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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5
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0answers
118 views

Characterizing discrete quantum groups

Let $M$ be a von Neumann algebra, and let $\Delta$ be a unital normal $*$-homomorphism $M \rightarrow M \mathbin{\bar\otimes} M$ that satisfies the coassociativity condition $(\Delta \mathbin{\bar\...
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126 views

Coinvariants of tensor products of Hopf algebras

Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way. The axioms of Hopf algebras imply that $$ G^{coinv(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \mathbb{C}1. $$ ...
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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 ...
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Hopf algebras structure and quantum affine algebras

I'm looking for some information about the Hopf algebras structure and the quantum groups. In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...
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Quantum analogue of certain property of compact groups

Let $\mathcal{A}$ be the category of $C^*$ algebras. For a group $G$ let $\tilde{G}$ be the space of all conjugacy classes of $G$. What is a precise description of a maximal ,or in some sense ...
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On reflexive bialgebras

Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...
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Abelian category from the category of Hopf algebras

The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf sub-algebra of $H_1$. Is there some replacement or alteration of the notion of a kernel in the Hopf algebra setting. Same ...
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1answer
87 views

Irreducibility of product bicomodules

Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right, $H$-comodule respectively. The tensor product $$ V \otimes W $$ has an obvious $H$-$H$-bicomodule structure. If $V$ and $W$ are ...
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Is $Tor_A(k,k)$ a bicommutative Hopf algebra?

Let $A$ be a commutative (or graded commutative) algebra over a field $k.$ In some sources, such as Mcleary's book on spectral sequences, Corollary 7.12, pg. 248, it is claimed that $\text{Tor}_A(k,k)$...
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Tensor algebras in the bicategory $\mathsf{2Vect}$

To my knowledge there are two main approaches to categorify the notion of a vector space. I will refer to them as BC-2-vector spaces (Baez, Crans) and KV-2-vector spaces (Kapranov, Voevodsky). Both ...
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What is a quantum analogue of the fact that the second fundamental group of every Lie group is trivial?

What is an appropriate version of the following fact in terms of Hopf algebras and quantum groups: "For every connected Lie group $G$ the second fundamental group $\pi_2(G)$ is trivial?" Is there ...
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83 views

Reduced compact quantum group and left and right multiplication

Let $(A,\Delta)$ be a compact quantum group in the sense of Woronowicz, and let $A_0$ be its dense Hopf subalgebra. We can construct from the Haar state $h:A \to \mathbb{C}$ an inner product $$ \...
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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 ...
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1answer
159 views

Rigidity for the category of comodules over a Hopf algebra

On this page https://ncatlab.org/nlab/show/rigid+monoidal+category there is a discussion of rigidity (left-right duality) for the catagory of modules over a Hopf algebra. What happens if we look at ...
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Tensor product of fields 2

Let $K_1, K_2$ be finite field extensions of a field $k$. Question: Is it true that $A=K_1 \otimes_k K_2$ is isomorphic to a product of group algebras over fields? Question 2: In case the ...
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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-...
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Relative projectivness and tensor products

Let $R$ be a commutative ring, $H<G$ a finite group pair. Let $P$ be a $RG$ module that is $RH$ projective. It is known that for any $RG$ module $X$ the tensor $P\otimes X$ is $RH$ projective. This ...
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Update on “Hopf algebras: their status and pervasiveness” by Hazewinkel

Hazewinkel wrote this article in 2005. Perhaps it's time for an update. For example, updating item 34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of ...
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W H Lin's thesis and Hopf subalgebras of the Steenrod algebra

If $B$ is a subalgebra of $A$, you can ask whether the $B$-module structure on $B$ can be extended to give an $A$-module structure on $B$. W H Lin, in his 1973 PhD thesis at Northwestern, showed that ...
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A weaker version of strongly graded algebras

Let $A = \oplus_{i \in \mathbb{Z}} A_i$ be a graded algebra. We say that it is strongly graded if $A_i.A_j = A_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that $$...
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Characters on Hopf algebras

For any algebra $A$, a character for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking ...
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Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
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Intuitive, elementary intros to Hopf algebras/monoids

Motivation: I'm interested in understanding the role that noncrossing partitions play in Hopf algebras/monoids (HAs) as the components of the power series of the compositional inverse of formal ...
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Classification of $\operatorname{Rep} D(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
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Submodules of $V\otimes V^*$

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...
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2answers
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Subfunctor of internal Hom

Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{...
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Embedding problems on quantum groups?

We work over the field of complex numbers. We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
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54 views

Integrals in noncommutative graded algebras which are not necessarily Hopf

Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\...
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2answers
383 views

Comultiplication of elements of partition of unity

Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra). Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
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Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?

Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...
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1answer
129 views

“Nice” bases for finite dimensional semisimple Hopf algebras

Let $H$ be a finite dimensional semisimple Hopf algebra over $\mathbb{C}$. Can one choose a basis $\{h_1, \dots, h_n \}$ of $H$, where $h_1 = 1$, such that if we write $$ \Delta(h_i) = \sum_{1 \leq j,...
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Weakly reflexive algebra vs proper (residually finite-dimensional) algebra

Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine): An algebra $A$ is called ...
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Images and Kernels of tensor products of homomorphisms of modules

Let $\mathcal{H}$ be a Hopf algebra and let $M_1,M_2,N_1,N_2$ modules over $\mathcal{H}$. If $f:M_1\rightarrow N_1$ and $g:M_2\rightarrow N_2$ are homomorphisms of modules, then are the following ...
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1answer
174 views

What properties of a finite group fibre functor give its endomorphisms a hopf algebra structure?

Tannaka duality for a finite group lets us recover the group algebra $\mathbb{C}[G]$ as the endomorphisms of the forgetful functor $F:RepG\rightarrow Vect$, and taking the monoidal automorphisms ...
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392 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 ...
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Noncrossing partitions in Hopf algebras/monoids via compositional inversion

Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
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Classification of one dimensional (non-commutative) formal group laws over $k[\epsilon]/(\epsilon^n)$

It's well-known that any one dimensional formal group law over a $\mathbb Q$-algebra or a reduced ring is commutative, but there are one dimensioal non-commutative formal group laws over rings like $k[...
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1answer
103 views

Cobraided and coquasitriangular Hopf algebras

In a Hopf algebra, are the notions "cobraided" and "coquasitriangular" the same? Kassel (Hopf algebras) uses "cobraided" and Montgomery (HAs and their actions on rings) uses the other -- both on page ...
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1answer
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When is $N^{*} \otimes_K M$ projective for a local Hopf algebra?

Given a finite dimensional local Hopf algebra $A$ over a field $K$ and two finite dimensional indecomposable modules $N$ and $M$. Is it known when the module $N^{*} \otimes_K M$ is projective? Can ...
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1answer
99 views

Some quantities associated to finite dimensional Hopf algebras

let $(H,\Delta,m,s)$ be a Hopf algebra. To this Hopf algebra one can associate two obvious linear maps $T_H, S_H: H \to H $ with $T_H=m\circ \Delta,\quad S_H=s$. Are there two finite dimensional ...
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1answer
116 views

Examples of basic coalgebras

For an algebraically closed field $k$, let $C$ be a $k$-coalgebra. Given a minimal injective cogenerator $E$, there is a so-called basic coalgebra $B_C=coend^C(E)$, s.t. the comodule categories $Mod^C$...
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The relationship between representations of groups and evaluation and coevaluation maps for $vect_{G}$ module categories

Let $G$ be a finite group and $vec_{G}$ be the monoidal category of finite dimensional $G$-graded vector spaces. Given any $vec_{G}$ module category $\mathcal{M}$ we can define a dual module category ...
10
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3answers
633 views

Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...
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4answers
835 views

The tensor product of two monoidal categories

Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way? The motivation I am thinking of is two categories that are representation ...
8
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1answer
324 views

Image of Comultiplication on Finite Quantum Groups/Hopf Algebras

Let $A=:F(G)$ be the algebra of functions on a finite quantum groups aka a finite dimensional C*-Hopf Algebra. Suppose that $F(G)$ is neither commutative nor cocommutative. In their 1966 paper Kac ...
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1answer
79 views

Is there a way to adjoin a counit to a non counital coalgebra?

Let $k$ be a field. If $A$ is a non unital $k$-algebra, there is a simple way to make it unital by taking $\tilde{A}:=A\oplus k$ and setting $$ (a+\lambda)(b+\mu):=ab+\lambda b +\mu a +\lambda \mu$$ ...
4
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1answer
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Invariance of Finite Dimensional Woronowicz $\mathrm{C}^*$-ideals under the Antipode

Let $(A,\Delta)=:F(G)$ be a finite dimensional $\mathrm{C}^*$-Hopf algebra, and so the algebra of functions on a quantum group $G$. Let $J$ be a closed ideal in $F(G)$ and $\pi:F(G)\rightarrow F(G)/J$...
2
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1answer
192 views

Milnor-Moore and the characteristic zero homology of H-spaces

In their work on Hopf-algebras: https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf> on the last page p.263, they say that the Hurewicz map ...
3
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1answer
158 views

Noncommutative Leray - Hirsch theorem in the context of noncommutative principal bundles

In the literature, are there some researchs on non commutative analogy of Leray-Hirsch theorem in the context of non commutative Principal bundles?
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Is every finite quantum group a quantum symmetry group?

This post is basically a quantum extension of Is every finite group a group of “symmetries”? Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra. Frucht's theorem ...

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