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Questions tagged [hopf-algebras]

For questions about Hopf algebras and related concepts, such as quantum groups. The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science.

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Finiteness of results in Connes-Kreimer approach

Remark: Although this is technically a physics-related post, the content heavily relies on pure mathematics, so I deemed it more appropriate here. When reading the papers by Connes and Kreimer (e.g. [...
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A diagrammatic proof of antipode being antihomomorphism in a Hopf algebra

Let $(H, \mu, \eta, \Delta, \epsilon, S)$ be a Hopf algebra with $S: H \to H$ denoting the antipode. By definition, $S$ is the convolution inverse of $1: H \to H$ in $\operatorname{End}(H)$, with the ...
Ray's user avatar
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On the orthogonality relations for quantum Clebsch-Gordan coefficients

I am currently reading Quantum groups by Christian Kassel, more precisely, section VII.7 which is about quantum Clebsch-Gordan coefficients. To make this question self-contained, let me introduce the ...
richrow's user avatar
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$\mathbb{C}G$-modules and $\mathbb{C}^{G}$-comodules

I know that representations of a group $G$ are essentially $\mathbb{C}G$-modules. How is it that every $\mathbb{C}G-$module is also equivalent to a $\mathbb{C}^{G}-$comodule. I have not found this ...
NoetherNerd's user avatar
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For the binomial Hopf algebra, what is the group of grouplike elements of the dual algebra?

The linear space of finite polynomials over a field $\mathbb{K}$ of zero characteristic has a structure of a graded connected bialgebra (meaning the zeroth subspace is isomorphic to the base field): $\...
Daigaku no Baku's user avatar
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1 answer
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Hopf algebra related to monoidal category

Recently, I heard that Braided rigid monoidal category corresponds to a quasi-triangular hopf algebra. I know in braided condition gives hexagonal equations and monoidal category gives pentagon/...
phy_math's user avatar
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How does base change affect group schemes?

I am reading through Waterhouse's Introduction to Affine Group Schemes and am having trouble understanding how I should think of base change as it relates to the way a groups scheme "looks". ...
integraletothexy's user avatar
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Exercise 3.14b of Waterhouse Affine Group Schemes

I have been working on Exercise 3.14b of Waterhouse Introduction to Affine Group Schemes and want to make sure I am on the right track. Suppose G is represented by A. Write down the map $\varphi: A \...
Chriswaluigi's user avatar
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1 answer
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Corepresentations of quantum subgroups

If $H$ is a Hopf subalgebra of the Hopf algebra $G$ and $\alpha:V\longrightarrow V\otimes H$ is right corepresentation of $H$ on the vector space $V$ (finite--dimensional), is there a way to induce a ...
AmSa's user avatar
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Tensor Product of Modules of Bialgebras

Lately I saw this post (in chinese) saying that the tensor product of modules relies on comultiplication, and the tensor product over a commutative algebra is a consequence of a canonical bialgebra ...
SalutaFungo's user avatar
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Morphisms between modules over a Hopf algebra

Let $U$ be a Hopf algebra over a field $K$ and $M,N$ be $U$-modules. Then the set of all $K$-linear maps $\textrm{Hom}_{K}(M,N)$ has a canonical $U$-module structure given by $$(u.f)(m) = \sum_{(u)} ...
Luka's user avatar
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The 'union of factors' comultiplication in a monoid ring?

let $\mathbb{Z}[M]$ be 'the' monoid ring of $M$ over $\mathbb{Z}$; that is to say (if my understanding is right) the set of finite linear combinations of elements of $M$, with product given by $\sum_n ...
Steven Stadnicki's user avatar
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Associated graded space as a (bi)algebra

A filtered bialgebra $(A,m,u,\delta,\epsilon)$ with $A_j,j\in\mathbb Z$ filtration is defined as a bialgebra so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$...
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Shuffle product formula for coproduct

I'm studying the coproduct $\Delta$ defined on a tensor algebra $T(V)$ and its action on tensor products of elements from a vector space $V$. The coproduct is given by $\Delta(v) = v \boxtimes 1 + 1 \...
Martin Geller's user avatar
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1 answer
60 views

Comultiplication on the tensor algebra

Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
Brendan Murphy's user avatar
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Affine group schemes

I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups. My question Let $k$ be an ...
Tommk's user avatar
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What are the primitive elements of tensor algebra

In a bialgebra $(H,m,u,\Delta,\epsilon)$, subspace of primitive elements are $P(H)=\{x\in H:\Delta (x)=x\otimes 1+1\otimes x\}$. I know that if $x,y$ are primitive, then $[x,y]=xy-yx$ is also ...
Eric Ley's user avatar
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4 votes
2 answers
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Grouplike Hopf algebras are group rings?

Let $H$ be a commutative and cocommutative Hopf algebra over an algebraically closed field $k$. I've read that if $H$ is grouplike in the sense that it has no nonzero primitive elements, then $H$ is ...
tcamps's user avatar
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Antipode of a Hopf algebra being an antihomomorphism: unable to follow the proof

A PhD thesis contains the following proof that antipode of a Hopf algebra is algebra antihomomorphism (page 22): Here $\nu = \eta \circ \varepsilon$, where $\eta$ is the unit map and $\varepsilon$ is ...
Daigaku no Baku's user avatar
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Solution of the Yang-Baxter equation not coming from quasi-triangular structure

Let $A$ be an associative, unital algebra over a field $\Bbbk$, and let $R \in A \otimes A$ be an invertible element which is a solution of the Yang-Bater equation in $A \otimes A \otimes A$ $$R_{12}...
Minkowski's user avatar
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1 answer
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Hopf algebra structure on the ring of functions from an infinite group

A basic algebra statement confuses me. This is in Greenlees' Equivariant Formal Group Laws and Complex Oriented Cohomology Theories, p. 229 (in the journal). If $k$ is a field, $A$ is an infinite ...
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Group isomorphism $\operatorname{Tgt}_e(G) \cong \operatorname{Hom}_{k-\text{linear}}(\ker(\epsilon)/\ker(\epsilon)^2, k)$ for algebraic groups.

I'm reading from Milne's text https://www.jmilne.org/math/CourseNotes/RG.pdf, in chapter 8 on Lie algebras of (affine) algebraic groups $G$ over $k$. In it, he claims in 8.6 that there is an ...
stupid_questions's user avatar
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1 answer
90 views

Understanding a proof of a lemma for rigid categories [closed]

I'm reading the proof of the lemma 3.4 in the Bruguieres' paper on Hopf monads which claims the following: Lemma Let $F,G: \mathcal{C} \rightarrow \mathcal{D}$ be two strong monoidal functors and $\...
Andres Felipe Vargas Mican's user avatar
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1 answer
60 views

Commutative diagrams in the definition of bialgebras, what do they mean?

I am reading the definition of Bialgebras over a field $\mathbb{K}$. The definition is the following: A bialgebra over a field $\mathbb{K}$ is a vector space $B$ over $\mathbb{K}$ equipped with $\...
Saikat's user avatar
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About the regular representation of weak hopf algebra

In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now ...
popo's user avatar
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Non unital Hopf relation

The following problem is an exercise in Loday-Vallette's Algebraic Operads. I hope I am understanding this correctly. Any suggestions or hints would be appreciated. Show that the restriction of the ...
Chanel Rose's user avatar
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Haar integral of a finite dimensional Hopf algebra: an explicit expression

Let $\mathcal{H}$ be a finite dimensional Hopf algebra. A nonzero element $\Omega\in \mathcal{H}$ is called an integral in $\mathcal{H}$ if $$x~\Omega=\epsilon(x)\Omega,~~\forall x\in \mathcal{H}.\tag{...
Lagrenge's user avatar
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1 answer
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Intuition for Coproduct of Grouplike vs Primitive elements in a Coalgebra?

I'm trying to understand Hopf Algebras as a physicist with a limited background in abstract algebra, and I might be in a little over my head. In particular I'm trying to wrap my head around the fact ...
Alex Kritchevsky's user avatar
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1 answer
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Are antipodes of free, finite rank Hopf algebras over general rings invertible?

It is a well-known result by Larson and Sweedler that, for finite-dimensional Hopf algebras over a field, the antipode is always a linear isomorphism. My question is whether this property still holds ...
Minkowski's user avatar
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1 vote
1 answer
25 views

Definition of equivalent representation of Hopf algebras

I have a question about equivalent representation of Hopf algebras because I am not unfamiliar with Hopf algebras. Here is my question: If $\rho_1$ and $\rho_2$ are two representations of a Hopf ...
fusheng's user avatar
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1 vote
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88 views

Shuffle product is skew-commutative?

In this old paper, Harrison develops what is today known as Harrison cohomology. On page 192 Harrison writes: Let $k$ be a field and $A$ a commutative associative algebra over $k$. Let $T$ be the ...
Peter's user avatar
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Hatcher hopf algebra sign convention

I am reading 3.C in Hatcher's algebraic topology, where I was introduced to a sign convention for the product in $A\otimes A$ $$(a\otimes b)(c\otimes d)=(-1)^{|b||c|}(ac\otimes bd)$$ Does anybody ...
DevVorb's user avatar
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0 votes
0 answers
41 views

Explicit 2-cocycle on a Hopf algebra

I'm studying the theory of cocycles on Hopf algebras, but I would like to see an explicity 2-coclycle, for example, in the Lauret algebra $A=\mathbb{C}[z,z^{-1}]$. Is there someone who can write one? ...
AmSa's user avatar
  • 139
5 votes
1 answer
56 views

Are the group-like elements of a finite dimensional Hopf algebra finite?

Let $H$ be a finite dimensional Hopf algebra over a field $k$. Let $G(H)$ be the set of group-like elements of $H$. Is $G(H)$ finite?
Mec's user avatar
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3 votes
0 answers
122 views

Equivalent definitions of Hopf algebras

Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two ...
ferolimen's user avatar
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4 votes
1 answer
109 views

Showing that the comultiplication map $E_*E\to E_*E\otimes E_*E$ is co-associative for a flat homotopy commutative ring spectrum

Let $(E,\mu,e)$ be a flat homotopy commutative ring spectrum, so we have an isomorphism $$\Phi_E:E_*E\otimes_{\pi_*E}E_*E\to E_*(E\wedge E)$$ sending homogeneous elements $x:S^n\to E\wedge E$ and $y:S^...
Isaiah Dailey's user avatar
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1 answer
58 views

Are any subalgebras of the steenrod algebra isomorphic to the group algebra over for some group? [closed]

Heading says it all. Wondering if there are any subalgebras of the steenrod algebra which are isomorphic as hopf algebras to $\mathbb{F}_2{G}$ for some group $G$? In particular interest to me are the ...
categorically_stupid's user avatar
2 votes
0 answers
106 views

Is the preimage of a Hopf subalgebra a Hopf subalgebra?

The following must be simple, but I have no intuition here, so excuse me. Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces ...
Sergei Akbarov's user avatar
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0 answers
25 views

Why the enveloping algebra of a Lie algebra is a bialgebra? [duplicate]

If $\mathfrak{g}$ is a Lie algebra (over the field $k$ ), then we can construct its universal enveloping algebra $U(\mathfrak{g})$. We can define $\Delta:U(\mathfrak{g})\rightarrow U(\mathfrak{g})\...
ckx's user avatar
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2 votes
1 answer
116 views

What is the dual Hopf algebra of T(V)?

I am reading "Geometric versus Non-Geometric Rough Paths" by Martin Hairer and David Kelly. I don't know much about Hopf algebras but $T(V)$ is the one example I'm comfortable with so far ...
Theo Diamantakis's user avatar
1 vote
0 answers
47 views

If $H$ is a Hopf algebra deformation of $U (\mathfrak g)$ then $H/h H \simeq U (\mathfrak g)$ as a Hopf algebra.

Question $:$ If $H$ is Hopf algebra deformation of the universal enveloping algebra $U (\mathfrak g)$ for some Lie algebra $\mathfrak g$ according to the above definition then how to show that $H/h H \...
Anacardium's user avatar
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1 vote
0 answers
21 views

Is there any way to show that the algebraic tensor product $A[[h]] \otimes A[[h]] \subsetneq (A \otimes A) [[h]]\ $?

Let $A$ be a Hopf algebra over $k.$ Consider the formal power series $A[[h]]$ in $h$ over $A$ endowed with the $h$-adic topology. Then how do we show that $A[[h]] \otimes_{\text {alg}} A[[h]] \...
Anil Bagchi.'s user avatar
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0 votes
1 answer
77 views

Compute $\operatorname{Ext}_{\mathcal{A}_2}^{s,t}(\mathcal{A}_2/ I(\mathcal A_2 . Sq^1), \mathbb F_2).$

I am trying to learn computations of the ASS by myself from "user's guide of spectral sequences" book and here is the thing I want to compute: Compute $\operatorname{Ext}_{\mathcal{A}_2}^{s,...
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0 answers
79 views

Sweedler's notation and commutators of Hopf subalgebras

Let $A$ be a cocommutative $K$-Hopf algebra, where $K$ is a field. Given Hopf subalgebras $X,Y$, one can define the commutator of $X,Y$ to be the subalgebra $[X,Y]$ of $A$ generated by the elements $$\...
Lios's user avatar
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0 votes
1 answer
19 views

Compute $S(E^i F^j K^l)$ in $U_q$

Here is the question I am trying to solve: Compute $S(E^i F^j K^l)$ in $U_q.$ Here is my thoughts: Definition: We define $U_q = U_q(\mathfrak{sl}(2))$ as the algebra generated by the four variables $E ...
Emptymind's user avatar
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0 votes
2 answers
331 views

Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$

Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$ (as defined in the previous exercise; assume that the field $k$ is of characteristic zero.) Here is the exercise before it: Let $L$ ...
Emptymind's user avatar
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3 votes
2 answers
230 views

Determining the grouplike elements of a Hopf algebra

Here is the question I am trying to solve: For any Lie algebra $L$ determine the group of grouplike elements of the Hopf algebra $U(L).$ Some definitions: 1-To any Lie Algebra $L$ we assign an (...
Emptymind's user avatar
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2 votes
0 answers
39 views

Adjunction between groups and complete Hopf algebras

I'm trying to read Quillen's "Rational Homotopy Theory" and at one point he uses that there is an adjunction between the category of groups and complete Hopf algebras, given by $\hat{K}:Grp \...
categorically_stupid's user avatar
0 votes
1 answer
135 views

Proving that a symmetric bilinear form on $L$ is invariant

Let $L$ be a Lie algebra and $\rho: L \to \mathfrak{gl}(V)$ a finite-dimensional representation of L. Define a symmetric bilinear form on $L$ by $$\langle x, y \rangle_{\rho} = tr (\rho(x) \rho(y))$$ ...
user avatar
0 votes
0 answers
46 views

Why are the maps $\eta, \mu, \Delta, \varepsilon$ linear?

Here is the question I am trying to solve: (Divided powers) Consider the vector space $C = k[t]$ of polynomials in one variable. Prove that there exists a unique coalgebra structure $(C, \Delta, \...
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