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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$U\left(\mathfrak a\right) \otimes_{U\left(\mathfrak a\cap\mathfrak b\right)} U\left(\mathfrak b\right) \cong U\left(\mathfrak a + \mathfrak b\right)$ over a ring containing $\mathbb{Q}$
While the Poincaré-Birkhoff-Witt theorem is usually proven (and sometimes even formulated) for free modules only, it is known (see also here) that it holds for arbitrary modules if the ground ring is ...
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Is there a coalgebraic definition of filtered algebras?
If $M$ is a monoid, then an $M$-graded algebra over $k$ is the same thing as a $k[M]$ comodule algebra. To see this, if $\delta$ is a coaction of $k[M]$ on an algebra $A$, for each $m \in M$ define
$$...
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Bialgebras with rigid representation theory
Repost from math.SE since no answer after two months, but feel free to close if not appropriate:
Everything is finite-dimensional over a field $k$.
Let $B$ be a bialgebra with $B\text{-mod}$ its ...
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Hopf algebras examples
Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course.
Some of the examples I know are well-...
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Is there a classical version of Yetter-Drinfeld modules?
One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$.
If we think ...
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Intuition for left Hopf-modules
I'm reading Radford's text on Hopf Algebras right now and I'm a bit confused about the definition of a left $A$-Hopf algebra. The definition given in the book is:
Let $A$ be a $\Bbbk$-bialgebra. A ...
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States "absorbed" by a Haar idempotent on a compact quantum group
Firstly, a small question of nomenclature. If $(S,\bullet)$ is a magma, is there good terminology to relate $a$ to $b$ when
$$a\bullet b=b=b\bullet a?$$
Can we say that $b$ absorbs $a$? Can we say ...
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Quiver and relations for Hopf algebras associated to quiver algebras
Let $A=KQ/I$ be a finite dimensional quiver algebra with admissible relations $I$.
$A$ can be made into a restricted Lie algebra over a field of characteristic $p$ via
$[x,y]=xy-yx$ and $x^{p}=x^p$. ...
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Lagrangian subcategories of (non-pointed) braided tensor categories
I am interested in generalising the following claim in On braided fusion categories I (Remarks 4.67.)
“A braided fusion category $\mathcal C$ may have more than one Lagrangian subcategory. E.g., if $\...
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What is an example of a Frobenius algebra that is not Koszul?
What is an example of a Frobenius algebra that is not Koszul? Are there reasonable requirements for a Frobenius to be Koszul?
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Is the Frobenius property invariant by Morita equivalence?
Kaplansky's sixth conjecture [Ka75] states that the dimension of a semisimple finite dimensional Hopf algebra over $\mathbb{C}$ is divisible by the dimension of its irreducible complex representations....
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Hopf algebras vs. Kac algebras
I recently came across Kac algebra. They are roughly Hopf algebras and $C^*$-algebras with compatible structures. It follows from Artin–Wedderburn theorem that every semisimple complex Hopf algebra ...
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Two (or less) filtrations on coenveloping coalgebra
Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) ...
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Bicrossed and bismash product of Hopf algebras
I have been recently studying different methods to construct Hopf algebras.
In Theorem IX.2.3 of "Quantum groups" by Kassel the bicrossed product of a pair of matched bialgebras (or Hopf ...
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Coproduct for a Frobenius algebra
The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
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A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?
A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
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Problem in understanding Theorem $6.2.9$ from Chari and Pressley
The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...
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Understanding definition of quantization of a Poisson-Hopf algebra
I am going through the chapter Quantization of Lie bialgebras from the book A Guide to Quantum Groups by Chari and Pressley. There I found a notion called Quantization which deals with deformations of ...
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Representation finite Hopf algebras up to stable equivalence
It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra.
Question: Is it true that every representation-finite Hopf algebra is stable ...
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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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Equivariant description of indecomposable elements in shuffle algebra
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Tor{Tor}$Let's suppose $V$ is a $k$-vector space equipped with its standard (left) $\GL (V)$-action. The shuffle algebra is the graded dual of the ...
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When are Brauer tree algebras Hopf algebras?
Question 1: Which Brauer tree algebras are Hopf algebras?
For example every representation-finite group algebra is a Brauer tree algebra and thus a Hopf algebra, but not every Brauer tree algebra ...
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About extensions between morphisms on the multiplier algebra
Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism
$$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \...
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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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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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Representation-finite blocks of Hopf algebras up to derived equivalence
Question: Which representation-finite selfinjective algebra is derived equivalent to a block of a (finite dimensional) Hopf algebra?
Famous examples are all Brauer tree algebras. Are there more ...
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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 every $1$-dimensional complex representation of $G$ is trivial.
proof: First if $...
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Timmerman's "An invitation to quantum groups and duality" corollary 3.2.8
Consider the following propositions of Timmerman's book "An invitation to quantum groups and duality":
I do not understand the equivalence
$$h(\mathfrak{C}(\delta_V)S(\mathfrak{C}(\delta_W))...
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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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Hopf algebras arising as Group Algebras
Every commutative $C^*$-algebra is isomorphic to the set of continuous functions, that vanish at infinity, of a locally compact Hausdorff space. Every commutative finite dimensional Hopf algebra is ...
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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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Primitive elements in the universal enveloping algebra of Lie superalgebra
Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
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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 ...
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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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Is there a condition such that the $A$ action of a $A \rtimes H$-module is a restriction of the $H$-action?
Let $H$ be a Hopf algebra and $A$ a subalgebra of $H$ such that $A$ is a left coideal of $H$ (that is $\Delta(A) \subset H \otimes A$) and $A$ is preserved by the adjoint action of $H$.
Consider now ...
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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 ...
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Why Drinfel'd-Jimbo-type quantum groups?
Hopf algebras are pretty easy to motivate, as a not-necessarily-commutative generalization of the ring of functions on an algebraic group (and there are many other ways in which they come up). I like ...
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Mayer–Vietoris sequence for coproduct of Hopf algebras
Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in Agore - Categorical Constructions for Hopf ...
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A finite $H$-module algebra is necessarily inner
Let $H$ be a finite-dimensional Hopf algebra over a field $k$, and $A$ an $H$-module algebra.
If $A$ is finite-dimensional over $k$, is the action of $H$ on $A$ necessarily inner?
If this is not true ...
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Natural knot homology
All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
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Hopf algebra with a non-invertible antipode
What is an example of a Hopf algebra with a non-invertible antipode?
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What kind of topological invariants can you get from just Hopf algebras?
In A minus sign that used to annoy me but now I know why it is there, Peter Tingley shows how to build knot invariants from the representations of the $U_q(\mathfrak{sl}_2) $ quantum group by ...
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How to interpret compositional diagrams for quantum sets algebraically
$\newcommand{\id}{\mathrm{id}}$My reference for this post is Musto, Reutter and Verdon's A compositional approach to quantum functions, arXiv:1711.07945. Questions are in bold below. Allow me to begin ...
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Exterior algebra of free modules over Hopf algebras
Let $H$ be a commutative, cocommutative Hopf algebra over a field $\mathbb{K}$, and $M$ a free Hopf module over $H$. Is the exterior algebra $\Lambda^k_\mathbb{K} M$ with the diagonal $H$-action
$$h \...
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Combinatorial proof of a matrix equation
I'm looking for combinatorial proofs (using, e.g., trees) of the following particular matrix equation $(I)$ and also combinatoric operational analogs of its solution via matrix inversion and/or Cramer'...
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Characterising algebraic compact quantum groups among Hopf $^*$-algebras
Let $(A, \Delta)$ be a Hopf $^*$-algebra. Assume that $\{u^\alpha\}_{\alpha \in I}$ is a maximal collection of pairwise inequivalent irreducible unitary corepresentation matrices. I want to show that
$...
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If $\operatorname{Hom}(\delta_V, \delta_W) = 0$, then $\mathfrak{C}(\delta_V) \cap \mathfrak{C}(\delta_W) = 0.$
Let $(A, \Delta)$ be a Hopf $^*$-algebra and $\delta_V: V \to V \otimes A$ and $\delta_W: W \to W \otimes A$ be two corepresentations of $(A, \Delta).$
Assume that the space of intertwiners $\...
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Orthogonality relations for Haar state and antipode (Timmerman)
Consider the following proposition from Timmerman's "An invitation to quantum groups and duality":
I am having trouble seeing why the boxed equations are true (Note that on the left the ...
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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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Different Bialgebra/Hopf algebra structures on coalgebras
Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.
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