**39**

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 ...

**6**

votes

**0**answers

96 views

### An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see here.
We are interesting in an alternative ...

**5**

votes

**0**answers

109 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 ...

**20**

votes

**2**answers

820 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 ...

**7**

votes

**1**answer

139 views

### Hopf Galois extensions and conditional expectations for C* algebras

Suppose that $H$ is a Hopf algebra with normalised invariant integral (appropriate side) $\int:H\to \mathbb{C}$. The $H$ right comodule algebra $P$ is a Hopf Galois extension, so the canonical map ...

**2**

votes

**1**answer

89 views

### dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32
Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...

**6**

votes

**0**answers

96 views

### What's the relation between half-twists and star structures on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...

**5**

votes

**1**answer

178 views

### group completion theorem of homology as Hopf algebras

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...

**8**

votes

**3**answers

594 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 ...

**0**

votes

**1**answer

90 views

### Yetter-Drinfeld modules as rigid category

I'm reading a proof of the following theorem
If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category.
In this proof we define ...

**3**

votes

**0**answers

59 views

### Dense Hopf $*$-subalgebra of compact quantum groups and cancellation laws

Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital ...

**2**

votes

**0**answers

43 views

### Co-quasitriangular Hopf algebra - notation

In one article I found the following statement :
If $A$ is a Hopf algebra over $k$ with co-quasitriangular structure $r$, then one can define map $r_{21}*r:A\otimes A\rightarrow k \ \ $ (...).
...

**4**

votes

**0**answers

62 views

### Is there a perfect integral fusion category with PFdim = 2 mod 4?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff any $1$-dimensional complex representation of $G$ is trivial.
proof: First if ...

**1**

vote

**0**answers

40 views

### Realization of braiding on infinite dimensional vector space via Yetter-Drinfeld structure over Hopf algebra

In one article I found the following statement
Let $V$ be finite dimensional vector space. A braiding $\Psi$ on $V$ can be realized via Yetter-Drinfeld structure over a Hopf algebra with ...

**1**

vote

**1**answer

95 views

### Difference between two definitions of graded coalgebra

I'm reading articles about applications of Hopf algebras in physics. In one of them there is following definition of graded coalgebra:
A coalgebra $C$ is called graded if $C=\bigoplus\limits_{n\ge ...

**13**

votes

**2**answers

592 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 ...

**8**

votes

**3**answers

898 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 ...

**3**

votes

**1**answer

60 views

### Normal Hopf ideals and Hopf subalgebras, quotient group schemes

There is a following theorem:
$H$ is a commutative Hopf algebra over a field $k$. Then there exists a bijective correspondence between
$$\{ \textrm{Hopf subalgebras }K\subset H \} \quad ...

**0**

votes

**0**answers

68 views

### Programmatically computing dual Hopf Algebras: State of the art:

Given a graded Hopf algebra of finite type, we know the (graded) linear dual is also a graded Hopf algebra. For instance the dual Hopf algebra to the polynomial algebra on an even degree generator, ...

**4**

votes

**3**answers

811 views

### Tensor product of linear mappings versus chain complexes

A chain complex of vector spaces $X_k$ is a sequence of linear mappings
$\dots \overset{d_{k-1}}{\longrightarrow} X_k \overset{d_{k}}{\longrightarrow} X_{k+1} \overset{d_{k+1}}{\longrightarrow} ...

**6**

votes

**1**answer

178 views

### Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...

**9**

votes

**1**answer

322 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 ...

**8**

votes

**0**answers

187 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 ...

**0**

votes

**0**answers

35 views

### Grouplike elements in dual weak Hopf algebras

It is said in D. Nikshych's paper On the structure of weak Hopf algebras (arXiv:math/0106010) that if $A$ is a finite dimensional weak Hopf algebra, then a functional $\gamma$ in the dual (weak Hopf) ...

**2**

votes

**0**answers

206 views

### dimension of induced comodule

Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...

**5**

votes

**0**answers

144 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 ...

**2**

votes

**0**answers

218 views

### Is there a non-trivial maximal Hopf algebra?

Let $H$ be a Hopf algebra over an algebraically
closed field $\mathbb{K}$ of characteristic $0$.
Maximal means without left coideal subalgebra $I$ (i.e. $\Delta(I) \subset H \otimes I$) other than ...

**10**

votes

**3**answers

411 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 ...

**1**

vote

**0**answers

70 views

### Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$.
Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...

**3**

votes

**0**answers

111 views

### Hopf-Borel theorem over polynomial rings

There is a theorem of Borel saying that:
For a connected cocommutative Hopf algebra $A$ over the perfect field $k$ such that underlying graded vector space is finite-dimensional over $k$ we have that ...

**6**

votes

**1**answer

250 views

### In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective?
Recall that an object ...

**1**

vote

**0**answers

85 views

### Exponential map on a unipotent group

Let $G$ by a unipotent linear algebraic group defined over a field of characteristic $0$, with Lie algebra $\mathfrak{g}$. The exponential map $\mathfrak{g}\to G$ is bijective, and we can recover the ...

**0**

votes

**1**answer

200 views

### Coaction of a group

Suppose $G$ is a finite group which acts on a $C^*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$.
I ...

**1**

vote

**0**answers

132 views

### The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...

**23**

votes

**2**answers

768 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 ...

**3**

votes

**1**answer

411 views

### Hopf-algebras in associative ring spectra

I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative ...

**4**

votes

**4**answers

953 views

### Apocryphal Maschke theorem?

This may be totally trivial or wrong. I am just posting this because I am sick and tired of trying to understand this myself and I am sure someone out here can just answer it out of his head in 2 ...

**0**

votes

**0**answers

46 views

### Analogue of Baer's injectivity criterion for comodule algebras

Let $H$ be a Hopf algebra and $A$ an $H$-comodule algebra; denote by $M^H_A$ the category of right $(H,A)$-Hopf modules [i.e. $A$-module, $H$-comodules, everything is compatible with everything else]. ...

**6**

votes

**0**answers

172 views

### Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...

**4**

votes

**1**answer

118 views

### Comodule analogue for statement that a faithful representation of an affine group scheme generates all

If $V$ is a faithful finite dimensional representation of an affine group scheme $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n ...

**2**

votes

**1**answer

192 views

### Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...

**5**

votes

**1**answer

227 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 ...

**3**

votes

**0**answers

82 views

### How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided Hopf algebras) understood?

As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a ...

**2**

votes

**0**answers

99 views

### Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$

I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167:
Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra ...

**2**

votes

**2**answers

128 views

### Identities that connect antipode with multiplication and comultiplication

I asked this initially in math.stackexchange:
The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities:
$$
S\otimes S\circ \Delta=\sigma\circ\Delta\circ S
$$
$$
...

**4**

votes

**0**answers

89 views

### Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$.
Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...

**2**

votes

**2**answers

230 views

### Is antipode unique for bialgebras in arbitrary monoidal categories?

If $B$ is a bialgebra in the category $\tt{Vect}$ of vector spaces (over $\mathbb C$, for example) then $B$ can't have two different antipodes.
Is this true for bialgebras in an arbitrary symmetric ...

**1**

vote

**0**answers

260 views

### Coideals of Hopf algebra coming from right (left) coideals K->K^+

If $H$ is a Hopf algebra over a field and $K$ is a right or left coideal of $H$ then $K^+=K\cap\ker\epsilon$ is a coideal of $H$. Does this hold when $k$ is a commutative ring? If not, what is a ...

**2**

votes

**1**answer

174 views

### When is an exponential functor a bialgebra?

My question is about the notion of exponential functors as they are frequently defined in the literature on (strict) polynomial functors, e.g., the paper "General Linear and Functor Cohomology" by ...

**0**

votes

**0**answers

160 views

### Flatness over Hopf subalgebra

Let $A\subset B$ be flat Hopf algebras over a Dedekind ring $R$.
J. Moore has proved in the article
Compléments sur les algèbres de Hopf, John C. Moore, Séminaire Henri Cartan (1959-1960), Volume: ...