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

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3
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1answer
19 views

Induced coaction on a vector space.

Let $H$ be a Hopf algebra and let $V$ be an $H$-module. Then we have an action $H \times V \to V$ given by $(h, v) \mapsto h.v$. This action induces an $H$-action on the dual vector space $V^*$ of $V$ ...
0
votes
1answer
17 views

What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$? It seems that the natural action comes from the following. We have a ...
1
vote
0answers
19 views

The completion of the ring of Laurent polynomials with respect to the augmentation ideal.

Let $A = \langle a\rangle$ be an infinite cyclic group on one generator. I'm trying to understand the completion $\widehat{\mathbb{Q}}A$ of the group algebra $\mathbb{Q}A$ with respect to the ...
1
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0answers
22 views

Composition of Dual Maps in (rigid) Monoidal Categories

If $X, Y$ are objects in monoidal category $\mathcal{C}$ which have left duals $X^∗, Y^∗$ and $f : X → Y $ is a morphism in $\mathcal{C}$, then the left dual map $f^∗ : Y^∗ → X^∗$ of $f$ is given by: ...
1
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0answers
18 views

Braided Hopf algebra - inverse of a given map

Let $(H,\nabla, \Delta, S,\eta,\varepsilon)$ be a Hopf algebra in a braided monoidal category $(\mathcal{C},\Psi)$. We also assume that $S$ is invertible. Let us define a map ...
0
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0answers
17 views

Is the category of H-bicomodules within the monoidal category of H-bimodules equivalent to the category of left H-comodules

Fix $\mathbb{k}$ a field. Let $H$ be a $\mathbb{k}$-quasi-bialgebra. Is there an equivalence $ {}_H^H \mathcal{M}_H^H \cong {}^H \mathcal{M}$ where ${}_H^H \mathcal{M}_H^H$ is the category of ...
1
vote
1answer
34 views

Braided Hopf algebra - properties of braiding

Let $(H,\nabla,\Delta,S)$ be a Hopf algebra in a braided category. I'm trying to simplify the following $(\nabla\otimes \mathrm{id}\otimes \mathrm{id})(\nabla \otimes \Psi \otimes ...
1
vote
1answer
25 views

Defining the dual-comodule of a comodule?

As is well known, every left module has a dual, which is a right module. How does this work for comodules? More explicitly, does there exist a notion of the ...
2
votes
1answer
12 views

Unit commutes with $H$-action.

Let $H$ be a Hopf algebra. Let $A$ be an $H$-module algebra. Then the unit map $\eta: k \to A$ commutes with the $H$-action. It is said that "$\eta: k \to A$ commutes with the $H$-action" is ...
2
votes
1answer
53 views

Braided Hopf algebra - properties

If $(H,\Delta,\nabla)$ is a Hopf algebra in the prebraided monoidal category $(\mathcal{C},\Psi)$ then $\Psi_{H,H}=\left(\nabla\otimes \nabla\right)\left(S\otimes\Delta \nabla\otimes ...
4
votes
1answer
28 views

Echalon decomposition in binary shuffle (Hopf) algebras

Consider a binary shuffle algebra $\mathcal{W}$ of two letters $a, b$. As usual the concatination of two words $u = u_1 \dots u_m$, $v = v_1 \dots v_n$ is defined as: $$u \bullet v := u_1 \dots u_m ...
4
votes
0answers
51 views

Do we have $\delta(ab)=\delta(a)\delta(b)$ implies $\Delta(cd)=\Delta(c)\Delta(d)$?

Assume that $B$ is an algebra which is also a coalgebras (we do not assume that $B$ is a bialgebra: we do not assume $\Delta(cd)=\Delta(c)\Delta(d)$). Assume that $A$ is $B$-comodule algebra. Then we ...
1
vote
0answers
32 views

From Hopf Algebras to quantum groups

I start with self study about quantum groups. Until now I covered Hopf $*$-algebras and there representations (for example by the book of Klymik and Schmüdgen). Now I want to understand the step from ...
4
votes
1answer
24 views

Quotient coalgebra by coideal

I want to show, that if $I$ is a coideal in a coalgebra $A$, then $A/I$ is a coalgebra. To be a coideal means $\Delta(I)\subset A\otimes I+I\otimes A$ and $\varepsilon(I)=0$. We can use the induced ...
2
votes
1answer
50 views

How do you evaluate elements of the Drinfeld double $D(H)$ against elements of $H^\ast$ or $H$?

Proposition 8.2 of Majid's primer on quantum groups says that if $H$ is a finite dimensional Hopf algebra with quantum double $D(H)$, then this is a factorizable Hopf algebra with quasi-triangular ...
2
votes
1answer
50 views

Why is the coproduct $\Delta$ of the quantum double $D(H)$ an algebra homomorphism?

If $H$ is a a finite dimensional Hopf algebra, $H^\ast\otimes H$ has a Hopf algebra structure with multiplication $$ (\phi\otimes h)(\psi\otimes g)=\sum \psi_2\phi\otimes h_2g\langle ...
2
votes
1answer
29 views

Why does $\operatorname{Ad}_h((S\otimes 1)(Q))=\epsilon(h)(S\otimes 1)(Q)$ in a quasi-triangular Hopf algebra?

I'm reading a proof that in a quasi-triangular Hopf algebra $H$, $(S\otimes 1)Q$ is $\operatorname{Ad}$-invariant. Here $Q=\tau(R)R$, where $R$ is the invertible element in $H\otimes H$ satisfying all ...
2
votes
0answers
33 views

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 $ ...
2
votes
1answer
50 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 ...
3
votes
1answer
33 views

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 ...
4
votes
1answer
40 views

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 $$ ...
2
votes
0answers
28 views

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 ...
0
votes
0answers
16 views

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 ...
4
votes
1answer
43 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 ...
8
votes
1answer
101 views

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 ...
1
vote
1answer
40 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 $$ ...
1
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0answers
15 views

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 ...
3
votes
1answer
36 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 ...
0
votes
0answers
9 views

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 ...
2
votes
0answers
43 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 ...
1
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0answers
37 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 \ \ ...
2
votes
1answer
25 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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0answers
28 views

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 ...
0
votes
0answers
25 views

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 ...
2
votes
0answers
31 views

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 ...
1
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0answers
22 views

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 ...
2
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0answers
42 views

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 ...
2
votes
0answers
25 views

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 ...
0
votes
0answers
26 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 ...
2
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0answers
12 views

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 ...
2
votes
0answers
29 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 ...
6
votes
1answer
52 views

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 ...
1
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1answer
36 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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0answers
14 views

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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0answers
20 views

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 ...
0
votes
1answer
22 views

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^*, ...
2
votes
1answer
37 views

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 ...
0
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0answers
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 ...
0
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0answers
34 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. ...
2
votes
0answers
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 ...