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.
521
questions
0
votes
1
answer
115
views
Invertibility of quasitriangular Hopf algebra element using Sweedler notation
The question concerns part of a theorem in the book Foundations of Quantum Group Theory, Shahn Majid (Cambridge University Press, 1995). More specifically, Theorem 2.3.4 (p.55-57) which I'll rewrite ...
- 15
1
vote
1
answer
85
views
Steenrod's interpretation of the Hopf invariant. [closed]
Where can I read about Steenrod's interpretation of the Hopf invariant?
Is there any reference?
user933798
2
votes
0
answers
34
views
Hopf-Galois structures of cyclic type on a dihedral or quaternionic extension
Let $L/K$ be a dihedral or quaternionic finite field extension, that is such that $Gal(L/K)$ is either a dihedral or a quaternion group.
How many Hopf-Galois structures of cyclic type are there on ...
- 33
2
votes
1
answer
295
views
About the structure of a Hopf algebra on universal enveloping algebras of Lie algebras
We know that the universal enveloping algebra construction provides a functor from Lie algebras to cocommutative Hopf algebras which is left adjoint to the primitive functor. Furthermore, if we ...
- 1,316
2
votes
0
answers
55
views
For an algebraic group $G$, a $G$-module induces a $Dist(G)$-module
I'm reading Representation of Algebraic Groups of Jantzen about Distribution of Algebra. In chapter 7, page 103, 7.11, the author is stating that if $G$ is a group scheme over $k$, then any $G$-module ...
- 4,825
-1
votes
1
answer
55
views
Question about left Hopf-modules
I'm reading Radford's text on Hopf Algebras right now and I'm a bit confused about the definition of a left A-Hopf algebra. The definition given in the book is:
Let $A$ be a $\Bbbk$-bialgebra. A $\...
- 11
1
vote
1
answer
163
views
Character group of diagonalizable group scheme
Suppose $k$ is an integral domain, $\Lambda$ is a commutative group, $k[\Lambda]$ is the corresponding group ring and $\text{Diag}(\Lambda) = \text{Spec}(k[\Lambda])$ is the diagonalizable group ...
- 4,825
1
vote
1
answer
72
views
Proof that any extension that is Galois in the classical sense is also Hopf Galois
The usual action of $G$ on $L$ is by automorphisms that fix $\mathcal{K}$. Explicitly, for any $g\in G, l,m\in L, k\in \mathcal{K}$
\begin{align*}
g(l+m)&=g(l)+g(m)\\
g(lm)&=g(l)g(n)\\
g(k)&...
2
votes
1
answer
155
views
Lie operator is left exact
In page 190, part (c) of the book "Algebraic groups, the theory of group schemes of finite type over a field" of Milne, there's a part stating that:
An exact sequence of algebraic groups $e ...
- 4,825
6
votes
1
answer
428
views
Counterexample to "kernel determines image"
Working over a base field, there is a typical homomorphism theorem for affine algebraic groups ensuring that any two homomporphisms $G \to H_1$, $G \to H_2$ with the same kernel in $G$ have isomorphic ...
- 7,397
0
votes
1
answer
45
views
Is the unit section of a finite flat commutative group scheme determined by sending group-like elements to 1?
Let $R$ be a ring and $\mathop{\mathrm{Spec}}A$ be a finite flat commutative group scheme over $\mathop{\mathrm{Spec}}R$ so that the theory of Cartier duality applies.
Denote $s:R\to A,\ m:A\to A\...
- 1,723
5
votes
2
answers
378
views
Distributions of a group scheme as differential operators.
I'm trying to understand the distributions on an affine group scheme act as differential operators. Let $X$ be a scheme and $G$ a group scheme both affine over some commutative ring $k$. Suppose $\...
4
votes
1
answer
55
views
For two modules over a Hopf algebra $H$, are the module homomorphisms the same as the $H$-invariant linear maps?
Let $H$ be a Hopf algebra over a field $k$ and $V, W$ two $H$-modules. The antipode and comultiplication on $H$ allow us to turn $\mathrm{Hom}_k(V, W)$ into a $H$-module by setting
$$
(h \cdot f)(v) = ...
- 3,587
1
vote
1
answer
77
views
Axioms of a coalgebra restated using Sweedler's notation
I'm struggling with understanding manipulation using Sweedler's notation at a very fundamental level. I don't understand the equivalence of the axioms of coalgebras in the standard notation
[Coproduct ...
0
votes
1
answer
42
views
Let B be a bialgebra. Show that the following are equivalent
Let B be a bialgebra. Show that the following are equivalent:
B is a Hopf algebra.
The maps $T_1$, $T_2$: $B \otimes B \to B \otimes B$ determined by $T_1(a \otimes b) = \sum a_{(1)} \otimes a_{(2)} ...
- 367
1
vote
0
answers
110
views
Hopf "algebroid" structure of a groupoid convolution algebra?
To male thinks simple as possible, lets say we have a discrete group $G.$ Then the then the group algebra $\mathbb{C}[G]$ (of finitely supported complex valued functions on $G$) has a convolution and ...
- 18.3k
1
vote
0
answers
51
views
Is every sub-Hopf algebra a Frobenius extension?
Recall that a ring extension $S \subset R$ (i.e. fancy words for saying that $S$ is a subring of $R$) is called a Frobenius extension if the restriction functor $\operatorname{Res}:R\text{-mod} \to S\...
- 2,075
2
votes
1
answer
126
views
Is the kernel of an action of a Hopf algebra on an algebra a biideal?
S.Dascalescu, C.Nastasescu and S.Raianu define the action of a Hopf algebra $H$ on an (associative) algebra $A$ as a map $H\times A\owns (h,a)\mapsto h\cdot a\in A$ which
is an action of $H$ on $A$ ...
- 2,445
4
votes
1
answer
186
views
If $H$ is a finite-dimensional Hopf algebra, then the antipode is bijective?
If I’m given a finite-dimensional Hopf algebra $H$, how do I show the antipode is bijective? It's obvious that if we prove either injective or surjective, we get the other one for free since $H$ is ...
- 367
1
vote
1
answer
94
views
The Coproduct of the Identity element
For a general Hopf algebra $H$, the coproduct of the identity element is
$$\Delta(1) = 1\otimes 1.$$
Now for a finite group $G$ and a field $k$, $k(G)$ forms a Hopf algebra with basis $\{\delta_{g}| g\...
- 23
1
vote
1
answer
111
views
How to prove the reduced comultiplication of a coaugmented coalgebra is coassociative?
a coalgebra over a field $k$ is a vector space $C$ over $k$ together with $k$-linear maps
$$\text{comultiplication } \Delta: C \to C\otimes_k C \text{ and} $$
$$\text{counit } \epsilon: C \to k$$ ...
- 1,054
3
votes
0
answers
94
views
Show that a certain element is a linear combination of tensors
Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that the map
$$T: A \otimes A \to A \otimes A: a \otimes b \mapsto \Delta(a)(1 \otimes b)$$
is surjective.
We write $\Delta(...
user839372
4
votes
0
answers
83
views
Do there exist any algebras in which we cannot take tensor product?
Let's suppose that we have an algebra $\mathcal{A}$ (I don't really care whether it's a unital one or associative). From studying Lie algebras and some of their generalizations, I am used to be able ...
- 403
7
votes
1
answer
246
views
Yoneda's lemma: group morphisms give Hopf-algebra morphisms
Let $k$ be a commutative ring. Let $\text{Alg}$ be the category of commutative $k$-algebras and $\text{CHopf}$ the category of commutative Hopf-algebras. Let us also write $[\text{Alg}, \text{Grp}]$ ...
user839372
4
votes
1
answer
869
views
Hopf algebra structure on universal enveloping algebra?
Let $\mathfrak g$ be a Lie algebra. Show that on $U(\mathfrak g)$ (universal enveloping algebra) there is a natural Hopf algebra structure induced by the Hopf algebra structure on the tensor algebra $...
- 367
2
votes
0
answers
53
views
Isomorphism between two Hopf algebras
Let $k$ be a field, over which we consider algebras and coalgebras. A $k$-coalgebra is a comonoid object in $k$-modules, and a $k$-algebra is a monoid object in $k$-modules. A $k$-bialgebra is ...
user900250
1
vote
0
answers
36
views
cofinite subspaces in weak* topology
I am learning hopf algebras by D.E Radford, questions about cofiniteness occur to me but I can't find out the answers.
My question is
Let $U$ be a vector space over a field $k$. Is there any subspace ...
- 103
1
vote
1
answer
84
views
Comultiplication arising as pullback for group representations
Let $V$ and $W$ be $k[G]$-modules for an algebraically closed field, $k$ and $G$ a group. Now I know that $V\otimes W$ has a $k[G]\otimes k[G]$-module structure, and because $k[G]$ is in fact a Hopf ...
- 1,766
1
vote
0
answers
85
views
What are other examples of pivotal non-spherical categories?
Apparently, an example of a pivotal non-spherical category can be found in the world of Hopf algebras/representation theory:
Let $(H, \omega)$ be a pivotal non-spherical Hopf algebra with pivot $\...
- 3,018
0
votes
1
answer
33
views
Showing that $U(\mathfrak{sl}_2)$ is a coalgebra
We know that there is a coalgebra structure on $U(\mathfrak{sl}_2)$ as follows for any $z\in \mathfrak{sl}_2$:
$$\Delta(z)=1\otimes z+z\otimes 1, \qquad \epsilon(z)=0.$$
Can someone be so kind to ...
- 560
2
votes
0
answers
45
views
Bialgebras with rigid representation categories
Everything is finite-dimensional over a field $k$.
Let $B$ be a bialgebra with $B\text{-mod}$ its category of modules.
Suppose now that we fix a rigid structure on $B\text{-mod}$ such that the dual of ...
- 2,075
5
votes
0
answers
178
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 ...
- 2,445
2
votes
1
answer
140
views
Automorphism of tensor product of Hopf algebras
Let $A$ be a commutative Hopf algebra over the commutative ring $R$, that is we have comultiplication $\Delta:A\to A\otimes_{R}A$, counit $\epsilon: A\to R$ and coinverse (or antipode) $S: A\to A$. I ...
- 304
1
vote
1
answer
111
views
Exercise 2.11 of Waterhouse about Hopf algebra
I asked part (c) of this exercise already in this link Categorization of Group Scheme of rank 2. But I still have some difficulty solving next parts of this exercise. So I hope anyone can help me to ...
- 4,825
2
votes
1
answer
93
views
Categorization of Group Scheme of rank 2
I got stuck on this Exercise 2.11 in the book Introduction to Affine Group Scheme of Waterhouse. I really appreciate if anyone can give me a hint.
Let $A$ be a Hopf algebra over $k$ (a base ring) ...
- 4,825
2
votes
1
answer
124
views
Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$ mod $I \otimes I $ for all $x \in I$.
I got stuck on this problem, so if anyone can give me a hint on this, I really appreciate.
Let $I$ be the augmentation ideal in Hopf algebra $A$. Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$
mod $...
- 4,825
4
votes
0
answers
82
views
Dimension counting in exact sequences of Hopf algebras
It is known that the category of commutative and cocommutative Hopf algebras over a field $k$ is an abelian category. So we can talk about exact sequences. Reading a paper I found the following ...
- 2,109
1
vote
0
answers
91
views
End of a category
I am trying to prove Proposition 7.2 from this paper:
$\textbf{Proposition 7.2:}$ Let $A$ be a finite-dimensional quasi-Hopf algebra over a field $\mathbb{K}$. The end object $\Gamma$ in $\textbf{Rep ...
- 2,361
2
votes
1
answer
213
views
Artin-Wedderburn: Decomposition of a semisimple Dual Hopf algebra
1. Context
My lecture notes prove that any cocommutative finite-dimensional Hopf algebra over a field $k$ of characteristic zero is semisimple and cosemisimple.
They try to argue from there that any ...
- 3,018
3
votes
1
answer
142
views
Is there a consistent way to get all possible coproducts?
Let's illustrate the problem with an example. Consider an algebra of polynomials in one variable $1,x,x^2,\ldots$ with the product $\nabla (x^i,x^j) = x^{i+j}$. Then, reversing arrows in the diagram
$\...
- 403
4
votes
0
answers
93
views
String diagram of left $H$-action on $\mathrm{Hom}(U,V)$
1. Context
Let $\mathbb k$ be a field. Let $H$ be a $\mathbb{k}$-Hopf algebra. Let $U, V$ be objects in the category $H\text{-}\mathrm{mod}$ of left $H$-modules. (In particular they are $\mathbb{k}$-...
- 3,018
2
votes
1
answer
182
views
Coproduct of the Drinfeld element?
1. Problem
Let $(H,R)$ be a quasitriangular, finite-dimensional Hopf algebra with antipode $S$ and coproduct $\Delta$. Using Sweedler notation define the Drinfeld element $u$ as $u:= S(R_{(2)})R_{(1)}$...
- 3,018
9
votes
0
answers
261
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.)...
- 3,018
1
vote
0
answers
51
views
Quasitriangular structure for infinitely dimensional Drinfeld double?
It is well known that the Drinfeld double $D(H)=H^{*op}\otimes H$ of an Hopf algebra $H$ admits a quasitriangular structure. When $H$ is finitely dimensional, the $R$-matrix can be given by
$$R=\sum e^...
- 393
2
votes
1
answer
221
views
Examples of finite-dimensional, semisimple, non-separable $k$-algebras?
1. Motivation
Let $k$ be a field.
Apparently, any separable $k$-algebra is finite-dimensional and semisimple.
Using Maschke's theorem for Hopf algebras, one can prove that for Hopf algebras the ...
- 3,018
5
votes
1
answer
209
views
Relation: Module structure on the dual and braiding?
1. Context
Let $H$ be a Hopf algebra over a field $\mathbb k$. Let $(V, p)$ be a finite dimensional (left) $H$-module.
We want to endow its dual vector space $V^*$ with the structure of a (left) $H$-...
- 3,018
1
vote
1
answer
280
views
Every finite-dimensional Hopf algebra is isomorphic to a dual Hopf algebra?
1. Context
Let $H$ be a Hopf algebra over a field $\mathbb{k}$. Denote by $I_l(H)$, $I_r(H)$ its space of left integrals/right integrals respectively.
I am studying a proof of the following ...
- 3,018
2
votes
1
answer
255
views
Classification of non-commutative, non-cocommutative Hopf algebras
1. Context
Apparently, Sweedler's Hopf algebra (presented in 1969) was the first known example of a non-commutative, non-cocommutative Hopf algebra.
More generally, the $N^2$-dimensional Taft-Hopf ...
- 3,018
1
vote
1
answer
131
views
Unimodularity: How are these notions related?
1. Definitions
We call a Hopf algebra $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right integrals $I_r(H)$.
We call a square integer matrix $M$ unimodular if $det(...
- 3,018
8
votes
1
answer
327
views
Example of a cocommutative, non-unimodular Hopf algebra?
1. Definitions: Unimodularity and cocommutativity
Let $H$ be a Hopf algebra over a field $\mathbb k$.
We call $H$ unimodular if the space of left integrals $I_l(H)$ is equal to the space of right ...
- 3,018