Articles of coalgebras

Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.

First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes: The motivation for this is the proof that $Ext_{\Gamma}(k, k) = P(y_1, y_2, …)$ where $\Gamma$ is a commutative, graded connected Hopf algebra of […]

How to proof equivalent condition of algebra morphism and coalgebra morphism about Hopf algebra

My question is why $\mu$ is multiplication preserving iff $\mu\circ m=(m\otimes m)\circ(1\otimes \tau \otimes1)\circ(\mu \otimes \mu)$ and this holds iff $m$ is comultiplication preserving, where $\tau$ is the switching morphism, $\tau(A\otimes B)=B \otimes A$. Any help will be appreciated.

A question on coalgebras(1)

Is there a complex coalgebra $C$ with dimension at least 2 for which the scalar operators $T(x)=\lambda x$ are the only operators which satisfy $$(T\otimes T)\circ \Delta= \Delta \circ T^{2}$$ This equation is motivated by the fact that the differentiation operator $T=d/dx$ on complex coalgebra $\mathbb{C}[x]$ satisfies this equation.

Meaning of the antipode in Hopf algebras?

What I understand so far is that Hopf algebra is a vector space which is both algebra and coalgebra. In addition to this, there is a linear operation $S$, which for each element gives a so-called ‘anitpode’. Can anyone give an intuitive explanation of what is the ‘antiopde’ element? Why is it essential for the […]

Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will this decomposition obey Krull-Schmidt, by which I mean will the type and multiplicity of the irreducible comodules appearing be the same in any […]