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Questions tagged [hopf-algebras]

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

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1answer
50 views

Kernel of an algebra map and module of Kahler Differentials

Let $A$ be a $k$-algebra, $f:A\rightarrow k$ an algebra map with kernel $I$. I'd like to prove that $\Omega_A\otimes_f k $ is canonically isomorphic to $I/I^2$. This is from W.C.Waterhouse Intro to ...
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30 views

Differentials of Hopf Algebras

Let $A$ be a $k$-Hopf Algebra over some ring $k$, with augmentation ideal $I$. I would like to prove that the module of Khaler Differentials $\Omega_A$ of $A$ over $k$ is isomorphic to the tensor ...
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1answer
32 views

motivation of coalgebra

I don't know why should we need coalgebra?What is the motivation?By changing all the arrows of algebra structure, it seems strange.What is the application of coalgebra? What is the relation with Lie ...
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23 views

Suggestion of research papers

I’m trying to find the research papers related to Hopf algebra over non commutative polynomial rings O( M_n(H)) to get concrete understanding about it. But unfortunately I couldn’t find any ...
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17 views

Hopf Algebra Structure of $U_q(\mathfrak {sl}(2))$.

I am working out of Christian Kassel Quantum Groups. Define $U_q=U_q(\mathfrak {sl}(2))$ as the algebra generated by elements $E,F,K,K^{-1}$ subject to the following relations. $$ KK^{-1}=K^{-1}K=1\\ ...
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35 views

Characters of $\mathbb{G}_m$

Fix a field $k$. Let $\mathbb{G}_m$ be the multiplicative affine group scheme over $k$. A $k-$character $\chi$ of $\mathbb{G}_m$ is an endomorphism of affine group schemes $\mathbb{G}_m \to \mathbb{G}...
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26 views

Matrices over quaternions make Hopf Algebra or not?

I am learning Hopf algebra now a days. I am still confused about it’s axioms. I don’t know how to define antipode structure. What are the basic rules to define it. ? Can any one help me to solve this ...
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1answer
45 views

Unnecessary finite dimensionality requirement in Theorem 5.1.8 of Sweedler’s “Hopf Algebras”

I’m currently reading Sweedler’s Hopf Algebras, but am confused by the finite dimensionality in the following theorem: Theorem 5.1.8 A finite dimensional Hopf algebra $H$ is semi-simple as an ...
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1answer
90 views

Sorting out a homological computation concerning Hall algebras

Let me recall some basic definitions concerning Hall algebras given in for example Schiffmann's notes, Hubery's notes or even the excellent master thesis by Sjoerd Beentjes which you can find online. ...
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1answer
60 views

Do Hopf algebra representations have internal hom?

Suppose $G$ is a group, we can define the tensor representation of $V \otimes W$ by $g(x \otimes y) = gx \otimes gy$. And we can define internal Hom by $iHom(V,W)=Hom_k(V,W)$. The action of $g$ is ...
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16 views

Geometry of Hopf Fibration

Z1=(e^i(a+b)/2))*sin(c), Z2=(e^i(a-b)/2))*cos(c) What is the geometric meaning of the angles a and b c in this hopf map formula?
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27 views

Understanding homomorphism from coalgebra to algebra

Could someone please explain what exactly a homomorphism from coalgebra to algebra (from this paper: 1, page 10, definition 5.1). I understand a homomorphism as a map between two structures which ...
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1answer
21 views

An identity related to antipode of a Hopf algebra

Let $H$ be a Hopf algebra with a bijective antipode $S$. Does the equality $\sum\limits_{(h)} h_2 \otimes S^{-1}(h_1) = \sum\limits_{(h)} h_1 \otimes S(h_2)$ hold for any $h \in H$, where $\Delta(h)=\...
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28 views

Reference about the proof of this proposition?

``Let $M$ be a finite dimensional Hopf algebra with antipode $S$ and let $M^*$ be the dual bialgebra. Recalls that $x^* \in M^*$ is a primitive in $M^*$, then $x^*$ is a derivation of $M$. '' I am ...
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38 views

Understanding the proof of Cartier duality

I'm trying to understand the above proof of Cartier Duality. The step I don't understand is the following It says $$ \phi\psi(a) = ((\phi \otimes \psi) \circ \Delta(a) = (\phi \otimes \psi)( a \...
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15 views

$C$-Yetter Drinfeld morphism

I'm reading the paper ''quantization of the Hopf algebras of decorated planar rooted trees'' written by L. Foissy. But the definition of $C$-Yetter Drinfeld morphism isn't given in the paper. I think ...
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0answers
20 views

The quotient of a connected graded bialgebra

Let $k$ be a field. $H$ is called a connected graded bialgebra, if there are k-submodules $H^{n}$, $n \geq 0$, of $H$ such that: $H^0=k$; $H=\oplus _{n=0} ^{\infty} H^n$; $H^p H^q \subseteq H^{p+q}, ...
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26 views

Ribbon Element, Equivalence of Definitions

In their book Quantum Groups Chari and Pressley define the ribbon element of a quasitriangular Hopf Algebra algebra $(H,R)$ as a special element $\nu\in H$ such that $\nu$ is central in $H$ $\nu^...
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27 views

The proof of $S(gh)=S(h)S(g)$ of an antipode in Hopf algebras

Let $H=(H, m, \Delta, \mu, \epsilon, S)$ be a Hopf algebra. Then there is a property of antipde $S$: $$S(gh)=S(h)S(g)$$ where $g,h \in H$. I have seen in some materials the proof of this property ...
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48 views

$U(\mathfrak{g})$ is identified as Hopf algebra to $\mathbb{C}[[G]]^*$?

I was reading section 4.5.1 of https://arxiv.org/pdf/1801.00123.pdf but I got stuck at the following because I don't know much about Hopf algebras. Let $G$ be a complex Lie group and let $\mathbb{C}[...
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22 views

Modularity vs Antipode In the category $Bord^{\operatorname{or}}_{1,2,3}$

Supposedly the category $Bord^{\operatorname{or}}_{1,2,3}$ carries with it the structure of a Hopf algebra. If that's the case I would like to understand what the antipode is. To that end, I've ...
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0answers
31 views

Computing the ribbon element of $\operatorname{U_q}(\mathfrak{sl}_2)$

I'm trying to understand the computation of the ribbon element of $\operatorname{U_q}(\mathfrak{sl}_2)$. I've laid out an argument by Andre Henriques below. I don't understand how the conclusion (...
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1answer
54 views

Dual of cocyclic module is cyclic

I am working on a paper, which states that the dual of a finite dimensional cocyclic module is cyclic. I tried to write down a proof, but I failed and I do not know if this is true in full generality ...
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56 views

Lie algebra of a compact Lie group

Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
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36 views

A “concrete” example of a left Hopf algebra

I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition. To be more ...
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31 views

Sub-modules translate in sub-co-modules of the dual?

Is it true that if $W$ is a submodule of $V$ then by duality $W^*$ is a sub-co-module of $V^*$? Can anybody confirm that or give an example when this doesn't happen? I think it might be necessary to ...
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0answers
49 views

When is a monoidal structure on $\mathrm{Mod}_A$ induced by a bialgebra structure on $A$?

Fix a field $k$, and let $A$ be a (commutative, coassociative, counital) $k$-bialgebra. Write $\otimes = \otimes_k$. The category $\mathrm{Mod}_A$ of $A$-modules admits the structure of a monoidal ...
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1answer
25 views

Are these 2 definitions of $K$ and $H$ on $U(sl_q(2))$ coherent?

I'm studing $U(sl_q(2))$ and studying how to recover $U(sl(2))$ from $U(sl_q(2))$ I found these two definition for both $H$ and $K$ as formal generators. $$H=\frac{K-K^{-1}}{q-q^{-1}}$$ $$K=q^{H}$$ I'...
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1answer
27 views

Relations between representations and corepresentations of dually paired Hopf algebras

It is well known that if two Hopf algebras $A, B$ are dually paired and $\phi$ is a corepresentation of $A$ then it canonically induces a representation $R_\phi$ of $B$. I have a few "converse" ...
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42 views

What is wrong with this argument on $U_q(sl(2))$? What is missing to make it precise?

Let us define $U_q(sl(2))$ as the algebra with four generator as usual $$K\,K^{-1}=K^{-1}K=1,$$$$K\,E\,K^{-1}=q^{2}E,\,K\,F\,K^{-1}=q^{-2}F,\\EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.$$ I want to easily show ...
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0answers
21 views

Definition of non-degenerate copairing

Given a Hopf algebra $H$ over a field $1$ and a map $\lambda: 1\to H\otimes H$. What are the conditions turning $\lambda$ into a non-degenerate copairing? So I am asking for the definition.
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1answer
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When is a Hopf Algebra isomorphic to a group ring k[G]?

Let $H$ be a Hopf algebra over a field $k$. What are some nice conditions for when $H$ is isomorphic to $k[G]$ for a finite group $G$? The co-multiplication structure on the group algebra $k[G]$ is ...
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Some concrete examples of $M_q(2)$ points

Given $q \in C$ invertible, Kassel says that a $M_q(2)$ point of an $R$ algebra is a $m=\left(\begin{array}{cc} A & B\\ C & D \end{array}\right)\in R^{4}$ such that $A,\,B,\,C,\,D \in R$ ...
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3answers
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Concrete cases where $YX=qXY$

I was reading Kassel on the quantum plane and he defines an $R$-point on this plane as a pair of $X$, and $Y$ elements of the non commutative algebra $R$ such that $$YX=qXY,$$ with $q$ invertible. Can ...
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3answers
70 views

Existence of integrals in f.d Hopf algebras

In THE HAAR MEASURE ON FINITE QUANTUM GROUPS, van Daele gives an implausibly short proof of the existence of integrals in a finite-dimensional Hopf algebra. I'm probably overlooking something obvious,...
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1answer
52 views

q-quantization of Lie bialgebras

I am trying to understand the difference between the "Drinfeld" and the "Lusztig" theory of quantum groups, more specifically with respect to the problem of quantization of Lie bialgebras/Poisson Lie ...
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2answers
104 views

Definition morphism of Hopf algebra

this is my first post, so let me know if you need some more information. I am currently studying Hopf Algebras and and exercise I have tells me to show that $f:H \rightarrow H'$ is a morphism of ...
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0answers
64 views

Universal enveloping algebra vs algebra of continuous functions

I'm a physicist studying quantum groups, but this question is about the usual classical Lie groups (though it is related to the language that is used to describe the deformation to quantum groups, ...
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1answer
95 views

group-like elements of a Hopf algebra and the group algebra

Suppose we are given a $n$-dimensional Hopf algebra $H$ over field $\mathbb K$. I want to prove that $H$ contains n distinct group-like elements if and only if there exists a group $G$ such ...
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1answer
37 views

Checking antipode on Enveloping algebra of a Lie Algebra

Let $U\left(\mathfrak{g}\right)$ be the enveloping algebra of $\mathfrak{g}$.Let's define comultiplication, counit and antipode as $$\triangle\left(X\right) =X\otimes1+1\otimes X,$$ $$\epsilon\left(X\...
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0answers
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Are “grouplike elements” in quasi-Hopf algebras still invertible?

Suppose $H$ is a quasi-Hopf algebra with non-trivial evaluation $\alpha$. I cannot find any sources about grouplike elements in $H$. What I mean by "grouplike" is $$ \Delta(g) = g\otimes g \quad\text{...
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2answers
73 views

Checking the antipode for the dual of the Group Hopf Algebra

Consider the vector space $\mathbb{C}G\,=\left\{ f:\,G\longrightarrow\mathbb{C}\right\}$ , and define the following multiplication: $\mu\left(f\otimes g\right)\left(x\right):=f\left(x\right)g\...
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0answers
22 views

Free module over $H$-module algebra

Let $H$ be a finite dimensional Hopf algebra, $R$ be a $H$-module algebra and $V$ be a finite dimensional $H$-module such that $R\otimes_{k} V$ is a finitely generated $R$ module under the action: $r.(...
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1answer
79 views

Whether we can define the finitely generated coideal?

In the question How to understand the coidea of a colagebra?, we had posed the definition of coideals. Let $R$ be a unitary commutative ring and $X$ be a finite subset of $R$. Then the ideal ...
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0answers
53 views

Kernels of Hopf algebra homomorphisms

Let $f:A \to B$ be a homomorphism of Hopf algebras. I think that the subset $N:= \{a \in A \, |\, f(a) \in \mathbb{C}\cdot 1_B \, \}$ is a normal sub-Hopf algebra of $A$ (even though I am not sure if $...
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0answers
24 views

Duality between comodules $SL_q(2)$ and $U_q(sl(2))$

There's a welle known Hopf pairing between $SL_q(2)$ and $U_q(sl(2))$, I have no problem in construction such pairing wich can be easly be done by the coefficients of a 2-dimensional irreducible ...
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1answer
127 views

Reference on correspondence between commutative Hopf Algebras and Groups

Is it true that every commutative Hopf algebra is related to a Group in such a way that the co-multiplication is originated from the multiplication of the group, the antipode from the inverse? Making ...
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1answer
194 views

Convolution in Hopf algebras

For each Hopf algebra $H$ its space ${\mathcal L}(H)$ of operators $A:H\to H$ is usually endowed with the operation of convolution by the identity $$ A*B = \mu \circ (A\otimes B)\circ \varDelta $$ ...
2
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1answer
54 views

A question on counit of a coalgebra

I am reading an very interesting paper ''The infinitesimal Hopf algebra and the poset of planar forests'' https://arxiv.org/pdf/0802.0442.pdf written by Pro. Foissy. In his paper, on page 4, I don't ...
2
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1answer
90 views

The kernel of a morphism of co-rings is a co-ideal

I would like to show that The Kernel of a coring morphism $\phi:C\rightarrow C'$ between two $R$-corings $(C,\Delta,\epsilon)$ and $(C',\Delta',\epsilon')$ is a coideal. The only point that I can'...