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 $G$.

Filter by
Sorted by
Tagged with
45 votes
16 answers
8k views

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
4k views

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
  • 12.7k
30 votes
7 answers
5k views

Shuffle Hopf algebra: how to prove its properties in a slick way?

Let $k$ be a commutative ring with $1$, and let $V$ be a $k$-module. Let $TV$ be the $k$-module $\bigoplus\limits_{n\in\mathbb N}V^{\otimes n}$, where all tensor products are over $k$. We define a $k$...
darij grinberg's user avatar
30 votes
2 answers
2k views

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
29 votes
6 answers
2k views

How to recognize a Hopf algebra?

Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra? I guess in ...
MTS's user avatar
  • 8,409
26 votes
2 answers
2k views

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
26 votes
1 answer
2k views

Have the Quantum Group Theorists taught the Group Theorists Anything?

I will start with the general before moving to the specific. Consider for a moment the two (very) soft definitions. An abstraction of an object $X$ is a category $\mathcal{C}_0$ such that $X$ ...
JP McCarthy's user avatar
25 votes
4 answers
2k views

Does the forgetful functor {Hopf Algebras}→{Algebras} have a right adjoint?

(Alternate title: Find the Adjoint: Hopf Algebra edition) I was chatting with Jonah about his question Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$. It's very closely related ...
Anton Geraschenko's user avatar
25 votes
1 answer
1k views

The Major Families of Quantum Groups

If we define a quantum group to be a quasi-triangular or coquasi-triangular Hopf algebra, then what are the major families of quantum groups? Of couse, to start with we have the h-adic completions ...
Dyke Acland's user avatar
  • 1,459
21 votes
3 answers
2k views

Open problems in Hopf algebras

I couldn't find a list of open problems in Hopf algebras. So my question is the following: In the theory of Hopf algebras, what are the (big) open problems? Any kind of problem/question will be ...
david's user avatar
  • 221
21 votes
2 answers
3k views

Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
Theo Johnson-Freyd's user avatar
20 votes
10 answers
4k views

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-...
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
19 votes
1 answer
925 views

Lang's Jacobian identity: slicker, elementary proof?

In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven: Theorem 1. Let $p$ be a prime. Let $\...
darij grinberg's user avatar
19 votes
2 answers
1k views

Hopf algebra reference

I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf ...
David E Speyer's user avatar
19 votes
1 answer
2k views

Connes-Kreimer Hopf algebra and cosmic Galois group

Hi, I'm interested in the relation between the two following constructions motivated by renormalization: Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of ...
Adrien's user avatar
  • 8,234
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
2 answers
1k 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 $(A, \Delta, \varepsilon, \Phi)$ ...
Olivia Borghi'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
  • 797
17 votes
2 answers
2k views

Examples of representations of quantum groups

I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
asv's user avatar
  • 21.1k
17 votes
2 answers
1k views

What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
skupers's user avatar
  • 7,923
16 votes
3 answers
717 views

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
  • 161
16 votes
3 answers
2k views

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
  • 275
16 votes
4 answers
1k views

If tensor product of representations is a representation, must we have a bialgebra?

Hopf algebras and bialgebras are sometimes introduced by saying that you've got an associative algebra $A$ and want to introduce the structure of an $A$-module on $V \otimes W$ where $V,W$ are $A$-...
Joshua Grochow's user avatar
16 votes
1 answer
399 views

Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with. Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...
Noah Snyder's user avatar
  • 27.8k
15 votes
3 answers
1k 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 (...
Sergei Akbarov's user avatar
15 votes
3 answers
837 views

Are supervector spaces the representations of a Hopf algebra?

Supervector spaces look a lot like the category of representations of $\mathbb{Z}/2\mathbb{Z}$ - the even part corresponds to the copies of the trivial representation and the odd part corresponds to ...
Qiaochu Yuan's user avatar
15 votes
3 answers
536 views

Direct sum of Hopf algebras

I realise that this question might be rather basic but however I was unable to find the answer in any textbook nor manage to figure out the answer. The question is the following: given two Hopf ...
truebaran's user avatar
  • 9,140
15 votes
6 answers
2k views

What structure on a monoidal category would make its 2-category of module categories monoidal and braided?

So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if ...
Ben Webster's user avatar
  • 43.9k
15 votes
3 answers
2k views

Hopf algebra structure on the universal enveloping algebra of a Leibniz algebra?

A Leibniz algebra L may be thought of as a noncommutative generalisation of a Lie algebra. One drops the requirement that the bracket be alternating and substitutes the Jacobi identity for the ...
José Figueroa-O'Farrill's user avatar
15 votes
2 answers
4k views

Hopf algebra duality and algebraic groups

Background: Let $G$ be a linear algebraic group over an algebraically closed field $k$ and let $I \subseteq k[G]$ be the ideal of the identity element. The hyperalgebra $U(G)$ of $G$ is defined to be ...
Chuck Hague's user avatar
  • 3,627
15 votes
1 answer
1k views

Lagrange's theorem for Hopf algebras

Under what conditions is a Hopf algebra free over any of its sub-Hopf algebras? I am reading "Hopf algebras and their actions on rings" by Susan Montgomery, specifically chapter 3. Lagrange's theorem ...
Gjergji Zaimi's user avatar
14 votes
2 answers
1k views

Quantum-Jimbo Algebras: Why Such Fuss About Roots of Unity?

Coming from a Lie algebraic background, I'm trying to branch onto quantum group theory. The divide I see all the time, is $q$ a root of unity or $q$ not a root of unity. I am wondering why is this? If ...
Milen Stamatov's user avatar
14 votes
2 answers
1k views

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
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
  • 365
14 votes
1 answer
350 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 ...
Irina's user avatar
  • 437
14 votes
1 answer
636 views

Is every finite quantum group a quantum symmetry group?

This post is basically a quantum extension of Is every finite group a group of “symmetries”? Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra. Frucht's theorem ...
Sebastien Palcoux's user avatar
14 votes
1 answer
652 views

Quantum braid group

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting ...
Ulrich Pennig's user avatar
13 votes
3 answers
988 views

Primitive elements of a tensor product of bialgebras

Given a field $k$ of characteristic $0$. For every $k$-bialgebra $A$, let $\mathrm{Prim} A$ denote the $k$-vector subspace of $A$ consisting of all primitive elements of $A$. What conditions can we ...
darij grinberg's user avatar
13 votes
4 answers
5k views

What is the universal enveloping algebra?

Let ${\mathfrak g}$ be a Lie algebra in a symmetric monoidal category enriched over $K$-vector spaces, i.e., in particular, hom-s are $K$-vector spaces (where $K$ is a field of characteristic zero). ...
Bugs Bunny's user avatar
  • 12.1k
13 votes
2 answers
943 views

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
  • 29.8k
13 votes
1 answer
1k views

2-cocycle twists of braided Hopf algebras

2-cocycle twists of Hopf algebras Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map $$ f: H \otimes H \to k$$ such that $$ f(x_{(1)},y_{(1)})f(x_{(2)} y_{(...
MTS's user avatar
  • 8,409
13 votes
1 answer
942 views

Hopf algebras as cohomology of $\mathbb{CP}^\infty$, $\Omega S^3$ and related $H$-spaces

Let me begin by a couple of questions : Consider a graded abelian group $V=\oplus_{i\geq 0} V_i$ such that $V_{2i}=\mathbb{Z}$ and $V_{\textrm{odd}}=0$. What are the possible Hopf algebra ...
Somnath Basu's user avatar
  • 3,403
13 votes
1 answer
713 views

The Inverse of a Universal R-Matrix for Quantized Universal Enveloping Algebra of sl2 and the Dual Pairing with SUq(2)

I have recently begun to study quasi-triangular structures and have come across a problem I can't resolve. Let ${\cal U}_q({\mathfrak sl}_N)$ denote the quantised enveloping algebra of ${\mathfrak sl}...
Abtan Massini's user avatar
13 votes
1 answer
636 views

Identifying a Hopf algebra cohomology theory

Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be ...
Edwin Beggs's user avatar
  • 1,213
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
  • 350
13 votes
0 answers
351 views

Analog of Haar element in an algebra

In a Hopf algebra $H $ (over some field $ k $), there is the notion of a Haar element $ h \in H$. This is an element of the algebra which has the property that if $ V $ is a representation of $ H $, ...
Joel Kamnitzer's user avatar
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
12 votes
3 answers
810 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 ...
jg1896's user avatar
  • 2,683

1
2 3 4 5
12