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A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a structure exists on the group algebra $k G$ of a finite group $G$.

1 vote

What is known about the category of Hopf algebras?

Concerning the second question, if the algebra A is finite dimensional, a universal bialgebra analogous to End(A) is easily constructed, by considering a suitable quotient of the tensor algebra on the …
Marco Farinati's user avatar
5 votes

Direct sum of Hopf algebras

This is a positive version of the answer "no". As already mentioned before, the disjoint union of groups is not a group, BUT it is a grupoid! In a similar way, the direct sum of two Hopf algebras is n …
Marco Farinati's user avatar
4 votes

Coinvariants of tensor products of Hopf algebras

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 …
Marco Farinati's user avatar
1 vote

Characters on 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 …
Konstantinos Kanakoglou's user avatar
1 vote
Accepted

Examples of basic coalgebras

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 …
Marco Farinati's user avatar
5 votes

When is a given quiver algebra a hopf algebra?

The paper "algebres de chemin quantiques" by Cibils and Rosso answers exactely that question. Adv in Math 125(2) 1997
Marco Farinati's user avatar
2 votes

Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative fun...

Some years ago we did something that -I think- answers the algebraic version of your question [FS]. If you have a Hopf algebra $H$, then the counit gives you a map $\varepsilon_* :\mathrm{Hom}(H,H)\t …
YCor's user avatar
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7 votes

Subalgebra of a group algebra

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). …
Marco Farinati's user avatar
0 votes

A "concrete" example of a one-sided Hopf algebra

Did you try $B=k\{\dots,x_{-1},x_1,x_2,\dots, x_n,\dots \}/(x_nx_{n+1}-1: n\in\mathbb Z)$ with $\Delta x_n=x_n\otimes x_n$ for all $n$?
Marco Farinati's user avatar
1 vote

Group-like Elements in a Coquasitriangular Bialgebra

Take a look at [T. HAYASHI , Quantum groups and quantum determinants, J. Algebra 152 (1992), pp 146–165.] I'm not sure if he answer completely your question but for sure he discuss it in the context o …
Marco Farinati's user avatar
1 vote

Is $T(V) \rtimes T(V^* \otimes V)$ a bialgebra?

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 …
Marco Farinati's user avatar
2 votes

How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided ...

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 …
Marco Farinati's user avatar
1 vote

Cotensoring by a Hopf Algebra

For any $H$-comodule $V$ you have its structure map $\rho:V\to V\otimes H$. The image of $\rho$ is the equalizer defining cotensor product.
Marco Farinati's user avatar
1 vote

Action is determined by a braiding

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 …
Marco Farinati's user avatar
2 votes

Hopf algebra in derived category vector spaces

If $C$ is a graded coalgebra (e.g. $C$= the homology of a d.g. Hopf algebra), then $C_0$ is not necesarily a subcoalgebra, because $$\Delta(C_0)\subset (C\otimes C)_0=\oplus_{n\in\mathbb Z}C_n\otimes …
Marco Farinati's user avatar

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