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About 1: If $M$ is a monoid, then the bialgebra $B=k[M]$ (with comultiplication $\Delta(m)=m\otimes m$) is a Hopf algebra if and only if $M$ is actually a group. The proof is the followig: The antip …

Please expand your definition of "Casimir". Is a $g$-invariant element in $g^{\otimes n}$? If $g$ is a simple algebra and $\kappa$ its Killing form, then the map $$g\times g\times g\to k$$ $$(x,y,z) …

I recomend you the book by Lambe and Radford "Introduction to the Quantum Yang-Baxter equation and Quantum Groups - An Algebraic Aproach". I think you will find a good material related to your questio …

Assuming $\infty$= the cardinality of some set $I$, then consider the Lie algebra $\mathfrak a^{(I)}$= the direct sum of $I$-copies of $\mathfrak a$, with bracket coordinatewise. Then take its univers …

In general this coend is not a Hopf algebra. To convince yourself, think of a finite dimensional example, let $C=H^*$, so that you look for a minimal projective generator $H$-module $P$ and then look …

The paper "algebres de chemin quantiques" by Cibils and Rosso answers exactely that question. Adv in Math 125(2) 1997

Another proof: The exterior algebra is Koszul, it's Koszul dual is the symmetric algebra, that is commutative, but NOT super commutative, unless Char=2. Assuming $char \neq 2$, the exterior algebra ca …

I think $\sum_{i,j,k}X_{ij}X_{jk}X_{ki}$ should work for $\mathfrak{gl}(3)$. Now if you want for $\mathfrak{sl}(3)$, maybe you can change $X_{ii}$ by $X_{ii}-(1/3)\mathrm{Id}$. The formula above is li …

As @Zahlendreher said, the good language is that of bialgebras and comodule algebras so that you can do the bicross-product. Identifying $V^*\otimes V\cong(End(V))^*=:C$ then $V$ is a right $C$-como …

Yes, there is a criterium. Assuming $G$ finite, $A$ is of the form $k[H]$ for some subgroup of $G$ if and only if $A$ is a sub-bialgebra (and since $G$ is finite, if and only if is Hopf subalgebra). …

Sorry for the self publicity, but for the especific example of $gl_n(k)$, you can view it as $gl_n(k)\cong sl_n(k)\times k$ and we did some work for trivial central extensions that allow you to produc …

If H is a Hopf super algebra then $H\#k[\mathbb Z_2]$ is an ordinary Hopf algebra. So, with some work, one should get the classification up to dim 30 from the classification of ordinary Hopf algebras …

Take an example of a finite dimensional Hopf algebra $A$, presented by generators and relations, generated by grouplikes and primitives. There are a lot of non-commutative non-cocommutative example …

If $H$ is Hopf, then $H\otimes H$ with diagonal action is free, of the same dimension of $H$. The map is very explicit, you can surely can find it in any book of Hopf algebras (e.g. S. Montgomery). Th …