Questions tagged [hopf-algebras]

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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9
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0answers
158 views

Is there a homotopy coherent analogue of Dieudonné modules?

Let $K$ be a perfect field with characteristic $>0$ and $\mathcal{H}$ the category of graded connected abelian hopf algebras over $K$. By a theorem of Schoeller there is a canonical equivalence ...
3
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2answers
167 views

Low dimensional noncommutative non-cocommutative Hopf algebras

Sweedler's Hopf algebra (see here) is the lowest dimsnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative Hopf ...
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Do chains send homotopy inverse limits of spaces to homotopy inverse limits of $E_\infty$-coalgebras?

Let $X_\bullet := ... X_2 \to X_1$ be a tower of connected and simple spaces with the following properties: The induced tower $H_\ast(X_\bullet; \mathbb{F}_p)$ of graded $\mathbb{F}_p$-vector spaces ...
3
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Milnor exact sequence for homology of hopf algebras

Let $K$ be a field and $\mathrm{Hopf}^K_{E_\infty}$ the $\infty$-category of homotopy-coherent hopf algebras over $K$ that are coherently commutative and cocommutative. Precisely, $\mathrm{Hopf}^K_{E_\...
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+150

The double cover in the classical limit of $U_q(\mathfrak{sl}_2)$

I am trying to learn about Drinfeld–Jimbo quantum groups and I am having trouble with the classical limit of $U_q(\mathfrak{sl}_2)$. When properly expressed the limit makes sense as $q\to 1$ — see for ...
2
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41 views

Primitive elements in Hopf algebras over the integers

Let $H$ be a Hopf algebra over $\mathbb Z$, and assume that $H$ is cocommutative, graded, generated in degree $1$, and connected (its degree-$0$ part is $\mathbb Z$). Are there nice, natural ...
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46 views

Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)

I'm interested in solutions to the Yang-Baxter equation $$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$ that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
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Superfluous axioms for ribbon Hopf algebra

In his book Foundations of quantum group theory, Majid defines (2.1.10) a ribbon Hopf algebra as a quasi-triangular Hopf algebra $(H, R)$ with a special central element $v \in H$ satisfying (1) $v^2 ...
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1answer
242 views

Trying to understand “a refinement of the Peter–Weyl theorem” by Lusztig

"A refinement of the Peter–Weyl theorem" is the title of Chapter 29 in Lusztig's "Introduction to quantum groups" (Birkhäuser 2010, reprint of the 1994 edition). This chapter is ...
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2answers
979 views

Why does Drinfeld Unitarization work?

In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
2
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1answer
63 views

About extensions between morphisms on the multiplier algebra

Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism $$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \...
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1answer
66 views

Non-degeneracy of comultiplication (multiplier Hopf algebras)

Consider the following fragment from the paper "Multiplier Hopf-algebras" by Van Daele. Can someone explain how the coassociativity in definition 2.2 (ii) and the requirement $(\Delta \...
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1answer
68 views

Antipode on a multiplier Hopf-algebra

Probably an easy question, but here goes: I'm reading the paper Multiplier Hopf algebras by Van Daele. Let $(A, \Delta)$ be a multiplier Hopf algebra. Let $L(A), R(A), M(A)$ be the left, right and ...
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Examples of multiplier Hopf algebras

A multiplier Hopf-algebra (introduced by Van Daele) is a pair $(A, \Delta)$ where $A$ is a non-degenerate algebra $A$ together with a non-degenerate algebra morphism $\Delta: A \to M(A \otimes A)$ ...
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1answer
109 views

A comodule algebra map from a Hopf algebra to itself

Let $H$ be a cosemisimple Hopf algebra (or just a bialgebra). Considering $H$ as a left $H$-comodule (i.e. take $\Delta_H$ to be the coaction), can there exist a (non-identity) algebra map $\sigma:H \...
6
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1answer
102 views

Morphisms between compact quantum groups

Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
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1answer
323 views

Braided monoidal category, example

Let $H$ be a cocommutative hopf algebra. Let $M$ be the category of $H$-bimodules. Does the category $M$ form a braided monoidal category with tensor product $\otimes_{H}$ ?
3
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1answer
148 views

Show that a certain element is a linear combination of tensors

I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here. Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that ...
4
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1answer
92 views

Hopf “algebroid” structure of a groupoid convolution algebra?

This question is already posted in math.stackexchange, but didn't receive any answer. I'm not sure if this question fits in here, but surely someone in here can guide me to the correct answer. To make ...
3
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1answer
135 views

Is the kernel of an action of a Hopf algebra on an algebra a biideal?

I think, this must be simple, but I am not a specialist in this field, so excuse me. I asked this a week ago at MSE, but without success. S.Dascalescu, C.Nastasescu and S.Raianu define the action of a ...
4
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1answer
73 views

Bialgebras with rigid representation theory

Repost from math.SE since no answer after two months, but feel free to close if not appropriate: Everything is finite-dimensional over a field $k$. Let $B$ be a bialgebra with $B\text{-mod}$ its ...
4
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1answer
115 views

References for Hopf Galois module theory

I am a PhD student and I am really interested in Galois module theory, both in a "classical" and in a "nonclassical" sense. In the last months I have been reading about Hopf Galois ...
5
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1answer
203 views

Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$

$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between $$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$ The proof in "Tensor Categories ...
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Galois descent of a Hopf algebra

In a question here on why the classification of commutative Hopf algebras requires the field to be algebraically closed, a brilliant answer is given discussing the notion of Galois descent. As I ...
2
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1answer
78 views

Definition of multiplier bialgebra

Consider the following fragments from "An invitation to quantum groups and duality" by Timmerman: Question: In remark 2.1.6 (ii), it is stated that the homomorphism $\Delta\otimes \text{id}:...
13
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1answer
248 views

Question about definition of Hopf algebra in Hatcher

I have two questions about the definition of a Hopf algebra in Hatcher's book on algebraic topology. He defines it as follows (see Section 3.C, page 283): Definition: A Hopf algebra is a graded ...
4
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1answer
92 views

Non-cosemisimple duals of pointed Hopf algebras

I take the following quote from an answer to this question A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. The quantized enveloping algebras and ...
2
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2answers
309 views

Hopf algebra with a non-grouplike invertible element

What is an example of a Hopf algebra $(H,\Delta,\epsilon)$ containing an invertible element $h$ which is not grouplike: An element $h \in H$ such that $$ \Delta(h) \neq h \otimes h\qquad\text{(not ...
4
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1answer
238 views

Invertible elements of the Hopf algebra quantum $SU(2)$

Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see https://en.wikipedia.org/wiki/Compact_quantum_group (Note that on the ...
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0answers
57 views

Hopf algebra antipodes and right left comodule equivalences

Given a Hopf algebra $H$, denote by ${}^H\mathrm{mod}$ the category of left $H$-comodules, and by $\mathrm{mod}^H$ the category of right $H$-comodules. If the antipode $S$ of $H$ is invertible then we ...
14
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3answers
961 views

What is known about the category of Hopf algebras?

Several weeks ago I asked this at MathStackExchange, and to my surprise nobody answered. Recently I understood that I know almost nothing about the category $\operatorname{HopfAlg}$ of Hopf algebras (...
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0answers
130 views

Derived category of an abelian monoidal category

For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with ...
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1answer
89 views

Example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra

It's all in the question! What is an example of a sub-bialgebra of a Hopf algebra that is not a Hopf subalgebra?
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65 views

Monoid objects constructed from duals

Let $(M,\otimes)$ be a rigid monoidal category, for which left and right duals coincide. For any object $X \in M$, we can define a monoid structure on $X \otimes X^*$: Multiplication is defined by ...
3
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0answers
102 views

Is the category of Yetter-Drinfeld modules abelian?

Is $YD(H)$ the category of Yetter--Drinfeld modules over a Hopf algebra (defined over a field $k$) necessarily abelian? If not then what is the simplest example of a Hopf algebra $H$ for which $YD(H)$ ...
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1answer
312 views

$\mathbb{Z}$-graded algebras and tensor products

Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded: $$ A_kA_l = A_{k+l}. $$ ...
2
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1answer
118 views

Comodule Morita equivalence for Hopf algebras

Let $A$ and $B$ be two Hopf algebras, and denote by $\mathcal{M}^A$ and $\mathcal{M}^B$ their respective categories of right comodules. If we have a monoidal equivalence between $\mathcal{M}^A$ and $\...
5
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1answer
175 views

Comparing Hochschild (co)homology for algebras and coalgebras

Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes $$M\otimes A^{\otimes n}$$ and $$\...
9
votes
2answers
390 views

Definition of subcoalgebra over a commutative ring

Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$. Notes I'm reading give the following definition: $D$ is called subcoalgebra of $C$ if the ...
4
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0answers
159 views

Are the finite quantum permutation groups, weakly group-theoretical?

Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...
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0answers
88 views

Categorical interpretation of the comodulus of a Hopf algebra?

Let $H$ be a finite-dimensional Hopf algebra. Then it has a right cointegral $\lambda \in H^*$ and a left integral $c \in H$, characterized uniquely (up to scalar) by \begin{align} (\lambda \...
3
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0answers
153 views

Where can I find Drinfeld's original papers on quantum groups?

Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $U_h(\mathfrak{g})$ be the Drinfeld-Jimbo quantum group, i.e. the $\mathbb{C}[[h]]$-algebra topologically generated by $X_i,Y_i,H_i$ where $1\leq i\...
5
votes
1answer
161 views

Simple quotients of a triple tensor product

Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let also $V_1, V_2, V_3$ finite-dimensional simple modules over $\mathcal{H}$ and $Q$ be a simple quotient of $V_1\otimes V_2\otimes V_3$. Is it ...
5
votes
1answer
119 views

Reference requests : Presentation of the braided dual of $U_q(\frak{sl_2})$

I am interested in the braided dual of the quantum group $U_q(\frak{sl_2})$. This is the algebra generated by the matrix coefficients but where the multiplication is twisted by an action of the $R$-...
5
votes
1answer
112 views

Covariant splittings of Hopf algebra projections

What is an example of a pair of Hopf algebras $(A,B)$ with a surjective Hopf algebra map $\phi:A \to B$ such that $\phi$ does not admit a $B$-bi-comodule splitting $s:B \to A$? To be clear, the right $...
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0answers
79 views

When is the action of the braid group on tensor powers of Yetter-Drinfeld modules faithful?

Let $V$ be a Yetter-Drinfeld module over a Hopf algebra $H$ with invertible antipode. Recall that $V$ is a braided vector space with braiding $\Psi\colon V\otimes V\to V\otimes V, v\otimes w\mapsto v_{...
5
votes
0answers
514 views

Weak Hopf algebra structure on twisted group algebra

A (normalized) $2$-cocycle on a finite group $G$ with values in $S^1$ is a map $\sigma:G\times G\rightarrow S^1$ such that $$\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk)$$ and $$\sigma(g,e)=\sigma(...
6
votes
2answers
244 views

Confusion around the reflection equation algebra

I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a ...
2
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1answer
92 views

Non-counital coalgebras

For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
11
votes
3answers
687 views

Axiomatic definition of quantum groups

This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group? There are ...

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