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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Do dualizable Hopf algebras in braided categories have invertible antipodes?
A classical result of Larson and Sweedler says that a finite dimensional Hopf algebra over a field has invertible antipode. Does this result extend to the setting of Hopf algebras in braided ...
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Easy example of a non-symmetric braiding of $\operatorname{Rep}(G)$?
What is the smallest group $G$ such that $\operatorname{Rep}(G)$ has a non-symmetric braiding (or just an easy example)?
I seem to remember a result classifying all universal $R$-matrices of $\mathbb ...
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Classification of quasitriangular Hopf algebras
The classification of hopf algebras is a big and open problem, containing various subproblems (such as: the classification of groups, of Lie algebras, the study of special classes such as (co)...
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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 ...
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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 ...
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What kind of structures allow Galois descent?
EDIT: Question solved.
Let me explain what I mean.
The classical formulation of Galois descent, e. g. in Crawley-Boevey's "Cohomology and central simple algebras", uses the following notion:
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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 ...
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How to translate multi-segments to Drinfeld polynomials?
Let $\hat{H}_m=\hat{H}_m(q)$ be the Iwahori-Hecke algebra of $GL_m$, see for example, Section 2. The simple $\hat{H}_m$-modules are parametrized by Zelevinsky's multi-segments, See Section 2.2 of the ...
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How to construct the quantum group $U_q(\mathfrak{sl}(2))$ from the quantum coordinate ring $\operatorname{SL}_q(2)$?
Let $k$ be the ground field, and $q$ be an invertible element with $q$ not being a root of unity.
Let $\operatorname{SL}_q(2)$ be the quantum coordinate ring of $\operatorname{SL}(2)$ given explicitly ...
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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 ...
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Triviality of Semisimple Hopf Algebras of Cyclic Dimension
A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277
Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...
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When is the character group scheme of a group scheme representable? (Affine Case)
While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...
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The augmentation filtration on a group ring
Let $G$ be a group and $\mathbb QG=\{\sum a_ig_i\colon a_i\in\mathbb Q, g_i\in G\}$ its group ring over $\mathbb Q$. It is a Hopf algebra. Let $I=\{\sum a_ig_i\colon\sum a_i=0\}$ be the augmentation ...
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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 [KSZ06].
We are interesting in an alternative ...
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Apocryphal Maschke theorem?
This may be totally trivial or wrong. I am just posting this because I am sick and tired of trying to understand this myself and I am sure someone out here can just answer it out of his head in 2 ...
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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). ...
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Is the Cartan matrix of a finite-dimensional (Hopf) algebra invertible over the rationals?
This is probably well-known to representation theorists, but this doesn't imply being well-known to me.
Let $k$ be a field, and let $A$ be a $k$-algebra that is finite-dimensional as a $k$-vector ...
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groupring morphisms and bialgebra
Let $G_{1}$ and $G_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G_{1}]\rightarrow \mathbb{Z}[G_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G_{...
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Deligne Tensor Product of Categories, Explicit Equivalence of $A\otimes_\mathbb{C} B\text{-Mod} \cong A\text{-Mod}\boxtimes B\text{-Mod}$
$\newcommand\Mod[1]{#1\text{-Mod}}$Does any one have a reference on a explicit equivalence between
$$\Mod{A\otimes_\mathbb{C} B} \cong \Mod A\boxtimes \Mod B?$$
The proof in "Tensor Categories ...
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Hopf monads in categorical probability theory
1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
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Morphisms between compact quantum groups
Let $(A, \Delta_A)$ and $(B, \Delta_B)$ be two compact quantum groups (in the sense of Woronowicz). I would be tempted to define a morphism $(A, \Delta_A) \to (B, \Delta_B)$ to be a unital $*$-...
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Confusion around the reflection equation algebra
I have encountered several occurrences of the so called reflection equation algebra (REA) but depending on where I find them, I feel like I get slightly different objects. In all cases there is a ...
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Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?
For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...
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Corepresentations of Tensor Products of Hopf Algebras
Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure ...
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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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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 ...
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Invertible elements of the Hopf algebra quantum $SU(2)$
Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called the quantum special unitary group. For details see
https://en.wikipedia.org/wiki/Compact_quantum_group
(Note that on the ...
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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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When are the braid relations in a quasitriangular Hopf algebra equivalent?
Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}...
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How do quantum knot invariants change when I pick a funny ribbon element?
So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ...
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Group schemes over ring of Witt vectors and their representing algebras
Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom_{Rng}(A,R)$ of ring ...
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Commutative and Cocommutative Quantum Groups
I am using this definition:
An algebra of functions on a finite quantum group $\mathbb{G}$ is a finite dimensional $C^\ast$-Hopf algebra $A=:F(\mathbb{G})$.
I have the following (very well known --...
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Tannaka–Krein duality
First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and ...
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Is the "renormalized third comultiplication" on $\mathbf{Symm}$ integral?
Background:
For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...
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Involutive solutions to the Yang-Baxter equation (and triangular Hopf algebras)
I'm interested in solutions to the Yang-Baxter equation
$$R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23},$$
that are involutive $R^2_{12}=1$. Or put it another way, I'm interested in representations of the ...
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Comparing Hochschild (co)homology for algebras and coalgebras
Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes
$$M\otimes A^{\otimes n}$$
and
$$\...
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Hopf algebra in derived category vector spaces
Let $H$ be a complex of vector spaces over some field $k$ which is endowed with the structure of a Hopf algebra object. I have heard several times that if $H$ is concentrated in positive or negative ...
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Is there a ''simple'' formula for the inverse of the Drinfeld associator?
The Drinfeld associator $\Phi(x_0, x_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x_0, x_1\rangle\rangle$ of formal ...
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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 ...
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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\...
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The Quantum Group ${\cal O}_q(SL(n))$, for $q>1$
For the quantum group ${\cal O}_q(SL(n))$, $q\in \mathbb{R}$, I have read, without a proof, that for $p>1$, there exists a $q\in (0,1)$ such that
$$
{\cal O}_p(SL(n)) \simeq {\cal O}_q(SL(n)).
$$
...
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Quantum group representations from (convolution) matrix units?
Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$
There is a convolution product on $A=F(\...
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Quantum Hilbert's fifth problem
Hilbert's fifth problem inquires whether every locally Euclidean group is necessarily a Lie group. Von Neumann demonstrated that this is indeed true for the compact case.
The definition of a quantum ...
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If a strong monoidal functor $F$ has an ambidextrous adjoint, then how close is the adjoint to being strong monoidal?
Let $F : C \to D$ be a strong (symmetric, say) monoidal functor. Suppose that $G : D \to C$ is both left and right adjoint to $F$ (an ambidextrous adjunction). Then by doctrinal adjunction $G$ is both ...
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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 ...
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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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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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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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Independence of characters with respect to polynomials
I came across the following property :
Let $\mathfrak{g}$ be a Lie algebra over a ring $k$ without zero divisors,
$\mathcal{U}=\mathcal{U}(\mathfrak{g})$ be its enveloping algebra. As such, $\mathcal{...
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Hopf algebra interpretation of hypergraph duality?
The work of Aguiar and Ardila (https://arxiv.org/abs/1709.07504) on Hopf monoids for generalized permutohedra gives a Hopf monoid structure on the collection of hypergraphs; see sections 19 and 20 of ...
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