All Questions
Tagged with hopf-algebras reference-request
43 questions
19
votes
2
answers
1k
views
Hopf algebra reference
I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf ...
- 156k
17
votes
2
answers
1k
views
What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?
Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
- 8,167
16
votes
1
answer
1k
views
Lagrange's theorem for Hopf algebras
Under what conditions is a Hopf algebra free over any of its sub-Hopf algebras?
I am reading "Hopf algebras and their actions on rings" by Susan Montgomery, specifically chapter 3. Lagrange's theorem ...
- 85.6k
14
votes
1
answer
1k
views
2-cocycle twists of braided Hopf algebras
2-cocycle twists of Hopf algebras
Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map
$$ f: H \otimes H \to k$$
such that
$$ f(x_{(1)},y_{(1)})f(x_{(2)} y_{(...
- 8,559
13
votes
2
answers
1k
views
Hopf Algebra for a physicist
Hello,
for my bachelor's thesis I need to understand the Hopf Algebra of Feynman Diagrams. As I have only litte knowledge in Algebra by now I wanted to ask where I could start and what preknowledge I ...
- 133
13
votes
0
answers
332
views
Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
11
votes
2
answers
628
views
A quantum Grothendieck group?
Question 1: Given a co-commutative bialgebra, does there exist a sort of Grothendieck group type construction? Presumably this should take the form of a functor from co-commutative bialgebras to hopf ...
- 1,411
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
10
votes
1
answer
925
views
The coproduct on the cohomology of a Hopf algebra
If $H$ is a Hopf algebra over a field $k$, the Hopf algebra cohomology $\mathrm{Ext}_H(k,k)$ is —as usual— an algebra, but the Hopfness of $H$ turns it into a Hopf algebra.
Is there a reference on ...
- 47.3k
10
votes
3
answers
856
views
Tannaka-Krein duality in Chari-Pressley's book
I am not sure that this was not discussed before, so excuse me in this case. This can be considered as a special case of my previous question here.
V.Chari and A.N.Pressley in their "Guide to Quantum ...
- 7,426
10
votes
1
answer
842
views
Is there a "correct" general setting for the principle: "tensoring any object with a projective object yields another projective"?
Apparently this principle was first formulated for left modules over the group algebra $A=kG$ of a finite group, where $k$ is a field of characteristic $p>0$ dividing $|G|$. (See Exercise 2 on p. ...
- 52.9k
9
votes
2
answers
421
views
Hopf algebra kernels vs. algebra kernels
Let $f: H_1 \rightarrow H_2$ be a map of graded connected cocommutative Hopf algebras over a perfect field. Let $H \subset H_1$ be the Hopf algebra kernel of $f$, and let $I \subset H_1$ be the ...
- 11.1k
9
votes
2
answers
650
views
Definition of subcoalgebra over a commutative ring
Let $k$ be commutative ring and $(C, \Delta)$ be a coalgebra over $k$. Let $D$ be a $k$-submodule of $C$.
Notes I'm reading give the following definition:
$D$ is called subcoalgebra of $C$ if the ...
user167952
9
votes
3
answers
602
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 ...
- 527
6
votes
1
answer
778
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 ...
- 993
6
votes
3
answers
442
views
Commutative and Cocommutative Quantum Groups
I am using this definition:
An algebra of functions on a finite quantum group $\mathbb{G}$ is a finite dimensional $C^\ast$-Hopf algebra $A=:F(\mathbb{G})$.
I have the following (very well known --...
- 1,037
6
votes
1
answer
362
views
Comparing Hochschild (co)homology for algebras and coalgebras
Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes
$$M\otimes A^{\otimes n}$$
and
$$\...
- 518
6
votes
1
answer
227
views
Quantum group representations from (convolution) matrix units?
Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...
- 1,037
6
votes
0
answers
332
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, $\mathcal{...
- 3,738
6
votes
0
answers
134
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 $...
- 1,639
5
votes
1
answer
318
views
Up to date summary on semisimple Hopf algebra over $\mathbb{C}$
Question: Is there an up to date summary of results on the classification of semisimple Hopf algebras over $\mathbb{C}$ (or a field of characteristic 0)?
Here are some questions I wonder about:
...
- 26.5k
5
votes
1
answer
579
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(\...
user30211
5
votes
1
answer
128
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 ...
- 15.8k
5
votes
1
answer
165
views
Reference requests : Presentation of the braided dual of $U_q(\frak{sl_2})$
I am interested in the braided dual of the quantum group $U_q(\frak{sl_2})$. This is the algebra generated by the matrix coefficients but where the multiplication is twisted by an action of the $R$-...
- 51
5
votes
1
answer
215
views
Classification of $\operatorname{Rep}D(H)$
Question
Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
- 5,230
5
votes
1
answer
578
views
Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative functions
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}...
- 718
5
votes
0
answers
73
views
Corepresentations on Hilbert modules
In the seminal paper "Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbres" by Baaj and Skandalis, we find the following very general definition of what a corepresentation ...
- 18.7k
5
votes
0
answers
218
views
Lusztig's completion for universal enveloping algebra
In Arkhipov, Bezrukavnikov and Ginzburg's paper "Quantum Groups, the loop Grassmannian and the Springer resolution", they mentioned that Lusztig introduced a certain completion for universal ...
- 677
5
votes
0
answers
217
views
DGLA related to the deformation of hopf Algebras
Recently I was considering Hopf algebras and Drinfeld's twists. I stumbled upon
a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by ...
- 397
4
votes
1
answer
233
views
Mayer–Vietoris sequence for coproduct of Hopf algebras
Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in Agore - Categorical Constructions for Hopf ...
- 1,374
4
votes
1
answer
763
views
Further developments of Cartier–Gabriel–Kostant–Milnor–Moore Structure Theorem for cocommutative Hopf algebras
A very well-known theorem in Hopf algebra theory (see, for example, Lorenz - A tour of representation theory or the EGNO book (Etingof, Gelaki, Nikshych, and Ostrik - Tensor categories)) states that ...
- 718
4
votes
0
answers
168
views
Representations of $C\left(SO_q(n)\right)$
A complete classification of irreducible representations of the $C^*$-algebra $C(G_q)$, where $G_q$ is the $q$-deformation of a classical simply connected semisimple compact Lie group, was provided by ...
- 73
3
votes
1
answer
344
views
Reference for the Hecke relation for the universal R-matrix
I've come across a reference in a paper to the
Hecke relation for the universal R-matrix of a quasi-triangular Hopf algebra.
I've looked around, standard references, online etc, but can't seem to ...
- 1,479
3
votes
1
answer
292
views
Is the kernel of an action of a Hopf algebra on an algebra a biideal?
I think, this must be simple, but I am not a specialist in this field, so excuse me. I asked this a week ago at MSE, but without success.
S.Dascalescu, C.Nastasescu and S.Raianu define the action of a ...
- 7,426
3
votes
2
answers
323
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 ...
- 5,565
3
votes
1
answer
148
views
Integrals and finite dimensionality in braided Hopf algebras
Let $H$ be a Hopf algebra with invertible antipode. Let $A$ be a braided Hopf algebra in the Yetter-Drinfeld category ${}_H^H\mathcal{YD}$ over $H$.
A nonzero left integral in $A$ is a nonzero ...
- 1,066
3
votes
0
answers
752
views
Where can I find Drinfeld's original papers on quantum groups?
Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $U_h(\mathfrak{g})$ be the Drinfeld-Jimbo quantum group, i.e. the $\mathbb{C}[[h]]$-algebra topologically generated by $X_i,Y_i,H_i$ where $1\leq i\...
- 1,066
2
votes
1
answer
153
views
The shuffle algebra over the rationals is isomorphic to the polynomial algebra in the Lyndon words
On this wikipedia page is stated that over the rational numbers, the shuffle algebra (over a set $X$) is isomorphic to the polynomial algebra in the Lyndon words (on $X$). I was wondering if you can ...
- 85
2
votes
0
answers
84
views
Representation finite Hopf algebras up to stable equivalence
It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra.
Question: Is it true that every representation-finite Hopf algebra is stable ...
- 26.5k
2
votes
0
answers
52
views
Involutions on Hopf algebra crossed products
I am interested in a (cocycle) crossed product of a Hopf algebra $H$ with a (twisted) $H$-module algebra $A$, often denoted $H\#_\sigma A$, where $\sigma$ is the associated cocycle. This construction ...
- 18.7k
1
vote
1
answer
436
views
References For Important Hopf Algebras
Where can I find references that discuss important classes of Infinite Hopf Algebras. By important classes, I mean heavily used in research and of relevance to Hopf Algebraist(s),Physicists, Analysts(...
- 33
1
vote
0
answers
130
views
Can't parse a statement in an article on coalgebras and umbral calculus
This question is cross-posted from MSE.
I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", ...
- 491
0
votes
1
answer
1k
views
What is Taft algebra?
What is a Taft algebra? Is there any references about the original conception?
- 81