Questions tagged [hopf-algebras]
For questions about Hopf algebras and related concepts, such as quantum groups. The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science.
192
questions with no upvoted or accepted answers
21
votes
0
answers
512
views
Why is the Quasitriangular Hopf algebra called "Quasitriangular"?
The precise definition of a Quasitriangular Hopf algebra can be found on wikipedia.
What is the reason behind the word "Quasitriangular"?
Is it because the R-matrix is a triangular matrix, ...
- 19.8k
19
votes
0
answers
380
views
Hopf-like monoid in $(\Bbb{Set}, \times)$
I am looking for a nontrivial example of the following:
Let a monoid $A$ be given with unit $e$, and two of its distinguished disjoint submonoids $B_1$ and $B_2$ (s.t. $B_1\cap B_2=\{e\}$), endowed ...
- 91.1k
14
votes
0
answers
394
views
Different notions of q-numbers
It seems that most of the literature dealing with q-analogs defines q-numbers according to
$$[n]_q\equiv \frac{q^n-1}{q-1}.$$
Even Mathematica uses this definition: with the built-in function QGamma ...
- 283
9
votes
0
answers
264
views
Monodromy element: Why that name?
Let $(H,R)$ be a quasitriangular Hopf algebra, i.e. $R$ is a choice of an universal $R$-matrix for the Hopf algebra H. (You can find a definition of the term quasitriangular Hopf algebra on wikipedia.)...
- 3,068
7
votes
0
answers
124
views
What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?
Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that ...
- 2,692
6
votes
0
answers
142
views
Is this a valid q-deformation of $SU(2)$ via the $SU(3)$ Lie algebra?
In my physics research, I stumbled upon a rather simple deformation of the $SU(2)$ algebra, and I was wondering whether it qualifies as a `$q$-deformed Lie algebra' (a notion which is unfamiliar to me)...
- 1,436
6
votes
0
answers
1k
views
What are $q$-deformations?
This question has already appeared in a lot of different ways and here is another one.
First of all, many people know the typical quantum group $U_q(\mathfrak{sl}_2)$ by generators and relations. ...
- 12.6k
6
votes
0
answers
325
views
Do complete Hopf algebras have an antipode?
I am reading Quillen's paper on rational homotopy theory.
In appendix A of this paper Quillen defines a notion he calls complete Hopf algebras. These are certain cocommutative bialgebra structures (...
- 1,439
5
votes
0
answers
181
views
Epimorphisms and monomorphisms in the categories of Hopf algebras
From this paper I learned that in the category $\operatorname{HopfAlg}$ of Hopf algebras over a field $k$ epimorphisms are not necessary surjective and monomorphisms are not necessary injective. Can ...
- 2,463
5
votes
0
answers
124
views
Coideals in the grouplike colagebra are spanned by differences
Let $k$ be a field, and let $S$ be a nonempty set. Let $k[S]$ be the grouplike coalgebra of $S$ over $k$, i.e. the free vector space with basis $S$ equipped with the coproduct $\Delta(s)=s\otimes s$ ...
- 1,279
5
votes
0
answers
156
views
On group graded algebras and Brauer groups
I was reading the paper "Algebras graded by groups" by Knus. I want to test and further my understanding of the paper by asking several questions.
Since the paper is not readily available I ...
- 12.6k
5
votes
0
answers
99
views
An interpretation of this construction giving an operad from a bialgebra?
Let $A$ be a cocommutative bialgebra object (or even a Hopf algebra) in a symmetric monoidal category. Define an operad $\mathtt{P}_A$ by $\mathtt{P}_A(r) = A^{\otimes r}$ (so that $\mathtt{P}_A(0) = ...
- 54.7k
5
votes
0
answers
577
views
Do we have $\delta(ab)=\delta(a)\delta(b)$ implies $\Delta(cd)=\Delta(c)\Delta(d)$?
Assume that $B$ is an algebra which is also a coalgebras (we do not assume that $B$ is a bialgebra: we do not assume $\Delta(cd)=\Delta(c)\Delta(d)$). Assume that $A$ is $B$-comodule algebra. Then we ...
- 14.6k
5
votes
0
answers
801
views
Integral Homology of $BU$
We know that the integral cohomology of $BU$, $H^*(BU) = Z[c_1,c_2,...]$ where $|c_i| = 2i$ is the $i$th Chern class, with coproduct $\Delta c_i = \Sigma c_j\otimes c_{i-j}$.
And at almost ...
- 206
4
votes
0
answers
56
views
A diagrammatic proof of antipode being antihomomorphism in a Hopf algebra
Let $(H, \mu, \eta, \Delta, \epsilon, S)$ be a Hopf algebra with $S: H \to H$ denoting the antipode. By definition, $S$ is the convolution inverse of $1: H \to H$ in $\operatorname{End}(H)$, with the ...
- 1,350