# Tagged Questions

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 ...

**6**

votes

**0**answers

111 views

+150

### Functoriality of the Hopf dual

Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...

**3**

votes

**0**answers

62 views

### It there a nice way to describe the structure of Malcev-complete groups?

Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...

**4**

votes

**1**answer

254 views

### Is there another quantum deformation of sl(2)?

By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are:
$$
[E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F,
$$
some ...

**5**

votes

**1**answer

220 views

### Modules over Hopf Algebras and $E_2$-algebras

Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf
I was wondering if anybody knows of a nice relationship between ...

**2**

votes

**0**answers

61 views

### How are the unit/counit of a Hopf algebra and of an categorical adjunction related?

For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an adjoint pair if we have ...

**5**

votes

**0**answers

142 views

### Independence of characters with respect to polynomials

I came across the following property :
Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors,
$\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\...

**3**

votes

**0**answers

99 views

### Lie algebra of a compact Lie group

Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...

**3**

votes

**1**answer

105 views

### Tannaka-Krein reconstruction and rigidity

Let $\mathcal{C}$ be a rigid monoidal category together with a quasi-monoidal functor $\omega:\mathcal{C}\to\mathsf{vec}_{\Bbbk}$ to finite-dimensional vector spaces over a field $\Bbbk$, i.e. we have ...

**5**

votes

**0**answers

85 views

### Hopf algebra interpretation of hypergraph duality?

The work of Aguiar and Ardila (https://arxiv.org/abs/1709.07504) on Hopf monoids for generalized permutohedra gives a Hopf monoid structure on the collection of hypergraphs; see sections 19 and 20 of ...

**16**

votes

**1**answer

543 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 $\...

**13**

votes

**3**answers

542 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 ...

**4**

votes

**1**answer

171 views

### The Ungraded Milnor-Moore Theorem

Let $k$ be a field of characteristic $0$.
There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\...

**3**

votes

**1**answer

196 views

### When is this map of Hopf algebras Surjective?

I'm reading Akhil Mathew's blog post on Formal Lie Theory in Characteristic Zero.
Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ ...

**11**

votes

**3**answers

262 views

### Unbiased Hopf algebras

In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...

**8**

votes

**4**answers

432 views

### The dual of a dual in a rigid tensor category

For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.

**1**

vote

**0**answers

42 views

### Free module over $H$-module algebra

Let $H$ be a finite dimensional Hopf algebra, $R$ be a $H$-module algebra and $V$ be a finite dimensional $H$-module such that $R\otimes_{k} V$ is a finitely generated $R$ module under the action: $r.(...

**4**

votes

**1**answer

81 views

### Quasi-symmetric generalizations of classical symmetric functions

I am looking for quasi-symmetric versions of the classical $e_\lambda$ and $h_\lambda$ (the elementary and complete homogeneous symmetric functions). Is there some reference for this?
I am aware of ...

**3**

votes

**0**answers

72 views

### Is there a ''simple'' formula for the inverse of the Drinfeld associator?

The Drinfeld associator $\Phi(x_0, x_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x_0, x_1\rangle\rangle$ of formal ...

**7**

votes

**2**answers

281 views

### Hopf Subalgebras of Quantized Algebras

As is well known, quantized enveloping algebras $U_q(\frak{g})$ admit far fewer sub-Hopf algebras than classical enveloping algebras $U(\frak{g})$. As one can check directly, for appropriate subsets ...

**6**

votes

**0**answers

81 views

### What additional property does the antipode give on the category of all modules over an Hopf algebra?

It is well known that many constructions involving bialgebras extends to monoidal categories, and often becomes more natural in that framework.
If one cares about the category of finite dimensional ...

**2**

votes

**0**answers

108 views

### Module algebras and comodule algebras

Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-...

**7**

votes

**1**answer

114 views

### Exceptional Quantum Groups as FRT-Algebras

Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is ...

**2**

votes

**0**answers

44 views

### For any finite-dimensional Hopf C*-algebra, can one make the multiplication and co-multiplication cyclically symmetric simultanously?

For any finite-dimensional *-algebra, one can choose a basis such that the coefficients tensor of the anti-linear map $(a,b)\rightarrow (ab)^*$ becomes cyclically symmetric. (Any *-algebra is ...

**1**

vote

**0**answers

91 views

### When is a Frobenius Algebra a Quasi-Frobenius Ring?

Let $F$ be Frobenius algebra in the monoidal category $\mathcal{C}$ of bimodules over a not-necessarily commutative algebra $A$. When is it true that $F$ is a quasi-Frobenius ring.
For example, this ...

**7**

votes

**1**answer

173 views

### On the isomorphism problem of enveloping algebras

Let $\mathfrak{g}$ and $\mathfrak{g}'$ be Lie algebras. It is known that if $U(\mathfrak{g})\cong U(\mathfrak{g}')$ as associative algebras, then it is not necessarily true that $\mathfrak{g}\cong \...

**0**

votes

**0**answers

77 views

### $Q(f+g)_*=Q(f_*+g_*)$ (The maps induced by the sum is the sum of induced maps modulo decomposables [Reference request]

Let $X, Y$, let's say, homotopy commutative $H$-spaces, $f,g$ maps from $X$ to $Y$. (Actually we only need $Y$ to be homotopy commutative $H$_space,
but the statement is easier if we also suppose $X$ ...

**5**

votes

**1**answer

76 views

### A diagram for understanding action/coaction compatibility in a Yetter-Drinfeld module

For a Hopf algebra $H$ with antipode $S$, let $M$ be a left $H$-module with the action $h \otimes m \mapsto \rho(h,m)$, and also a left $H$-comodule with coaction $\delta \colon m \mapsto m^{(-1)} \...

**5**

votes

**2**answers

161 views

### Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?

For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...

**11**

votes

**0**answers

150 views

### Hopf-Galois extensions where the “extension” is a module?

For $H$ a Hopf-algebra, an $H$-Hopf-Galois extension is a map of rings $\phi\colon\thinspace A\to B$ such that $H$ coacts on $B$ over $A$, $B\otimes_AB\cong B\otimes H$, and the cofixed points, or the ...

**6**

votes

**1**answer

283 views

### Group schemes over ring of Witt vectors and their representing algebras

Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom_{Rng}(A,R)$ of ring ...

**2**

votes

**0**answers

45 views

### turning left modules into right modules over bialgebroids

Let $\mathcal{B}$ be a left R-bialgebroid as defined in https://arxiv.org/pdf/1403.3597.pdf on page 87. Let $M$ be a left $\mathcal{B}$-module. Can $M$ be made a right $\mathcal{B}$-module (as for the ...

**6**

votes

**1**answer

269 views

### The difference between $q$-deformations and $h$-deformations

What is the difference between $q$-deformations and $h$-deformations of universal enveloping algebras?
In chapter XVI of Quantum groups by Kassel, a very precise definition of a quantum enveloping ...

**2**

votes

**0**answers

107 views

### Quantum invariant: The canonical $2$-tensor

In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...

**1**

vote

**0**answers

71 views

### Detecting skew-primitives in representation categories

Suppose $H$ and $H'$ are two (possibly infinity dimensional) Hopf algebras which are not isomorphic as Hopf algebras, but are isomorphic as algebras. More specifically they are not isomorphic as Hopf ...

**2**

votes

**0**answers

49 views

### Relative version of Hopf cyclic cohomology

In this paper Connes and Moscovici introduced the Hopf algebra of transverse differential operators in order to compute the index formula for the diffeomorphism invariant geometry. They developed the ...

**2**

votes

**0**answers

69 views

### Algebras for (Koszul) Hopf operads

If necessary, we can restrict the following to the case where we consider only Hopf (co)operads in the category of chain complexes over fields of characteristic zero.
In case of ordinary operads, ...

**6**

votes

**1**answer

272 views

### Inner automorphisms of Hopf algebras

Is there a reasonable notion of an inner automorphism of a Hopf algebra $H$ which in the case of a group ring $H=\mathbb KG$ for a group $G$ reduces to a conjugation by $g\in G$?

**3**

votes

**1**answer

147 views

### What are the primitive elements in a polynomial Hopf algebra with primitive indeterminates?

Asked on math.stackexchange https://math.stackexchange.com/questions/2510606/what-are-the-primitive-elements-in-a-polynomial-hopf-algebra-with-primitive-inde but didn't get response (in fact got a ...

**4**

votes

**0**answers

106 views

### Category of (co)commutative Hopf monoids in an exact category

I'm transferring this question over from SE, since it didn't get much attention over there.
Let $(C, \otimes)$ be an exact monoidal category, and let $H(C)$ be the category of cocommutative and ...

**8**

votes

**1**answer

223 views

### Name for the action of a bialgebra on an algebra

Give an algebra $A$, a bialgebra $B$, and an action, that is, a bilinear map $\triangleright: B \times A \to A$ such that
$$
(b_1b_2) \triangleright a = b_1\triangleright(b_2 \triangleright a).
$$
...

**2**

votes

**0**answers

29 views

### Looking for an automorphism constructed from Hopf algebraic data

I am in an automorphism quest.
Let $H$ be a quasitrianglar Hopf Algebra with R-matrix $\mathcal{R} \in H \otimes H$. I know that $\mathcal{R}_{21}^{-1 }$ is a solution of the Yang Baxter equation and ...

**17**

votes

**2**answers

705 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 ...

**3**

votes

**1**answer

170 views

### Number of Isomorphism Classes of Corepresentations of A Compact Quantum Group

Given a compact quantum group $(G,\Delta)$, with dense Hopf algebra $H$, is it always true that, up to isomorphism, $H$ will have a countable number of irreducible comodules?

**16**

votes

**2**answers

777 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 ...

**2**

votes

**0**answers

61 views

### group action on Tor groups of modules and smash product

I am trying to understand theorem 3.4.2 from the paper "Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $\mathbf Z_{(p)}$ for representations with $p$-small weights" by ...

**13**

votes

**0**answers

261 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 $, ...

**1**

vote

**0**answers

139 views

### The order of the antipode in a Hopf algebra

As a result of Radford, any finite-dimensional Hopf algebra an antipode of finite order.
My question: How can we classify all finite-dimensional Hopf algebras whose antipode is identity?
Here are ...

**8**

votes

**0**answers

164 views

### Categorical interpretation of quantum double $D(A,B,\eta)$

It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties:
The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the ...

**2**

votes

**1**answer

90 views

### Is Sweedler's Hopf algebra factorizable?

Sweedler's 4-dimensional Hopf algebra admits a one-parameter family of triangular structures given by
\begin{equation}
R_{\lambda}:=1\otimes1+1\otimes g+g\otimes 1-g\otimes g-\frac{\lambda}{2}(x\...

**2**

votes

**0**answers

75 views

### Modules over quantum complete intersections

Let $a_i \geq 2$ be natural numbers and $q_{ij}$ field elements of the field $k$ for $i>j$.
A quantum complete intersection is the algebra $A:=k<x_1,...,x_n>/(x_i^{a_i},x_i x_j - q_{ij} x_j ...