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Results tagged with hopf-algebras
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user 98863
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 $G$.
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does finite dimensional representations of bialgebras separate elements?
let $B$ be a bialgebra over a field (i.e. associative, coassociative, unitary and counitary, maybe it has an antipode or maybe not). If $b\in B$ acts by zero on every finite dimensional representatio …
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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 …
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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 …
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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 …
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4
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Hopf algebras examples
My favorite Hopf algebra is the following:
take $x$, $d$ two free variables, fix a field $k$ and consider $H$ the $k$-algebra generated by $x^{\pm 1}$, $d$ with relations
$xd+dx=0$,
$d^2=0$
This …
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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 …
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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 …
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5
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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 …
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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
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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.
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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 …
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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 …
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7
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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).
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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 $n$?
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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 …
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