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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Classification of quasitriangular Hopf algebras

The classification of hopf algebras is a big and open problem, containing various subproblems (such as: the classification of groups, of Lie algebras, the study of special classes such as (co)...
Konstantinos Kanakoglou's user avatar
11 votes
2 answers
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Cartier-Kostant-Milnor-Moore theorem

If $k$ is an algebraically closed field of characteristic zero and $H$ is a cocommutative Hopf algebra, then $$ H \cong U(P(H)) \ltimes kG(H). $$ What happens if the field is not algebraically closed? ...
a213f's user avatar
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Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has ...
Antonio Nogueria's user avatar
8 votes
2 answers
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What is a pointed Hopf algebra?

Hi, I would like to know what pointed Hopf algebras are and why it is that they are important. Thank you.
Albert's user avatar
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30 votes
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quantum groups... not via presentations

Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a_{ij}$, the quantum group $U_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K_i$, $E_i$, $F_i$ (the ...
André Henriques's user avatar
10 votes
3 answers
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About the classification of commutative and of cocommutative, fin. dim. Hopf algebras

I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
Konstantinos Kanakoglou's user avatar
5 votes
0 answers
161 views

Are the integer index finite depth irreducible subfactors Kac-coideal?

Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra? In other words, of the following form (...
Sebastien Palcoux's user avatar
16 votes
3 answers
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Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras

Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. In which cases the conjecture is known ...
warren's user avatar
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13 votes
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Can one define quantized universal enveloping algebras in a basis-free way?

(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&...
Gro-Tsen's user avatar
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When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims. On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams operations ...
darij grinberg's user avatar
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5 answers
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Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation. Question 1: what is known about this representation ...
Alexander Chervov's user avatar
8 votes
3 answers
473 views

Classification of $\operatorname{Rep} D(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
Student's user avatar
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Is there a non-trivial Hopf algebra without left coideal subalgebra?

Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra. A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$. $H$ is called maximal if it has no left coideal $\...
Sebastien Palcoux's user avatar
6 votes
2 answers
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Classification of plethories over $\mathbb{Q}$

Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). ...
Martin Brandenburg's user avatar
6 votes
1 answer
215 views

Quantum group representations from (convolution) matrix units?

Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$ There is a convolution product on $A=F(\...
JP McCarthy's user avatar
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Existence of a Kac algebra for a given fusion ring in a particular class

A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., d_{r}...
Sebastien Palcoux's user avatar
5 votes
3 answers
485 views

On a dual of Kaplansky's $2^{nd}$ conjecture: admissible algebras?

Kaplansky's second conjecture (on Hopf algebras) deals with "admissible" coalgebras: He calls a coalgebra admissible, if there is an algebra structure making it a Hopf algebra. The conjecture states ...
Konstantinos Kanakoglou's user avatar
4 votes
2 answers
135 views

If $\operatorname{Hom}(\delta_V, \delta_W) = 0$, then $\mathfrak{C}(\delta_V) \cap \mathfrak{C}(\delta_W) = 0.$

Let $(A, \Delta)$ be a Hopf $^*$-algebra and $\delta_V: V \to V \otimes A$ and $\delta_W: W \to W \otimes A$ be two corepresentations of $(A, \Delta).$ Assume that the space of intertwiners $\...
Andromeda's user avatar
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1 answer
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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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45 votes
16 answers
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What does the generating function $x/(1 - e^{-x})$ count?

Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series $$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty \frac{(-x)^n}{(n+1)!}...
Theo Johnson-Freyd's user avatar
34 votes
6 answers
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Why Drinfel'd-Jimbo-type quantum groups?

Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...
Greg Muller's user avatar
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26 votes
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Structure of Hopf algebras - trouble understanding an old paper

UPDATE: I am grateful to Peter May for the accepted answer, which makes most of the details below irrelevant. However, I will leave them in place for the record. I am trying to understand the proof ...
Neil Strickland's user avatar
20 votes
2 answers
2k views

Stable homotopy type theory?

This is a combined question, strictly speaking I am asking three questions concerning, respectively, homotopy type theory, stable homotopy theory and Yetter-Drinfeld modules. But I believe in the ...
მამუკა ჯიბლაძე's user avatar
20 votes
10 answers
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Hopf algebras examples

Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course. Some of the examples I know are well-...
18 votes
3 answers
2k views

Hopf dual of the Hopf dual

Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some ...
Nadia SUSY's user avatar
18 votes
1 answer
582 views

Bialgebras with Hopf restricted (or Sweedler) duals

It is known from the general theory that, given a bialgebra (over a field $k$) \begin{equation} \mathcal{B}=(B,\mu,1_B,\Delta,\epsilon) \end{equation} the Sweedler's dual $\mathcal{B}^0$ (called also ...
Duchamp Gérard H. E.'s user avatar
17 votes
4 answers
1k views

Braided Hopf algebras and Quantum Field Theories

It is well-known, that there are a lot of applications of classical Hopf algebras in QFT, e.g. Connes-Kreimer renormalization, Birkhoff decomposition, Zimmermann formula, properties of Rota-Baxter ...
mikis's user avatar
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Classification of Hopf algebras (state of the art)

I assume that the classification of (certain families of) Hopf algebras is still an open problem, am I right? My question is the following: What is the current state of the art? What is known about ...
warren's user avatar
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14 votes
2 answers
1k views

Is a bialgebra with all group-like elements invertible a Hopf algebra?

We know that in a Hopf algebra all group-like elements are invertible. Is the converse also true? Here is the precise formulation of my question : Let $B$ be a bialgebra and $GLE$ = { $g \in B ~|~ g \...
Steve B's user avatar
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14 votes
2 answers
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Hopf Algebra for a physicist

Hello, for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...
understandhopf's user avatar
13 votes
1 answer
420 views

Hopf algebras vs. Kac algebras

I recently came across Kac algebra. They are roughly Hopf algebras and $C^*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra ...
dm82424's user avatar
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12 votes
1 answer
785 views

What kind of algebraic object is $\mathcal{D}_X$? (algebra of diifferential operators). What's special about modules over it?

Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}_X$ the algebra of differential operators over it. The overall vague question is what kind of algebraic object is $...
Saal Hardali's user avatar
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12 votes
6 answers
2k views

Hopf algebras arising as Group Algebras

Every commutative $C^*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is ...
John McCarthy's user avatar
11 votes
0 answers
277 views

Is there a non-Kac complex finite dimensional semisimple Hopf algebra?

A complex (finite-dimensional) Hopf algebra is said to be a Kac algebra if it is a ${\rm C^{\star}}$-algebra in such a way that the comultiplication $\Delta$ is a $\star$-homomorphism. Obviously, a (...
Sebastien Palcoux's user avatar
10 votes
4 answers
1k views

Twist of a group Hopf-algebra

Let $G$ be a finite group with identity element $e$, and $C[G]$ the ring of complex-valued functions on $G$, with pointwise addition and multiplication. Then $C[G]$ is naturally a Hopf algebra, with ...
Marty's user avatar
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10 votes
0 answers
258 views

Are local finite dimensional Hopf algebras symmetric?

Recall that a finite dimensional algebra $A$ over a field $K$is called a Frobenius algebra in case $A \cong D(A)$ as right modules, where $D(A) \cong Hom_K(A,K)$. In case $A \cong D(A)$ as bimodules, ...
Mare's user avatar
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9 votes
1 answer
678 views

Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)

A quick Question: Is there some duality known between the quasi Hopf algebra $D^\omega(H)$ of a finite group $H$ to an orbifold model (such as SU(2)/$G$ or SO(3)/$G$ orbifold of some group $G$)? What ...
wonderich's user avatar
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9 votes
2 answers
901 views

If associated-graded of a filtered bialgebra is Hopf, does it follow that the original bialgebra was Hopf?

Warning: older texts use the word "Hopf algebra" for what's now commonly called "bialgebra", whereas now "Hopf" is an extra condition. So as to avoid any confusion, I'll ...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
426 views

$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$

While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
darij grinberg's user avatar
9 votes
2 answers
1k views

Algebra in a category

I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \...
Jianrong Li's user avatar
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9 votes
2 answers
393 views

Hopf algebra kernels vs. algebra kernels

Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field. Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the ...
Nicholas Kuhn's user avatar
9 votes
4 answers
2k views

The tensor product of two monoidal categories

Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way? The motivation I am thinking of is two categories that are representation ...
Nadia SUSY's user avatar
8 votes
1 answer
732 views

Modules over Hopf Algebras and $E_2$-algebras

Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf I was wondering if anybody knows of a nice relationship between ...
Matthew Levy's user avatar
8 votes
2 answers
758 views

Is a Hopf algebra a group object of some category?

The page of ncatlab on group object states that: A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf algebra. Question: Is a (noncommutative) Hopf algebra a group object of some ...
Sebastien Palcoux's user avatar
7 votes
1 answer
371 views

Compatibility conditions for Yetter-Drinfeld modules

In the paper, page 28, Definition 4.2.1, the compatibility condition for a Yetter-Drinfeld module over $H$ is $$ h_{(1)} v_{(-1)} \otimes h_{(2)}.v_{(0)} = (h_{(1)}.v)_{(-1)}h_{(2)} \otimes (h_{(1)}....
Jianrong Li's user avatar
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7 votes
2 answers
381 views

Low dimensional noncommutative non-cocommutative Hopf algebras

Sweedler's Hopf algebra (see here) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, ...
Quin Appleby's user avatar
7 votes
2 answers
608 views

Abelian category from the category of Hopf algebras

The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf sub-algebra of $H_1$. Is there some replacement or alteration of the notion of a kernel in the Hopf algebra setting. Same ...
Jake Wetlock's user avatar
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7 votes
2 answers
498 views

Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
Student's user avatar
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6 votes
0 answers
194 views

Integral Milnor-Moore theorem

Given a field K of char. zero the theorem of Milnor Moore states that taking the enveloping hopf algebra defines an embedding $\mathcal{U} $ from Lie algebras over K into hopf algebras over K. Taking ...
Hadrian Heine's user avatar
6 votes
1 answer
342 views

Is the "renormalized third comultiplication" on $\mathbf{Symm}$ integral?

Background: For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...
darij grinberg's user avatar