All Questions
Tagged with gr.group-theory hopf-algebras
24 questions
26
votes
1
answer
2k
views
Have the Quantum Group Theorists taught the Group Theorists Anything?
I will start with the general before moving to the specific.
Consider for a moment the two (very) soft definitions.
An abstraction of an object $X$ is a category $\mathcal{C}_0$ such that $X$ ...
- 1,027
12
votes
3
answers
849
views
Subalgebra of a group algebra
Let $k$ be a field, $G$ a finite group, and $k[G]$ the group algebra.
Let $A$ be a subalgebra of $k[G]$. In general, $A$ is not the group algebra of some subgroup $H$ of $G$.
Question: Is there any ...
- 5,230
11
votes
5
answers
2k
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 ...
- 24.9k
10
votes
4
answers
1k
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 ...
- 13.3k
9
votes
3
answers
3k
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 ...
- 887
8
votes
2
answers
852
views
Is a Hopf algebra a group object of some category?
The page of ncatlab on group object states that:
A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf
algebra.
Question: Is a (noncommutative) Hopf algebra a group object of some ...
7
votes
1
answer
365
views
Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?
What is the smallest group $G$ such that $\operatorname{Rep}(G)$ has a non-symmetric braiding (or just an easy example)?
I seem to remember a result classifying all universal $R$-matrices of $\mathbb ...
- 301
7
votes
0
answers
385
views
When is the character group scheme of a group scheme representable? (Affine Case)
While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...
- 171
6
votes
2
answers
210
views
groupring morphisms and bialgebra
Let $G_{1}$ and $G_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G_{1}]\rightarrow \mathbb{Z}[G_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G_{...
- 1,613
5
votes
1
answer
319
views
Malcev completion of free groups
Let $K$ be a field with $\operatorname{char} K=0$, $\hat{L}_n$ the complete free Lie algebra of $n$ variables $x_1,\dotsc,x_n$ and $\exp(\hat{L}_n)$ its associated group with the product given by BCH ...
- 985
5
votes
0
answers
92
views
$\text{Rep}(D_4)$ and its three fiber functors
It is well-known that the fusion category $\text{Rep}(D_4)$ of representations of the dihedral group $D_4$ of order 8 admits three distinct fiber functors. Therefore, there are three different Hopf ...
5
votes
0
answers
172
views
Are the symmetric groups integrable as Hopf algebras?
Let $G$ be a group. For $g,h \in G$, let $[g,h]=g^{-1}h^{-1}gh$ be a commutator. The normal subgroup $G' = \langle [g,h] \ | \ g,h \in G \rangle$ is called the commutator subgroup or derived subgroup. ...
4
votes
0
answers
67
views
Is the associated grouplike $\gamma=uS(u)^{-1}$ of a quasi-triangular Hopf algebra always the square of another grouplike?
Let $(H,R)$ be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements (I actually need it for $H$ fin. dim. pointed with $G(H)$ abelian). Let $...
- 1,813
3
votes
1
answer
426
views
Is Malcev completion an embedding?
The Malcev completion of an abstract group $G$ over a field $k$ of characteristic zero is defined by
$$\hat G = \mathbb{G}(\widehat{k[G]}) ,$$
the group-like part of the completed (by the augmentation ...
- 985
3
votes
1
answer
293
views
Cocyles for abelian extensions
Suppose we have an abelian extension of Hopf algebras,
$$k \rightarrow k^G \rightarrow A \rightarrow kF \rightarrow k.$$
According to the general theory there is a left action of $F$ on $G$ and a $...
- 887
3
votes
1
answer
298
views
In what way do exact sequences of Lie ideals integrate to the category of groups?
Please excuse, very naive question:
Suppose $g$ is a topological Lie algebra over Q and $G$ = $exp(g)$ the associated group
(take free group on formal symbols $exp(X)$, X $\in$ $G$, and impose all ...
- 904
3
votes
0
answers
325
views
Intuitive, elementary intros to Hopf algebras/monoids
Motivation:
I'm interested in understanding the role that noncrossing partitions play in Hopf algebras/monoids (HAs) as the components of the power series of the compositional inverse of formal ...
- 10.5k
3
votes
0
answers
122
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 ...
- 273
3
votes
0
answers
515
views
What happens geometrically when you take associated-graded (or complete, ...) of a group ring at its augmentation ideal?
I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
- 54.6k
2
votes
1
answer
1k
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 ...
- 743
1
vote
1
answer
108
views
Some quantities associated to finite dimensional Hopf algebras
let $(H,\Delta,m,s)$ be a Hopf algebra. To this Hopf algebra one can associate two obvious linear maps $T_H, S_H: H \to H $ with $T_H=m\circ \Delta,\quad S_H=s$.
Are there two finite dimensional ...
- 356
1
vote
1
answer
106
views
Cosemisimple pointed Hopf algebras
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Every cosemisimple pointed Hopf $\mathbb{K}$-algebra $A$ is easily seen to be cocommutative. Does this imply that $A$ is the ...
- 1,479
0
votes
1
answer
123
views
Orthogonal idempotents with sum equal to 1 in $k[G]$ span sub-Hopf algebra
Let $G$ be a finite group. Let $B$ be a set of orthogonal non-zero idempotents with $|B| \leq |G|$, s.t. $\sum_{b \in B}b =1_{kG}$. Is it known if $B$ spans a sub-Hopf algebra $kH \subseteq kG$?
- 1,813
0
votes
1
answer
184
views
Hopf algebra of representative k-valued functions of an abstract group
Let $G$ be an abstract group. If we can embed $G$ into a group $H$, in a way that we had $G$ and $H$ of the same Hopf algebra of representative $k$-valued functions ($R(G)\sim R(H)$ as Hopf algebras). ...
- 29