**1**

vote

**0**answers

53 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 ...

**6**

votes

**1**answer

191 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

72 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 ...

**2**

votes

**0**answers

145 views

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

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

**0**

votes

**1**answer

180 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 ...

**0**

votes

**0**answers

41 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]. ...

**3**

votes

**1**answer

377 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 ...

**5**

votes

**0**answers

154 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

94 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 ...

**1**

vote

**0**answers

124 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 ...

**3**

votes

**0**answers

65 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

81 views

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

Is it known whether or not the integer index finite depth irreducible subfactors (planar algebra) are Kac-coideal subfactors: $(R^{\mathbb{A}} \subset R^{\mathbb{I}})$, with $\mathbb{A}$ a finite ...

**2**

votes

**0**answers

91 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

112 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:
$$
S\otimes S\circ \Delta=\sigma\circ\Delta\circ S
$$
$$
...

**3**

votes

**0**answers

79 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

197 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 ...

**2**

votes

**1**answer

135 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

45 views

### When does the Spectrum of a Commutative Hopf Algebra Separate Points?

Let $H$ be a (finitely generated) commutative Hopf algebra over the complex numbers. When is it true that, for every $g \in H$, we can always find an algebra map $f_g:H \to \mathbb{C}$ such that ...

**7**

votes

**2**answers

221 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 ...

**6**

votes

**0**answers

221 views

### Topological quotient Hopf-algebras and “change-of-rings”

One way of constructing coalgebra objects in the homotopy theorist's category of spectra is to take the suspension spectrum of a space, with the diagonal providing a cocommutative comultiplication. ...

**3**

votes

**2**answers

253 views

### Algebraic Groups, Modules, and Comodules

Background:
Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For
$$
\widehat{H} := \text{Alg}_k\{H; k\},
$$
we recall (see Abe Chapter 4 ...

**0**

votes

**0**answers

43 views

### Does the standard Podlés sphere have a quasitriangular Hopf algebra structure? Do quantum homogeneous spaces have one in general?

Function spaces on (classical) homogeneous spaces can have a bialgebra structure:
Take $S^2$ to be the unital, associative algebra generated by $x, y, z$ with the relation $x^2 + y^2 + z^2 = 1$ and ...

**6**

votes

**1**answer

162 views

### Problem with Eisenbud's Lemma “Symmetry of Diagonalization”?

This question was first asked on MathSE but nobody answered.
In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like ...

**2**

votes

**0**answers

159 views

### (Non trivial) coidempotents(Co-$K$-theory)

I was interested to know about coalgebraic version of "Idempotents".
So I seached the web and I found the following interesting post :
...

**3**

votes

**0**answers

78 views

### dual notion to hopf galois extension and properties thereof

Let $H$ be a finite dimensional hopf algebra and $B \subset A$ be an $H$-extension of algebras. We know that the following are equivelant
1) $A \cong B \times_\sigma H$ is a cocycle crossed product ...

**4**

votes

**1**answer

287 views

### Tannaka–Krein duality

First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and ...

**3**

votes

**0**answers

69 views

### Is multiplication continuous in quantum-SU(2) with respect to the $L^2$-norm

For the Hopf $\ast$-algebra ${\cal O}_q(SU(2)$, with its unique Haar measure $h$, we have an inner product, and a norm, on ${\cal O}_q(SU(2))$ defined by
$$
<x,y> : = h(xy^*), ...

**1**

vote

**0**answers

91 views

### Action of Landweber-Novikov algebra on infinite polynomial ring

Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...

**2**

votes

**0**answers

119 views

### quantum deformations of tensor category

I was told that, if I understand correctly, that the enveloping algebra of semisimple Lie algebra admits one family of quantum deformation as Hopf algebra, which was proved by Drinfeld. Anyone can ...

**5**

votes

**4**answers

588 views

### 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 ...

**7**

votes

**1**answer

252 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 ...

**2**

votes

**0**answers

131 views

### Is an integral simple fusion ring, categorifiable?

A fusion ring $\mathcal{F}$ (see here p 28) is integral if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers. Its rank is $r$ and its dimension is $\sum ...

**1**

vote

**0**answers

97 views

### faithful modules of algebraic group

Let $G$ be a linear algebraic group over a field $k$. $k[G]$ is the
coordinate ring of $G$. $k[G]^{*}$ is the dual algebra of the
coalgebra $k[G]$. $H=k[G]^{\circ}$ is the finite dual of the Hopf
...

**2**

votes

**1**answer

103 views

### Understanding Sweedler's notation for the structure map of a comodule

I was hoping someone might be able to shed some light on the choice of indices for expressing the coaction using Sweedler notation.
For example, in the paper of Andruskiewitsch About ...

**3**

votes

**0**answers

132 views

### How a unitary corepresentation of a Hopf C*-algebra, deals with the antipode?

Let $\mathcal{A}$ be a Kac algebra (Hopf C*-algebra), with comultiplication $\delta$, counit $\epsilon$ and antipode $S$.
The following definition comes from this paper (p51-52) of ...

**2**

votes

**0**answers

96 views

### integrals in Hopf algebras

Let $H$ be a Hopf algebra and $g$ be a group-like element in
$H^{*}$. Define $L_{g}=\{x\in H\mid hx=g(h)x,~\forall h\in H\}$. If $H$
is not semisimple, when can we find a group-like element $g$ in
...

**0**

votes

**0**answers

141 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: ...

**9**

votes

**4**answers

392 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 ...

**4**

votes

**0**answers

174 views

### 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}, ..., ...

**1**

vote

**0**answers

106 views

### How simplify the pentagonal equation from two fusion rings?

A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...

**5**

votes

**1**answer

152 views

### Hopf algebra structure on the ring of quasisymmetric functions

I'm looking for a particular description of the Hopf algebra structure on the ring of quasisymmetric functions.
Let me illustrate by giving this kind of description for the Hopf algebra of symmetric ...

**6**

votes

**0**answers

149 views

### Is it possible to define Hopf algebra as an object in the category of (associative) algebras?

I think this must be well-known, but I can't find references, so my apologies from the very beginning.
Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ...

**3**

votes

**0**answers

71 views

### Do purification and equivariantization commute?

Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that ...

**5**

votes

**0**answers

207 views

### How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...

**4**

votes

**1**answer

124 views

### How to make Schubert calculus into a Hopf (actually a PCH) algbera?

The parallels between the formulas in Schubert calculus and in the theory of the representations of symmetric groups (par Geissinger-Zelevinsky) are so apparent (e.g. Giambelli formula), that one must ...

**14**

votes

**0**answers

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 ...

**0**

votes

**0**answers

179 views

### polynomial representation of $sl_{2}(k)$

Let $k$ be an algebraic closed field of characteristic 0. We write
$$X=\left(
\begin{array}{ccc}
0 & 1\\
0 & 0\\
\end{array}
\right),~~
Y=\left(
\begin{array}{ccc}
0 & 0\\
1 & 0\\
...

**1**

vote

**0**answers

179 views

### universal enveloping algebras and commutator subalgebras

Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$,
$U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$
and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...

**11**

votes

**2**answers

561 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 ...

**3**

votes

**0**answers

222 views

### Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types

Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible ...