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 ...

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3k 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 ...
21
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6answers
2k 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 ...
21
votes
2answers
667 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 ...
21
votes
6answers
850 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 ...
17
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4answers
1k 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 ...
17
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2answers
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 ...
16
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7answers
2k 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 ...
16
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1answer
825 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 ...
16
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1answer
900 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 ...
14
votes
1answer
624 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 ...
14
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0answers
154 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 ...
13
votes
1answer
828 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 ...
12
votes
3answers
712 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 ...
12
votes
6answers
986 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 ...
11
votes
2answers
551 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 ...
11
votes
2answers
779 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 ...
11
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0answers
501 views

Given an algebra, can it be realized as a block of a Hopf algebra?

During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in ...
10
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7answers
2k 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 ...
10
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1answer
451 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 ...
10
votes
2answers
438 views

An explicit description of $\operatorname{gr}(k \cdot G)$ for the filtration induced by the augmentation ideal?

Let $A$ be any bialgebra (associative, unital, etc.) over a ring $k$. Then among other things it has a counit $\epsilon : A \to k$, and hence an augmentation ideal $I = \ker \epsilon$, which is a ...
10
votes
2answers
625 views

Is there a canonical Hopf structure on the center of a universal enveloping algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define ...
10
votes
1answer
577 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)} ...
10
votes
1answer
448 views

Generators of the Odd Dimensional Quantum Spheres

As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the ...
10
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1answer
486 views

Is there a “correct” general setting for the principle: “tensoring any object with a projective object yields another projective”?

Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$. (See Exercise 2 on p. ...
10
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1answer
701 views

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 ...
9
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1answer
538 views

The coproduct on the cohomology of a Hopf algebra

If $H$ is a Hopf algebra over a field $k$, the Hopf algebra cohomology $\mathrm{Ext}_H(k,k)$ is —as usual— an algebra, but the Hopfness of $H$ turns it into a Hopf algebra. Is there a reference ...
9
votes
2answers
1k 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 ...
9
votes
1answer
476 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 ...
9
votes
1answer
412 views

Geometric interpretation of integrals of coordinate rings

If $X$ is an affine scheme over the field $k$ than algebraic invariants of the coordinate ring $k[X]$ usually have a geometric interpretation in terms of $X$ (and vice versa). As an example, the ...
8
votes
4answers
2k 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). ...
8
votes
2answers
606 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 ...
8
votes
1answer
233 views

Hopf algebras and bijective antipodes

By a theorem of Larson and Sweedler, the antipode of every finite-dimensional Hopf algebra is bijective. My question is the following: Is it true that in every noetherian Hopf algebra the antipode ...
8
votes
4answers
383 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 ...
8
votes
3answers
521 views

Faithfully flat descent over Hopf algebras in terms of comodule structures

Let A be a (finite-dimensional graded cocommutative) Hopf algebra over a field k, E be a Hopf subalgebra, and R=A \otimes_E k. Then the comultiplication on A induces a coalgebra structure on R. ...
8
votes
1answer
681 views

Connected graded involutive bialgebras (Hopf algebras) sought (for Dynkin idempotent checking)

Again, there is a general and a concrete question: Concrete question. Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N$-graded). Then, it ...
8
votes
0answers
294 views

Bicommutative Hopf algebras have internal hom objects. What are they?

Let $k$ be a field and $\mathcal H$ the category of conilpotent bicommutative Hopf algebras over $k$. (Conilpotent means that every element eventually becomes zero if you apply the reduced ...
7
votes
3answers
2k views

Faithful characters of finite groups

Related to a previous question I am asking furthermore a proof for the following: Question 1: If $\chi$ is a faithful irreducible character of a finite group $G$ then the regular character of $G$ is ...
7
votes
4answers
2k views

Hopf Algebras and Quantum Groups

I have studied graduate abstract algebra and would like to learn about Hopf algebras and quantum groups. What book or books would you recommend? Are there other subjects that I should learn first ...
7
votes
1answer
510 views

Are groups in (Var/k, rational maps) necessarily algebraic groups?

Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant]. Let's consider a ...
7
votes
2answers
322 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 ...
7
votes
2answers
215 views

An identity in the free associative algebra

Let $V$ be a finite dimensional vector space over a field of characteristic $0$, and let $T(V)$ be the tensor algebra (also called the free associative algebra) on $V$. This is actually a Hopf ...
7
votes
4answers
626 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 ...
7
votes
1answer
385 views

Comparing two similar procedures for quantizing a Casimir Lie algebra

My primary reference for this question is the very good book Quantum Groups and Knot Invariants by C. Kassel, M. Rosso, and V. Turaev. I'm also drawing from P. Etingof and O. Schiffmann, Lectures on ...
7
votes
1answer
298 views

What are the primitives of Lusztig's twisted bialgebra $\mathbf{f}$?

I'm trying to understand the coproduct on Lusztig's $\mathbf{f}$ and, apart from the Chevalley generators, I don't know of any more primitive elements. In the $q=1$ case, the Weyl group acts as a Hopf ...
7
votes
1answer
231 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 ...
7
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1answer
592 views

Yetter--Drinfeld Modules and Braidings

Let $H$ be a Hopf algebra, $M$ a right $H$-comodule for a coaction $\Delta_R$, and $\triangleleft$ a right $H$-action on $M$ such that $M$ is a Yetter--Drindfeld module. We know from general theory ...
7
votes
0answers
297 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 ...
6
votes
3answers
446 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 ...
6
votes
1answer
585 views

Distributions on product spaces

I hope this is suitable to MO. Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
6
votes
2answers
377 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 give my definitions before ...