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 $G$.

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How a unitary corepresentation of a Hopf C*-algebra, deals with the antipode?

Let $\mathcal{A}$ be a Kac algebra (Hopf C*-algebra), with comultiplication $\delta$, counit $\epsilon$ and antipode $S$. The following definition comes from this paper (p51-52) of Izumi-Longo-...
Sebastien Palcoux's user avatar
3 votes
0 answers
134 views

integrals in Hopf algebras

Let $H$ be a Hopf algebra and $g$ be a group-like element in $H^{*}$. Define $L_{g}=\{x\in H\mid hx=g(h)x,~\forall h\in H\}$. If $H$ is not semisimple, when can we find a group-like element $g$ in $H^{...
sife's user avatar
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0 answers
263 views

Flatness over Hopf subalgebra

Let $A\subset B$ be flat Hopf algebras over a Dedekind ring $R$. J. Moore has proved in the article Compléments sur les algèbres de Hopf, John C. Moore, Séminaire Henri Cartan (1959-1960), Volume: ...
Phung Ho Hai's user avatar
16 votes
4 answers
1k views

If tensor product of representations is a representation, must we have a bialgebra?

Hopf algebras and bialgebras are sometimes introduced by saying that you've got an associative algebra $A$ and want to introduce the structure of an $A$-module on $V \otimes W$ where $V,W$ are $A$-...
Joshua Grochow's user avatar
6 votes
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236 views

Existence of a Kac algebra for a given fusion ring in a particular class

A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., d_{r}...
Sebastien Palcoux's user avatar
2 votes
0 answers
165 views

How simplify the pentagonal equation from two fusion rings?

A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...
Sebastien Palcoux's user avatar
5 votes
1 answer
328 views

Hopf algebra structure on the ring of quasisymmetric functions

I'm looking for a particular description of the Hopf algebra structure on the ring of quasisymmetric functions. Let me illustrate by giving this kind of description for the Hopf algebra of symmetric ...
Julian Rosen's user avatar
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Is it possible to define Hopf algebra as an object in the category of (associative) algebras?

I think this must be well-known, but I can't find references, so my apologies from the very beginning. Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ${...
Sergei Akbarov's user avatar
6 votes
0 answers
129 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 $...
Eric Rowell's user avatar
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How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two. Version 1: Let $(C,\otimes)$ be any monoidal ...
Theo Johnson-Freyd's user avatar
5 votes
1 answer
148 views

How to make Schubert calculus into a Hopf (actually a PCH) algbera?

The parallels between the formulas in Schubert calculus and in the theory of the representations of symmetric groups (par Geissinger-Zelevinsky) are so apparent (e.g. Giambelli formula), that one must ...
George's user avatar
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16 votes
1 answer
399 views

Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with. Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...
Noah Snyder's user avatar
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1 vote
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polynomial representation of $sl_{2}(k)$

Let $k$ be an algebraic closed field of characteristic 0. We write $$X=\left( \begin{array}{ccc} 0 & 1\\ 0 & 0\\ \end{array} \right),~~ Y=\left( \begin{array}{ccc} 0 & 0\\ 1 & 0\\ \...
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universal enveloping algebras and commutator subalgebras

Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$, $U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$ and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...
sife's user avatar
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14 votes
2 answers
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Quantum-Jimbo Algebras: Why Such Fuss About Roots of Unity?

Coming from a Lie algebraic background, I'm trying to branch onto quantum group theory. The divide I see all the time, is $q$ a root of unity or $q$ not a root of unity. I am wondering why is this? If ...
Milen Stamatov's user avatar
3 votes
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312 views

Quantum Coordinate Algebras at Roots of Unity and Non-Standard Irrep Types

Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible finite-...
Janos Erdmann's user avatar
3 votes
1 answer
313 views

Hopf Duals and Matrix Coefficients

One defines the finite dual of a Hopf algebra $H$ as $$ H^o := \{f \in H^* ~|~ f(I) = 0, \text{ for some ideal $I$ of $H$ with } \dim_C(H/I) < \infty \}. $$ As is well-known, $H^o$ has a well-...
Janos Erdmann's user avatar
2 votes
1 answer
241 views

Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations

The representations of the quantized enveloping algebra $U_q(\frak{sl}_n)$ are labeled by the positive weight lattice $P^+$ of the classical Lie algebra $\frak{sl}_n$. Moreover, the dual Hopf algebra $...
Antonio Nogueria's user avatar
11 votes
2 answers
1k views

Quantized Enveloping Algebras at $q=1$

As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation $$ [E,F] = \frac{K-K^{-1}}{q-q^{-1}}. $$ To address this problem, one has ...
Antonio Nogueria's user avatar
1 vote
0 answers
95 views

Degree of a commutator in a hyperalgebra or enveloping algebra

Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...
Chuck Hague's user avatar
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3 votes
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Hopf Algebra Pairings and Module-Comodule-Equivalences

Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left ...
Mihail Matrix's user avatar
1 vote
0 answers
171 views

Classification of Hopf algebra with exactly two 1-dimensional modules

Is there a classification of indecomposable non-semisimple finite dimensional Hopf algebras with exactly two 1 dimensional modules? If not, is there one when all simple modules are 1 dimensional and ...
trew's user avatar
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3 votes
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223 views

Hopf-Borel theorem over polynomial rings

There is a theorem of Borel saying that: For a connected cocommutative Hopf algebra $A$ over the perfect field $k$ such that underlying graded vector space is finite-dimensional over $k$ we have that ...
Alexander's user avatar
6 votes
1 answer
342 views

Is the "renormalized third comultiplication" on $\mathbf{Symm}$ integral?

Background: For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...
darij grinberg's user avatar
4 votes
2 answers
389 views

Different Hopf algebra structures on same graded algebra

I'm reading Hatcher's book on algebraic topology. In Section 3.C, he proves as Theorem 3C.4 that if $A$ is a graded commutative associative Hopf algebra over a field of characteristic $0$ and $A^n$ ...
Roger's user avatar
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5 votes
2 answers
503 views

Free cocommutative commutative Hopf monoids

I have some questions about generalizations of abelian groups, relative to symmetric monoidal categories. 1) Let $C$ be a cocomplete cartesian monoidal category with equalizers. I can show that the ...
Martin Brandenburg's user avatar
14 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 ...
understandhopf's user avatar
3 votes
2 answers
179 views

On local parameters at the origin in an algebraic group

Let $k$ be an algebraically closed field and $G$ an algebraic group over $k$ which is also a $k$-variety (so $G$ is integral, etc). Let $I$ be the ideal defining the identity $e \in G$ and let $\{ t_1,...
Chuck Hague's user avatar
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6 votes
2 answers
590 views

Classification of plethories over $\mathbb{Q}$

Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). ...
Martin Brandenburg's user avatar
2 votes
1 answer
326 views

Explicit Descriptions of $q-SO(2)$, and $q-Sp(2)$?

When ever I hear noncommutative geometers talking about quantum groups, it is usually $q-SU(2)$ that they are discussing. As a result there are many good and explicit generator and relation ...
Ago Szekeres's user avatar
16 votes
3 answers
717 views

Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras

Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. In which cases the conjecture is known ...
warren's user avatar
  • 161
11 votes
1 answer
568 views

Hopf algebras and bijective antipodes

By a theorem of Larson and Sweedler, the antipode of every finite-dimensional Hopf algebra is bijective. My question is the following: Is it true that in every noetherian Hopf algebra the antipode ...
warren's user avatar
  • 275
16 votes
3 answers
2k views

Classification of Hopf algebras (state of the art)

I assume that the classification of (certain families of) Hopf algebras is still an open problem, am I right? My question is the following: What is the current state of the art? What is known about ...
warren's user avatar
  • 275
2 votes
1 answer
105 views

Zero Actions on a Hopf Module Preserved Under the AntiPode?

Let $H$ be a Hopf algebra, and $(M,\triangleleft)$ a $H$-module. Now for $m \in M$, and $h \in H$, then is it true in general that $$ m \triangleleft h = 0 ~~~ \implies ~~~~m \triangleleft S(h) = 0? $...
Dyke Acland's user avatar
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7 votes
1 answer
506 views

Fusion category and Hopf algebra

Let $H$ be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. Further, let $K\subseteq H$ be a normal Hopf subalgebra. As we all know, $H$ then can be reconstructed ...
Jingcheng Dong's user avatar
2 votes
1 answer
335 views

Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...
Mihail Matrix's user avatar
30 votes
2 answers
2k views

quantum groups... not via presentations

Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a_{ij}$, the quantum group $U_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K_i$, $E_i$, $F_i$ (the ...
André Henriques's user avatar
3 votes
1 answer
275 views

Non-Faithfully Flat Quantum Homogeneous Spaces

Let $G$ and $H$ be two Hopf algebras, and $\pi: G \to H$ a Hopf algebra map. We will call an algebra of the form $$ M:= \lbrace m \in G ~ | ~ m_{(1)} \otimes \pi(m_{(2)}) = m \otimes 1 \rbrace $$ a ...
Réamonn Ó Buachalla's user avatar
2 votes
1 answer
188 views

$H$-Hopf modules equal the tensor products of their coinvariants with H

In a comment for this old question, it was said that > There is a theorem that Hopf modules are (up to isomorphism) the tensor products of their coinvariants with H. (This theorem is usually worded ...
Abtan Massini's user avatar
1 vote
1 answer
334 views

Category of Hopf algebras.

Can you tell me, where I can find a proof of the following fact: Let $R$ be a commutative ring. Consider the category of commutative Hopf algebras over $R$. Then this category is equivalent to the ...
philipp's user avatar
  • 147
5 votes
0 answers
280 views

Is the "Toeplitz algebra" the representation ring of a Hopf algebra related to SU(2)?

More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional ...
Qiaochu Yuan's user avatar
4 votes
3 answers
2k views

Mistake in Wikipedia Entry "Coalgebra"

Consider the following quote from the Wikipedia entry Coalgebra: The kernel of every coalgebra morphism $f : C_1 \to C_2$ is a coideal in $C_1$, and the image is a subcoalgebra of $C_2$. I can't ...
Ago Szekeres's user avatar
1 vote
1 answer
230 views

Trivial Hopf Coinvariant Subspace Example

For $G,H$ Hopf algebras, and $\pi:G \to H$ a Hopf algebra map, can some-one give me an example of a (right) $H$-comodule $(V,\Delta_R)$, such that $$ (G \otimes V)^{\text{co}H} = \lbrace g_{(1)} \...
Dyke Acland's user avatar
  • 1,459
2 votes
0 answers
68 views

2-cocycles/Bigalois-objects over nontrivial liftings

It is easy to extend group-2-cocycles to smash-products with Nichols algebras over the group (just trivially). The same certainly doesn't work for nontrivial liftings. As I would like to check a ...
Simon Lentner's user avatar
1 vote
1 answer
160 views

Coinvariant Subalgebras of Hopf Comodules and Quotients

For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta_R(W) \subseteq W \otimes H$, and note that this implies that $...
Ago Szekeres's user avatar
19 votes
1 answer
2k views

Connes-Kreimer Hopf algebra and cosmic Galois group

Hi, I'm interested in the relation between the two following constructions motivated by renormalization: Connes-Kreimer gives an interpretation of the renormalization procedure in the framework of ...
Adrien's user avatar
  • 8,234
2 votes
1 answer
474 views

Convolution inverse

I'm currently reading some stuff on Hopf algebras, specifically Hopf Algebras: An Introduction by Sorin Dascalescu, Constantin Nastasescu and Serban Raianu. One proof involves showing that for a Hopf ...
Ryan's user avatar
  • 545
6 votes
1 answer
228 views

Dimension formula for Cartan-type abelian.group Nichols algebra?

Existence of a root system has been established for Nichols algebras $B(V)$ of a Yetter-Drinfel'd-module $V$ (resp. braided vectorspaces $V$) over abelian groups (resp. with diagonal braiding $x_i\...
Simon Lentner's user avatar
9 votes
3 answers
2k views

Tensor product of linear mappings versus chain complexes

A chain complex of vector spaces $X_k$ is a sequence of linear mappings $\dots \overset{d_{k-1}}{\longrightarrow} X_k \overset{d_{k}}{\longrightarrow} X_{k+1} \overset{d_{k+1}}{\longrightarrow} \dots$...
shuhalo's user avatar
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6 votes
1 answer
219 views

Liftings of Nichols algebras over racks via Doi twist

As a more nontrivial example for my Dissertation thesis, I'd require some example of the following type (of course I'll "cite" ;-) ), so thanx in advance: Andruskiewitsch/Grana have by a new ...
Simon Lentner's user avatar