**40**

votes

**15**answers

5k 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 ...

**24**

votes

**6**answers

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 ...

**24**

votes

**2**answers

819 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 ...

**22**

votes

**6**answers

914 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 ...

**20**

votes

**2**answers

867 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 ...

**18**

votes

**4**answers

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 ...

**18**

votes

**7**answers

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 ...

**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 ...

**16**

votes

**1**answer

705 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 ...

**16**

votes

**1**answer

880 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**

votes

**1**answer

1k 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 ...

**15**

votes

**0**answers

172 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

**2**answers

622 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 ...

**13**

votes

**6**answers

1k 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 ...

**13**

votes

**1**answer

877 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

**3**answers

846 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

**2**answers

880 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**

votes

**2**answers

635 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

**1**answer

585 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 ...

**11**

votes

**1**answer

854 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 ...

**11**

votes

**0**answers

530 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**

votes

**7**answers

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**

votes

**3**answers

417 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 ...

**10**

votes

**2**answers

2k 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 ...

**10**

votes

**1**answer

484 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

**2**answers

681 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

**4**answers

455 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 ...

**10**

votes

**2**answers

467 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

**1**answer

676 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

**1**answer

480 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**

votes

**1**answer

512 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. ...

**9**

votes

**1**answer

652 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

**1**answer

329 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 ...

**9**

votes

**1**answer

425 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

**3**answers

480 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 ...

**8**

votes

**4**answers

3k 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

**3**answers

612 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 ...

**8**

votes

**2**answers

687 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

**3**answers

918 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 ...

**8**

votes

**1**answer

282 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

**3**answers

553 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

**1**answer

723 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

**0**answers

226 views

### What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...

**8**

votes

**0**answers

200 views

### Bialgebras with Hopf restricted 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 ...

**8**

votes

**0**answers

332 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

**3**answers

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

**4**answers

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

**1**answer

543 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

**2**answers

238 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

**4**answers

663 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 ...