For questions about Hopf algebras and related concepts, such as quantum groups.

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tensor products of Hopf algebras

Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ ...
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32 views

If $\Delta$ is cocommutative in $k(G)$, why does that imply $G$ is abelian?

Suppose $G$ is a finite group and $k(G)$ is it's group function Hopf algebra. I read that for $k(G)$ is quasi-triangulated requires that $G$ be abelian. If $R$ is the distinguished element of ...
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localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$

Let $X$ be an $H$-space. Let $F$ be a field. Then $H_*(X;F)$ is a Hopf algebra over $F$. According to group-like elements in the Hopf algebra of the homology of H-spaces, $$ \pi_0(X)=\{g\in ...
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group-like elements in the Hopf algebra of the homology of $H$-spaces

Let $X$ be an $H$-space with product $\mu$. Then $H_*(X)$ is a Hopf algebra with product $\mu_*$. Let $\psi$ be the coproduct of the Hopf algebra $H_*(X)$. Define a subset $S$ of $H_*(X)$ as $$ ...
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Do complete Hopf algebras have an antipode?

I am reading Quillen's paper on rational homotopy theory. In appendix A of this paper Quillen defines a notion he calls complete Hopf algebras. These are certain cocommutative bialgebra structures ...
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Counit of the universal comeasuring of an algebra?

Suppose $A$ is an algebra, and $(M(A),\beta_U)$ is the initial comeasuring of $A$. It's known that $M(A)$ is a bialgebra, say with coproduct $\Delta$. If $k$ is the base field, then ...
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30 views

Hopf algebra of graphs

Let $B$ be the set of isomorphism classes of finite graphs. Let $V$ be the $k$-vector space freely generated by $B$. I have heard that $V$ carries the structure of a Hopf algebra, and would like to ...
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different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283, the notion Hopf algebra is defined as follows: (ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
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34 views

Coadjoint action $\operatorname{Ad}^*_\phi(h)$ respects coproduct $\Delta$?

In Majid's quantum group primer at the beginning of Chapter 3, page 18, he's proving that if $H'$ and $H$ are dually paired bialgebras or Hopf algebras, the coadjoint action $$ ...
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What does it mean for $\operatorname{Ad}_g(-)$ to be trivial on commutative Hopf algebra $H$?

Just curious, what does it mean for the adjoint action to be trivial on a commutative Hopf algebra $H$? Does it mean $\operatorname{Ad}_g(f)=\epsilon(g)f$, where $\epsilon$ is the counit, or ...
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29 views

Notions of free (and/or cofree) Hopf algebras?

I don't know if this somewhat vague question has a precise answer, but I'd appreciate thoughts and references. Let $k$ be a field. I am looking at Hopf algebras over $k$ by which I mean an algebra ...
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How does the dual pairing map for Hopf algebras extend to tensor products?

If $H$ and $H'$ are Hopf algebras, they may be dually paired by a map $\langle,\rangle\colon H'\otimes H\to k$ satisfying various properties, one being, $\langle \alpha\beta,h\rangle=\langle ...
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39 views

Algebra structure on dual to coalgebra

I'm trying to prove the following theorem using braided diagrams: Let $(C,\Delta,\varepsilon)$ be a finite-dimensional coalgebra. There is an algebra structure on $C^*$ given by multiplication ...
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34 views

Yetter-Drinfeld modules over Hopf algebra

Let $H$ be a bialgebra and assume that $M$ is left $H$-module and left $H$-comodule. $M$ is called Yetter-Drinfeld module iff $\forall h\in H, \ \forall m \in M \ \ ...
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1answer
22 views

Universal enveloping algebra as bialgebra

If $\mathfrak{g}$ is a Lie algebra (over $k$ ), then we can construct its universal enveloping algebra $U(\mathfrak{g})$. We can define $\Delta:U(\mathfrak{g})\rightarrow U(\mathfrak{g})\otimes ...
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Isomorphism between sum of tensor products

In Sweedler book "Hopf algebras" (p.229) there is theorem that, if $(C_n)_{n\ge 0}$ is a filtration of coalgebra $C$ ( i.e. $C_0\subset C_1\subset \cdots\subset C, \ C=\bigcup_{n}C_n$ and ...
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The lowest term of coalgebra filtration contains coradical

In one article about Hopf algebras I found a statement that if $C$ is a coalgebra and $(C_n)_{n\ge 1}$ is a coalgebra filtration of $C$ ( i.e. filtration of $C$ as a vector space, such that ...
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Coradical of coalgebra

I'm reading a book about Hopf algebras and one definition is confusing for me. If $C$ is a coalgebra then we can define a coradical of $C$ as a sum of all simple subcoalgebras of $C$. Is there sum ...
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References about an action $U_q(n) \times \mathbb{C}_q[N] \to \mathbb{C}_q[N]$.

Let $N$ be the unipotent subgroup of a Lie group $G$ consisting of all upper triangular matrices and $n$ the Lie algebra of $N$. I found in some paper that there is an action $U_q(n) \times ...
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A question about the dual map of $A \otimes B \to B$.

Let $A$, $B$ be two algebras. Suppose that we have an action $\varphi: A \otimes B \to B$ and there is a pairing $\psi: A \otimes B \to \mathbb{C}$. The action $\varphi$ induces a map $\delta: B \to ...
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pontrjagin ring of the homology of iterated loop suspension

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243) I totally do not understand the proofs of these two theorems from page 228 to page 243 ...
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24 views

Intrinsic action on dual basis of a quantum group module

TL;DR. Is there a way to describe the action of a quantum group (that is, the action of the $E$'s and $F$'s) on the linear dual of a module (with action given by the antipode) explicitly in terms of a ...
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Hopf algebra associated to $GL_V$ for $V$ infinite rank?

Let $V$ be a free $\mathbb{Z}$-module of rank $n$, and let $F$ be the functor associating to a ring $R$ the group $Aut_R(V \otimes R)$. If $V$ was finite, say $Z^n$, this functor is representable by ...
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27 views

Equivalence between modules and comodules

Let $C$ be a coalgebra over a commutative ring $R$, if $C$ is cauchy (f.g. and projective), then there is an equivalence of categories between $\operatorname{Comod}(C)$ and the category ...
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Non-computational proof that $det(A)$ is a unit in $R$ implies $A$ is a unit, for $A \in M_n(R)$.

Quoting from Waterhouse's Introduction to Affine Group Schemes: "...suppose we have a representable functor $G$ [from $k$-algebras to groups], and a map $\Delta: A \to A \otimes A$ giving a ...
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35 views

what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
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Indecomposables generate the algebra

The following result seems intuitive, but I'm having hard time proving it, or determining if it is, in fact, false. Let $(A, \mu, \eta, \epsilon)$ be an augmented algebra, with augmentation ideal ...
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Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
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A computation for Manin triple.

I am reading the book. I have some questions about the computations in (4.1) on page 40. The computation are in the following. I don't know why $$ ([[e_r^*, e_k], e_s^*]+[e_r^*, ...
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graded Hopf algebra and its dual

I am learning Hopf algebras, and there are two questions as follows: Is the tensor product of two Hopf algebras still a Hopf algebra? Let $A$ be an infinite dimensional algebra. Is the dual ...
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9 views

The Hopf algebra of symmetric functions and quasi-symmetric functions

The algebras Sym of symmetric functions and Qsym of quasi-symmetric functions are Hopf algebras. As we know, Sym and Qsym are subalgebra of the form power series ring Q[[x_1,x_2,...]]. Are they the ...
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33 views

The maps $\delta: V \to B \otimes V$ and $\delta': V \otimes V^* \to B$.

Let $B$ be a coalgebra and $V$ a vector space. Suppose that we have a coaction $\delta: V \to B \otimes V$. Is the map $\delta$ equivalent to a map $\delta': V \otimes V^* \to B$? Thank you very much. ...
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40 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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108 views

An R-matrix in a quasitriangular Hopf algebra

I am new to the theory of Hopf algebra. So I am sorry if the following question has really trivial answer. Suppose that we have a quasi triangular Hopf algebra with the universal R-matrix $R$. It ...
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35 views

Motivation behind Quasitriangular Hopf algebra

I would like to know why it is interesting to define the quasi-triangular structure on a Hop algebra. I understand that the pseudo-co-commutative (the existence of an intertwining operator between the ...
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A triangular Hopf algebra and its unitary R-matrix

Why is the R-matrix of a Hopf algebra called unitary when it satisfies the relation $$R^{-1}=R_{12},$$ I would say invertible, why then call it unitary? Is that a nomenclature that maybe comes from ...
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Cartier dual of an exact sequence

Suppose we have an exact sequence of affine finite flat commutative group schemes over an arbitrary ring $R$: \begin{equation} 0\rightarrow H\xrightarrow{i} G\xrightarrow{j} K\rightarrow 0 ...
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Defining the quantum group $U_q(\mathfrak{sl}_2)$

I've seen two defining relation for $U_q(\mathfrak{sl}_2)$ by the Serre relations $$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$ or by taking $K=q^H$ $$KK^{-1}=K^{-1}K=1,\quad ...
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What does the notation $U(\frak{g})[[\hbar]]$ mean?

I'm reading the following motivation for studying quantum group but I'm unfamiliar with the double bracket notation in $$U(\frak{g})[[\hbar]].$$ Is this a special set of polynomials with coefficient ...
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40 views

Extension of homomorphism in multiplier algebra

I'm reading Timmermann's "An invitation to quantum groups and duality" and I have a problem in understanding following theorem: Every non-degenerate homomorphism $\phi : A \rightarrow M(B)$extends ...
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Example of Hopf categories and conatural transformation

Let $\mathcal{C}$ be a category. It is well known how to internalize the notion of category. Let $(C_0,C_1)$ be an internal category, with source $s$, target $t$, composition $c$ and unit $e$. One can ...
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Comultiplication in graded Hopf Algebras

Let $H$ be a graded Hopf algebra over some commutative ring $k$. I'm looking for a proof of the following result, which seems to be stated in various locations. For $x$ in $H$ of degree $n$ ...
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Reference request: product in $\mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$.

Let $a \otimes b, a'\otimes b' \in \mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$, where $X, Y$ are two algebraic varieties. Suppose that algebraic group $T$ acts on $X, Y$. Then there are coactions ...
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1answer
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Is the coaction $\delta: A \to H \otimes A$ injective?

Let $A$ be an algebra and let $H$ be a bialgebra. Suppose that $A$ is an $H$-comodule. Then we have a coaction $\delta: A \to H \otimes A$. Is the coaction $\delta: A \to H \otimes A$ always ...
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Cocommutative k-Hopf algebra finite as k-vector space represents a constant group functor

I am working through some of my first exercises regarding Hopf algebras and I am kind of stuck with this one: Given an algebraically closed field $k$ and a cocommutative $k$-Hopf algebra $A$ finite ...
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1answer
37 views

Describe invariants using coaction.

Let $X$ be an algebraic variety and $G$ be an algebraic group which acts on $X$. We know that an invariant function $f$ in the coordinate ring $\mathbb{C}[X]$ is a function such that $g(f) = f$ for ...
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Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
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Tensor product via the diagonal action of a Hopf algebra

Let $H$ be a Hopf algebra and $V$ and $W$ two left $H$-modules, then $V\otimes W$ is also a left $H$-module via the comultiplication of $H$. I now want to consider the functor $-\otimes_H (V\otimes ...
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Quotients and Hopf algebras

Let $A,B$ be $k$-Hopf algebras, for an algebraically closed field $k$. It follows that $h^A = $Hom$(A,-)$ and $h^B = $Hom$(B,-)$ are group functors. Now consider a natural transformation $\phi: h^A ...
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56 views

Frobenius map and Hopf algebras

I was wondering if I can get some help understanding a problem. Namely, consider the Frobenius map of $\mathbb{F}_p$-algebra $f:A \rightarrow A$ given by $a \mapsto a^p$. I showed that if $A$ is an ...