Questions tagged [hopf-algebras]
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 structure exists on the group algebra $k G$ of a finite group $G$.
541
questions
4
votes
1
answer
204
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 ...
- 71
3
votes
1
answer
114
views
An algebra map between Hopf algebras that does not commute with the counit
Let $(G,\Delta,\epsilon,S)$ be a Hopf algebra. Can there exist an algebra map $\phi:H \to H$ such that
$$
\epsilon(\phi(g)) \neq \epsilon(g), ~~~~~ \textrm{ for some } g \in G?
$$
Does the anti-pode ...
1
vote
0
answers
20
views
Stable Picard group of the tensor product of two Hopf algebras
Suppose we know the stable Picard groups (=Picard group of the stable module category) of two cocommunicative Hopf $k$-algebras $G$ and $H$. Is it possible to deduce the stable Picard group of $G\...
- 415
0
votes
0
answers
49
views
A name for "anti-symmetric" Frobenius algebras?
Let $V$ be an $N$-dim vector space and denote its exterior algebra by $\Lambda^{\ast}(V)$. The algebra $\Lambda$ has an obvious Frobenius algebra structure $B(-,-)$ given by wedgeing and then ...
4
votes
1
answer
174
views
Hopf algebra and coideal question
Let $A$ be a Hopf algebra (over a field). Consider a unital subalgebra $B\subseteq A$ with $\Delta(B)\subseteq B\otimes A$. Put
$$B^+:= B\cap \ker(\epsilon).$$
It can be shown that $B^+$ is a two-...
- 83
0
votes
0
answers
92
views
Comodules category
Let $G$ be an abstract group, under which conditions we may have equivalent (resp. isomorphic) categories $Mod_{G}$ and $Comod_{R(G)}$, of $G-$modules and $R(G)-$comodules, where, $R(G)$ stands for ...
- 29
1
vote
0
answers
78
views
Brauer trees that are Hopf algebras
Let $T$ be a Brauer tree with associated Brauer tree algebra $KT$ for some field $K$.
Question 1: For which Brauer trees does there exist a field $K$ such that $KT$ is a Hopf algebra (or more ...
- 24.4k
3
votes
1
answer
196
views
Classification of periodic Hopf algebras
Let $A$ be a finite dimensional algebra over a field $K$.
$A$ is called periodic if $A$ as an $A$-bimodule is a periodic module, that is $\Omega^n(A) \cong A$ for some $n \geq 1$. Being periodic ...
- 24.4k
0
votes
1
answer
143
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
1
vote
0
answers
67
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:
...
- 24.4k
1
vote
0
answers
84
views
Classifying of low-dimensional Frobenius algebras
Does there exist a classification of finite-dimensional Frobenius algebras in low dimensional cases. This question is motivated by the following analogous question for Hopf algebras.
1
vote
0
answers
57
views
Is there a coalgebraic definition of filtered algebras?
If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define
$$...
- 234
2
votes
0
answers
91
views
Is there a classical version of Yetter-Drinfeld modules?
One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$.
If we think ...
- 2,382
5
votes
0
answers
120
views
Equivalence of Yetter-Drinfeld modules to Drinfeld center: is there a purely categorical proof?
Let $H$ be an Hopf algebra over a field $k$, and let $\mathcal{C}$ be the monoidal category of left $H$-modules.
It is known that the Drinfeld center of $\mathcal{C}$ is equivalent (as a braided ...
- 2,382
5
votes
1
answer
162
views
States "absorbed" by a Haar idempotent on a compact quantum group
Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ absorbs $a$? Can we say ...
- 755
2
votes
0
answers
52
views
Quiver and relations for Hopf algebras associated to quiver algebras
Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$.
$A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via
$[x,y]=xy-yx$ and $x^{p}=x^p$. ...
- 24.4k
1
vote
0
answers
76
views
Lagrangian subcategories of (non-pointed) braided tensor categories
I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...
- 113
3
votes
0
answers
93
views
Is the Frobenius property invariant by Morita equivalence?
Kaplansky's sixth conjecture [Ka75] states that the dimension of a semisimple finite dimensional Hopf algebra over $\mathbb{C}$ is divisible by the dimension of its irreducible complex representations....
- 24.7k
0
votes
0
answers
163
views
Cup-product in cohomology and Hopf algebra
Let $X$ be a manifold and let its cohomology
$H^*(X;\mathbb{Z})=\bigoplus_{q=0}^\infty H^q(X;\mathbb{Z})$
be a module of finite type without $p^2$-torsion
for any prime integer $p$.
Assume that on
...
- 169
3
votes
2
answers
333
views
What is an example of a Frobenius algebra that is not Koszul?
What is an example of a Frobenius algebra that is not Koszul? Are there reasonable requirements for a Frobenius to be Koszul?
4
votes
1
answer
82
views
Bicrossed and bismash product of Hopf algebras
I have been recently studying different methods to construct Hopf algebras.
In Theorem IX.2.3 of "Quantum groups" by Kassel the bicrossed product of a pair of matched bialgebras (or Hopf ...
- 208
1
vote
1
answer
102
views
Two (or less) filtrations on coenveloping coalgebra
Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) ...
- 3,875
2
votes
1
answer
99
views
Coproduct for a Frobenius algebra
The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
13
votes
1
answer
389
views
Hopf algebras vs. Kac algebras
I recently came across Kac algebra. They are roughly Hopf algebras and $C^*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra ...
- 208
4
votes
2
answers
546
views
A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?
A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
1
vote
0
answers
55
views
Problem in understanding Theorem $6.2.9$ from Chari and Pressley
The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...
- 179
3
votes
1
answer
133
views
Understanding definition of quantization of a Poisson-Hopf algebra
I am going through the chapter Quantization of Lie bialgebras from the book A Guide to Quantum Groups by Chari and Pressley. There I found a notion called Quantization which deals with deformations of ...
- 179
2
votes
0
answers
81
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 ...
- 24.4k
1
vote
1
answer
130
views
Equivariant description of indecomposable elements in shuffle algebra
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Tor{Tor}$Let's suppose $V$ is a $k$-vector space equipped with its standard (left) $\GL (V)$-action. The shuffle algebra is the graded dual of the ...
- 381
7
votes
0
answers
298
views
How to define $U_q \mathfrak{g}$ without generators and relations?
I'm trying to learn something about quantum groups. The related definitions tend to consist of formulas which are not extremely intuitive, on the first glance. So I wonder how the amount of formulas ...
- 475
1
vote
0
answers
92
views
When are Brauer tree algebras Hopf algebras?
Question 1: Which Brauer tree algebras are Hopf algebras?
For example every representation-finite group algebra is a Brauer tree algebra and thus a Hopf algebra, but not every Brauer tree algebra ...
- 24.4k
6
votes
0
answers
118
views
What about Hopf algebra and fusion structures for intertwiner algebras?
Let $G$ be a complex, reductive group and let $V_1, \dotsc, V_r$ be a collection of finite dimensional, irreducible complex
representations of $G$. Let $\mathcal{A} = \mathrm{End}_G(V_1 \otimes \dotsb ...
- 1,757
1
vote
0
answers
38
views
Representation-finite blocks of Hopf algebras up to derived equivalence
Question: Which representation-finite selfinjective algebra is derived equivalent to a block of a (finite dimensional) Hopf algebra?
Famous examples are all Brauer tree algebras. Are there more ...
- 24.4k
7
votes
1
answer
267
views
Does the category of commutative and cocommutative Hopf algebras have enough injectives?
It is well-known that the category of commutative and cocommutative Hopf algebras is abelian (see https://arxiv.org/abs/1502.04001v2 and its references). But does it have enough injectives? What about ...
- 2,991
3
votes
0
answers
55
views
Is there a condition such that the $A$ action of a $A \rtimes H$-module is a restriction of the $H$-action?
Let $H$ be a Hopf algebra and $A$ a subalgebra of $H$ such that $A$ is a left coideal of $H$ (that is $\Delta(A) \subset H \otimes A$) and $A$ is preserved by the adjoint action of $H$.
Consider now ...
- 377
2
votes
1
answer
222
views
Primitive elements in the universal enveloping algebra of Lie superalgebra
Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
1
vote
0
answers
38
views
A finite $H$-module algebra is necessarily inner
Let $H$ be a finite-dimensional Hopf algebra over a field $k$, and $A$ an $H$-module algebra.
If $A$ is finite-dimensional over $k$, is the action of $H$ on $A$ necessarily inner?
If this is not true ...
- 851
4
votes
1
answer
223
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 ...
- 980
11
votes
0
answers
162
views
Natural knot homology
All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
9
votes
1
answer
421
views
Hopf algebra with a non-invertible antipode
What is an example of a Hopf algebra with a non-invertible antipode?
- 125
7
votes
1
answer
231
views
Group-like elements in quotients of group rings
$\DeclareMathOperator\Gr{Gr}$Let $R$ be a local ring, let $A$ be a finite abelian group, and let $I$ be a Hopf ideal of the ring $R[A]$. The quotient $R[A]\twoheadrightarrow R[A]/I$ induces a map on ...
- 91
3
votes
0
answers
93
views
How to interpret compositional diagrams for quantum sets algebraically
$\newcommand{\id}{\mathrm{id}}$My reference for this post is Musto, Reutter and Verdon's A compositional approach to quantum functions, arXiv:1711.07945. Questions are in bold below. Allow me to begin ...
- 59
1
vote
0
answers
88
views
Exterior algebra of free modules over Hopf algebras
Let $H$ be a commutative, cocommutative Hopf algebra over a field $\mathbb{K}$, and $M$ a free Hopf module over $H$. Is the exterior algebra $\Lambda^k_\mathbb{K} M$ with the diagonal $H$-action
$$h \...
1
vote
0
answers
343
views
Combinatorial proof of a matrix equation
I'm looking for combinatorial proofs (using, e.g., trees) of the following particular matrix equation $(I)$ and also combinatoric operational analogs of its solution via matrix inversion and/or Cramer'...
- 9,149
2
votes
1
answer
138
views
Timmerman's "An invitation to quantum groups and duality" corollary 3.2.8
Consider the following propositions of Timmerman's book "An invitation to quantum groups and duality":
I do not understand the equivalence
$$h(\mathfrak{C}(\delta_V)S(\mathfrak{C}(\delta_W))...
- 83
3
votes
1
answer
195
views
Characterising algebraic compact quantum groups among Hopf $^*$-algebras
Let $(A, \Delta)$ be a Hopf $^*$-algebra. Assume that $\{u^\alpha\}_{\alpha \in I}$ is a maximal collection of pairwise inequivalent irreducible unitary corepresentation matrices. I want to show that
$...
- 83
4
votes
2
answers
128
views
If $\operatorname{Hom}(\delta_V, \delta_W) = 0$, then $\mathfrak{C}(\delta_V) \cap \mathfrak{C}(\delta_W) = 0.$
Let $(A, \Delta)$ be a Hopf $^*$-algebra and $\delta_V: V \to V \otimes A$ and $\delta_W: W \to W \otimes A$ be two corepresentations of $(A, \Delta).$
Assume that the space of intertwiners $\...
- 83
1
vote
1
answer
85
views
Orthogonality relations for Haar state and antipode (Timmerman)
Consider the following proposition from Timmerman's "An invitation to quantum groups and duality":
I am having trouble seeing why the boxed equations are true (Note that on the left the ...
- 83
3
votes
1
answer
251
views
Is there a definition of Heisenberg double for quasi-Hopf algebras?
$\newcommand{\dmod}{\text{-}\mathrm{mod}}$Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $k$ and $A\dmod$ be the category of finite dimensional left $A$-modules.
Since $A\dmod$ is a ...
- 113
7
votes
2
answers
283
views
Different Bialgebra/Hopf algebra structures on coalgebras
Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
- 305