t=0}^m x_t \otimes x_{m-t}$$ and counit $$\epsilon(x_m) = \delta_{0, m}$$ Determine the…