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 ...

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Structure theorem for non connected graded Hopf algebras

Let $V^{\bullet}$ be a graded vector space over a field of char $0$. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected cocommutative Hopf algebra and in ...
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65 views

Cosemi-simple FRT Hopf Algebras

This question is a somewhere similar variation on an old question of mine. Given an FRT-algebra, which is to say, roughly, that it can be constructed from an R-matrix in the Yang--Baxter sense (see ...
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1answer
82 views

bialgebras on quotient polynomials

Is there a general procedure for constructing a bi-algebra out of a quotient polynomial ring? In particular, how do I construct a bi-algebra corresponding to quotient polynomial ring ...
4
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1answer
102 views

Hopf-Galois Structure Maps

A Hopf-Galois extension of commutative rings, as defined by Montgomery here, is a morphism of commutative rings $\phi:A\to B$ with a Hopf-algebra $H$ coacting on $B$ by a ring map $c:B\to B\otimes H$ ...
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110 views

When is a given quiver algebra a hopf algebra?

Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? ...
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58 views

Homomorphisms of quantum spaces

Suppose we have a look at the Hopf $*$-algebra $U_q(sl(2))$ and the Hopf $*$-algebra $A_q(\widetilde{S}^3)$ introduced in the paper: http://arxiv.org/pdf/q-alg/9605017v1.pdf. I want to find a relation ...
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94 views

Compact Quantum Groups and FRT-Algebras

As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which ...
5
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1answer
218 views

Graded Hopf algebras and H-spaces

Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both ...
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170 views

Center of quantum affine algebras

Are there some references about the center of quantum affine algebras? I searched on google and only find the paper. In particular, what is the center of $U_q(\widehat{\mathfrak{sl}_2})$. Thank you ...
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115 views

An alternative Cauchy theorem on Hopf algebras

Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra. There already exists a generalization of Cauchy theorem using exponent, see here. We are interesting in an alternative ...
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868 views

Structure of Hopf algebras - trouble understanding an old paper

UPDATE: I am grateful to Peter May for the accepted answer, which makes most of the details below irrelevant. However, I will leave them in place for the record. I am trying to understand the proof ...
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1answer
160 views

Hopf Galois extensions and conditional expectations for C* algebras

Suppose that $H$ is a Hopf algebra with normalised invariant integral (appropriate side) $\int:H\to \mathbb{C}$. The $H$ right comodule algebra $P$ is a Hopf Galois extension, so the canonical map ...
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1answer
91 views

dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32 Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...
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105 views

What's the relation between half-twists and star structures on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
4
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1answer
183 views

group completion theorem of homology as Hopf algebras

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...
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71 views

Dense Hopf $*$-subalgebra of compact quantum groups and cancellation laws

Recall the notion of the compact quantum group, in the sense of Woronowicz: it is a pair $(A,\Delta)$ where $A$ is a unital $C^*$-algebra and $\Delta:A \to A \otimes_{\min} A$ is a unital ...
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45 views

Co-quasitriangular Hopf algebra - notation

In one article I found the following statement : If $A$ is a Hopf algebra over $k$ with co-quasitriangular structure $r$, then one can define map $r_{21}*r:A\otimes A\rightarrow k \ \ $ (...). ...
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42 views

Realization of braiding on infinite dimensional vector space via Yetter-Drinfeld structure over Hopf algebra

In one article I found the following statement Let $V$ be finite dimensional vector space. A braiding $\Psi$ on $V$ can be realized via Yetter-Drinfeld structure over a Hopf algebra with ...
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1answer
98 views

Yetter-Drinfeld modules as rigid category

I'm reading a proof of the following theorem If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category. In this proof we define ...
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1answer
106 views

Difference between two definitions of graded coalgebra

I'm reading articles about applications of Hopf algebras in physics. In one of them there is following definition of graded coalgebra: A coalgebra $C$ is called graded if $C=\bigoplus\limits_{n\ge ...
4
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70 views

Is there a perfect integral fusion category with PFdim = 2 mod 4?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$. Proposition: A finite group $G$ is perfect iff any $1$-dimensional complex representation of $G$ is trivial. proof: First if ...
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2answers
622 views

Does projective imply flat?

Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
3
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1answer
78 views

Normal Hopf ideals and Hopf subalgebras, quotient group schemes

There is a following theorem: $H$ is a commutative Hopf algebra over a field $k$. Then there exists a bijective correspondence between $$\{ \textrm{Hopf subalgebras }K\subset H \} \quad ...
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0answers
95 views

Programmatically computing dual Hopf algebras: state of the art

Given a graded Hopf algebra of finite type, we know the (graded) linear dual is also a graded Hopf algebra. For instance the dual Hopf algebra to the polynomial algebra on an even degree generator, ...
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1answer
199 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 ...
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226 views

What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...
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49 views

Grouplike elements in dual weak Hopf algebras

It is said in D. Nikshych's paper On the structure of weak Hopf algebras (arXiv:math/0106010) that if $A$ is a finite dimensional weak Hopf algebra, then a functional $\gamma$ in the dual (weak Hopf) ...
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1answer
329 views

Identifying a Hopf algebra cohomology theory

Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be ...
8
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200 views

Bialgebras with Hopf restricted duals

It is known from the general theory that, given a bialgebra (over a field $k$) \begin{equation} \mathcal{B}=(B,\mu,1_B,\Delta,\epsilon) \end{equation} the Sweedler's dual $\mathcal{B}^0$ (called also ...
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0answers
72 views

Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$. Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...
6
votes
1answer
261 views

In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?

Question: In a closed monoidal abelian category such that the unit object is compact projective, must the tensor product of compact projective objects be compact projective? Recall that an object ...
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0answers
94 views

Exponential map on a unipotent group

Let $G$ by a unipotent linear algebraic group defined over a field of characteristic $0$, with Lie algebra $\mathfrak{g}$. The exponential map $\mathfrak{g}\to G$ is bijective, and we can recover the ...
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274 views

Is there a non-trivial Hopf algebra without left coideal subalgebra?

Let $H$ be a Hopf algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$. A subalgebra $I$ of $H$ is called a left coideal subalgebra if $\Delta(I) \subset H \otimes I$. $H$ is ...
0
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1answer
236 views

Coaction of a group

Suppose $G$ is a finite group which acts on a $C^*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$. I ...
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0answers
48 views

Analogue of Baer's injectivity criterion for comodule algebras

Let $H$ be a Hopf algebra and $A$ an $H$-comodule algebra; denote by $M^H_A$ the category of right $(H,A)$-Hopf modules [i.e. $A$-module, $H$-comodules, everything is compatible with everything else]. ...
3
votes
1answer
426 views

Hopf-algebras in associative ring spectra

I'm interested in a definition of cocommutative Hopf-algebra objects in the $\infty$-category of associative (read: $A_\infty$) ring spectra. One thought I had was to think of cocommutative ...
6
votes
0answers
184 views

Tannakian formalism for topological Hopf algebras

Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of ...
4
votes
1answer
125 views

Comodule analogue for statement that a faithful representation of an affine group scheme generates all

If $V$ is a faithful finite dimensional representation of an affine group scheme $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n ...
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135 views

The convolution on a semisimple finite quantum groupoid

Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual. Let the Fourier transform $\mathcal{F}: \mathbb{A} \to ...
3
votes
0answers
87 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
votes
0answers
111 views

Are the integer index finite depth irreducible subfactors Kac-coideal?

Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra? In other words, of the following form ...
2
votes
0answers
113 views

Isomorphism Between $U_{h}(\frak{sl}_3)$ and $U({\frak sl}_3)[[h]]$

I take the following from Klimyk and Schmudgen's book Quantum Groups Page 167: Proposition For $\frak{g}$ a complex semi-simple Lie algebra, there is an isomorphism $\phi$ of the $h$-adic algebra ...
2
votes
2answers
150 views

Identities that connect antipode with multiplication and comultiplication

I asked this initially in math.stackexchange: The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities: $$ S\otimes S\circ \Delta=\sigma\circ\Delta\circ S $$ $$ ...
4
votes
0answers
95 views

Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$. Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
2
votes
2answers
253 views

Is antipode unique for bialgebras in arbitrary monoidal categories?

If $B$ is a bialgebra in the category $\tt{Vect}$ of vector spaces (over $\mathbb C$, for example) then $B$ can't have two different antipodes. Is this true for bialgebras in an arbitrary symmetric ...
2
votes
1answer
196 views

When is an exponential functor a bialgebra?

My question is about the notion of exponential functors as they are frequently defined in the literature on (strict) polynomial functors, e.g., the paper "General Linear and Functor Cohomology" by ...
7
votes
2answers
238 views

An identity in the free associative algebra

Let $V$ be a finite dimensional vector space over a field of characteristic $0$, and let $T(V)$ be the tensor algebra (also called the free associative algebra) on $V$. This is actually a Hopf ...
6
votes
0answers
235 views

Topological quotient Hopf-algebras and “change-of-rings”

One way of constructing coalgebra objects in the homotopy theorist's category of spectra is to take the suspension spectrum of a space, with the diagonal providing a cocommutative comultiplication. ...
3
votes
2answers
286 views

Algebraic Groups, Modules, and Comodules

Background: Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For $$ \widehat{H} := \text{Alg}_k\{H; k\}, $$ we recall (see Abe Chapter 4 ...
6
votes
1answer
182 views

Problem with Eisenbud's Lemma “Symmetry of Diagonalization”?

This question was first asked on MathSE but nobody answered. In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like ...