Articles of hopf algebras

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, the algebra structure of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C} \oplus M_2(\mathbb{C})$. Let $\{ e_1,e_2,a_{11}, a_{12}, a_{21}, a_{22} \}$ be a matrix […]

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 conceptual understanding of transmutations (and bosonizations) of (braided) Hopf algebras

Consider a coquasitriangular Hopf-algebra $(H,\mu,\eta,\Delta,\epsilon, S)$ over a field $\mathbb F$ with characteristic zero and the braided monoidal category $\mathcal C$ of $H$-right-comodules. We explicitly denote the coquasitriangular form of $H$ by $r$ and its convolution inverse by $r’$. A lengthy calculation yields the result that $H$ can be “transmutated” into an Hopf-algebra object in […]

Convolution product on the linear dual of the polynomial algebra

Let $\Bbbk$ be a field and let us consider the $\Bbbk$-algebra $\Bbbk[X]$ of polynomials in one indeterminate with its natural Hopf algebra structure, i.e. $$\Delta(X)=X\otimes 1+1\otimes X, \quad s(X)=-X\quad \text{and} \quad \varepsilon(X)=0 .$$ On its linear dual $\Bbbk[X]^*$ we may define the convolution product by $$(f*g)(X^n)=\sum_{i+j=n} \binom{n}{i}f(X^i)g(X^j)$$ for all $f,g\in\Bbbk[X]^*$ and $n\in\mathbb{N}$. Let $$\Bbbk[X]^\circ=\left\{f\in\Bbbk[X]^*\mid f\left(X^kp(X)\right)=0\textrm{ […]

Two different comultiplications on a Hopf algebra

I am pretty sure this statement is false : let $K$ be a field and let $(A, \eta, \mu, \Delta, \epsilon, c)$ be a Hopf Algebra. If we forget the comultiplication $\Delta$, is it forced by $(A, \eta, \mu, \epsilon, c)$ ? In other words, can we put two Hopf algebra structures on a $K$-algebra […]

What are Quantum Groups?

I am going to do a semester project (kind of a little thesis) this spring. I met a professor and asked him about some possible arguments. Among others, he proposed something related to quantum groups. I am utterly ignorant of the subject, so could someone help me out and tell me What is a quantum […]

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 […]

How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation theory of $G$ over $K$, as for instance if $K=\mathbb{C}$, by Maschke’s Theorem and Wedderburn’s Theorem we can write $\mathbb{C}[G] […]