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Questions tagged [hopf-algebras]

For questions about Hopf algebras and related concepts, such as quantum groups. The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science.

Hopf algebras, named after Heinz Hopf, was first introduced in the theory of algebraic topology, while studying cohomology of Lie groups, but in recent years has been developed by many mathematicians and applied to other areas of mathematics such as algebraic groups, combinatorics, mathematical physics and Galois theory.

It is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism, called antipode, satisfying a certain property.

The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

There is a wide variety of variations of the notion of Hopf algebra, relaxing properties or adding structure. Examples are weak Hopf algebras, quasi-Hopf algebras, (quasi-)triangular Hopf algebras, quantum groups, hopfish algebras etc.

For more details you may find the following references:
$1.~~$ "Introduction to Hopf algebras and representations" by Kalle Kytola
$2.~~$ "Hopf Algebras in Combinatories" by Darij Grinberg & Victor Reiner
$3.~~$ "Hopf Algebra" from Wikipedia
$4.~~$ "A Very Basic Introduction to Hopf Algebras" by J.M. Selig

522 questions
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Internal Homs in Modules over a Hopf Algebra

Given a Hopf algebra $H$, I wonder when the monoidal category of $H$-modules is left-closed, right-closed, and finally, under what circumstances right and left internal hom are isomorphic. If I'm not mistaken, it seems that for $H$-modules $M,N$ one…
Hanno
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Why is the element in the definition of a Drinfeld twisted antipode of a Hopf algebra invertible?

I'm writing a piece about Drinfeld twists, and I realized I'm missing one piece. It should be easy to solve, but I seem to be stuck anyway. Let $H$ a Hopf algebra and an invertible element $F \in H \otimes H$ a Drinfeld twist, satisfying the twist…
B. Hekkelman
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Hopf algebra: Identity under convolution

In Hopf algebra texts, it is usually stated that $1=\eta\epsilon\in$Hom($H^C,H^A$) is the identity under convolution. $\eta$ is the unit, $\epsilon$ is the counit. My question is, is that a definition, or can it be proved? Sincere thanks for any…
yoyostein
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If $H$ is a finite-dimensional Hopf algebra, then the antipode is bijective?

If I’m given a finite-dimensional Hopf algebra $H$, how do I show the antipode is bijective? It's obvious that if we prove either injective or surjective, we get the other one for free since $H$ is finite-dimensional. Can someone nudge me in the…
fosterc4
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Modules over the Drinfeld double vs Yetter-Drinfeld modules

Let $H$ be a finite Hopf algebra. How can I prove that Yetter-Drinfeld modules over $H$ (see https://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category) coincide with modules over the Drinfeld double of $H$ (see Equality of two definitions of…
user09127
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If a Hopf Algebra has a nontrivial, finite-dimensional right ideal, then it is finite dimensional

Let $H$ be a Hopf Algebra (over a field $K$), with comultiplication $\Delta$, counit $\varepsilon$, and antipode $S$. A $K$-subspace V is said to be: A right ideal if $VH \subseteq V$ A right coideal if $\Delta (V) \subseteq V \otimes H$ (some…
de wa nai
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Equivalence between modules and comodules

Let $C$ be a coalgebra over a commutative ring $R$, if $C$ is cauchy (f.g. and projective), then there is an equivalence of categories between $\operatorname{Comod}(C)$ and the category $\operatorname{Mod}(C^*)$ of modules over the dual algebra…
matiasdata
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Primitive space of connected Hopf algebras

Let $H$ be a connected Hopf algebra. Let $C_1, C_2$ be two subcoalgebras of $H$. Is $P(C_1+C_2)=P(C_1)+P(C_2)$? Any comments are welcome.
G.-S. Zhou
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Commultiplication and antipode in Hopf algebra?

Let $H$ be a Hopf algebra with antipode $S$. For $h \in H$, we have $(S \otimes S) \circ \Delta(h) = \tau \circ \Delta \circ S(h)$, where $\tau(a \otimes b) = b \otimes a$ and $\Delta$ is the commultiplication. How to prove this identity? Thank you…
LJR
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The exterior product in the Hopf algebra

Here is a exercise in the Eiichi Abe's book Hopf Algebras, Ex2.4,Ch2, p.83. If $X$ is a left coideal, show that $X\wedge Y$ is a right coideal and that $X\subset X\wedge Y$. Maybe you can explain the $X\wedge Y$ in the Hopf algebra, does it has any…
Strongart
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Skew primitive elements of the Taft Hopf algebra

Is there any reference where I can find the skew primitive elements of the Taft Hopf algebra? The Taft algebra is defined here: https://en.wikipedia.org/wiki/Taft_Hopf_algebra By a skew primitive element we mean some x of the Hopf algebra such…
Milan
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Identities that connect antipode with multiplication and comultiplication

The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities: $$ S\otimes S\circ \varDelta=\sigma\circ\Delta\circ S $$ $$ \nabla\circ S\otimes S=S\circ\nabla\circ\sigma $$ where $S$ is the antipode, $\Delta$, the…
Sergei Akbarov
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Proving antipode of Hopf algebra property

So I am trying to show $\epsilon(S(a)) = \epsilon(a)$, where $\epsilon$ is the co-unit and $S$ is the antipode. \begin{align} \epsilon(S(a)) &= \epsilon( S( \sum_{(a)} a^{(1)} \epsilon(a^{(2)}) ) ) \\ &= \epsilon( S( \sum_{(a)}…
Anteaters
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Nonstandard Hopf algebra of polynomials

Consider the polynomial algebra $\mathbb{C}[x,y]$ with operation $\Delta(x)=x\otimes x- y\otimes y $ and $\Delta(y)=x\otimes y+ y\otimes x$. Find the operations counit $\epsilon$ and antipode $S$ such that $\mathbb{C}[x,y]$ with the $\Delta$ will…
A.Kazakov
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An identity related to antipode of a Hopf algebra

Let $H$ be a Hopf algebra with a bijective antipode $S$. Does the equality $\sum\limits_{(h)} h_2 \otimes S^{-1}(h_1) = \sum\limits_{(h)} h_1 \otimes S(h_2)$ hold for any $h \in H$, where $\Delta(h)=\sum h_1 \otimes h_2$? Thank you.
1985
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