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

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2
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1answer
28 views

Structure theorem for non connected graded Hopf algebras

Let $V^{\bullet}$ be a graded vector space. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected Hopf algebra and in particular its Lie algebra of primitive ...
1
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1answer
21 views

Antipode map, Hopf algebra

A Hopf algebra $H$ (over field $k$) is a bialgebra $(H,m,u,\Delta,\epsilon)$ (H, product, unit, coproduct, counit), with an antipode map, $S:H\to H$ such that $$\sum (Sh_{(1)})h_{(2)} = \epsilon(h) = ...
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0answers
27 views

Does the antipode of a f.d. Hopf algebra have finite order

In a lecture I have heard that the antipode of a finite-dimensional Hopf algebra is conjectured to be finite, and that this has only been proven in characteristic $0$ by Larson and Radford in 1988. I ...
2
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0answers
31 views

$G$ equivariant quasicoherent sheaves on $X$ as compatible $G$ actions on the total spaces?

Let $G$ be an algebraic group, and $X$ a scheme on which $G$ acts: i.e the $S$ points of $G \times X \to X$ is a group for each affine $S$. Let $F$ be a quasicoherent sheaf on $X$. There is a notion ...
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31 views

More Hopf algebra confusion: Verifying an equation between matrix coefficients.

I am thinking of the following situation: The lie algebra $g = sl_2(\mathbb(C))$, and $V(1)$ is the unique dimension 2 irreducible representation (the defining representation). Let $U$ be the ...
2
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1answer
32 views

$V(1)$ generates the tensor category of representations of $sl_2(\mathbb{C})$ - what exactly does this mean?

Does anyone know what it means for a $V(1)$ to generate the tensor category of (finite dimensional) representations of $sl_2(\mathbb{C})$. Here $V(1)$ is the 2 dimensional irreducible Lie algebra ...
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1answer
24 views

Algebra of matrix coefficients in the dual of a hopf algebra, confusing verification

Let $H$ be a Hopf-algebra, and let $V$ be a finite dimensional $H$-module (a module for the algebra structure of $H$). For $f \in V^*$ and $v \in V$, we get $c^V_{f,v} \in H^*$ via $c^V_{f,v}(u) = ...
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8 views

Classification of group schemes of order 2: Why is $sx=x\otimes 1+1\otimes x-cx\otimes x$?

In their paper Group Schemes of Prime Order, Tate and Oort state Suppose $$G=\operatorname{Spec}(A)\text,\quad S=\operatorname{Spec}(R) \text,$$ and suppose the augmentation ideal ...
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37 views

Why does a skew-linear form on kG determine a triangular structure on k[G]?

I'm trying to understand braidings on finite group representations. They are the same as quasitriangular structures on the group algebra $k[G]$. The original reference seems to be ...
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1answer
33 views

$\text{Hom}_k(M,N)\cong M^*\otimes_k N$ as Hopf-algebra modules.

I'm reading Representations and Cohomology by D.J. Benson. At the beginning of the third chapter the following is explained: Let $\Lambda$ be a bialgebra over $R$ and $M,N$ left $\Lambda$-modules. We ...
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0answers
17 views

Skew primitive elements of the Taft Hopf algebra

Is there any reference where I can find the skew primitive elements of the Taft Hopf algebra? The Taft algebra is defined here: https://en.wikipedia.org/wiki/Taft_Hopf_algebra By a skew primitive ...
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0answers
18 views

Generators of locally finite part of $U_q(sl(2))$

I want to know the following: If one has the quantum group $U_q(sl(2))$ one can have a look at the locally finite part as introduced in http://arxiv.org/pdf/1307.3642v3.pdf (Definition $3.1$). From ...
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17 views

Two different comultiplications on a Hopf algebra

I am pretty sure this statement is false : let $K$ be a field and let $(A, \eta, \mu, \Delta, \epsilon, c)$ be a Hopf Algebra. If we forget the comultiplication $\Delta$, is it forced by $(A, \eta, ...
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40 views

What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?

Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that ...
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0answers
21 views

Homomorphisms of quantum spaces

Suppose we have a look at the Hopf $*$-algebra $U_q(sl(2))$ and the Hopf $*$-algebra $A_q(\widetilde{S}^3)$ introduced in the paper: http://arxiv.org/pdf/q-alg/9605017v1.pdf. I want to find a relation ...
3
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1answer
37 views

If $N$ is a subcomodule of $M$, then the H-submodule generated by $N$ is a H-Hopf-module of $M$.

Let $H$ be a Hopf algebra with $S$ its antipode and let $M$ be a right $H$-Hopf-module. We will write $\rhd$ the action of the left $H^*$-module on $M$ induced by the structure of right $H$-module, ...
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15 views

Analog of Hopf algebra structure for field of fractions

Let $G$ be a linear algebraic group. Then there is an additional structure on $k[G]$ called structure of Hopf algebra. Question: Is there an extra structure on field of fractions $k(G)$?
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1answer
27 views

Hopf algebras and “unifying” representation theory

We know that representations of a lie algebra $\mathfrak{g}$ can be studied by looking at the representations of the associative algebra $\mathcal{U}(\mathfrak{g})$, the universal enveloping algebra. ...
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1answer
32 views

Graphical calculus for braided Hopf algebras

I am trying to understand the graphical calculus presented in Ohtsuki's book Quantum Invariants. I think if I understand these first few examples it will help me greatly. Let ...
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0answers
31 views

Motivation for looking at the coalgebra structure of incidence algebra resp. group algebra

In basic combinatorics course we learn about the incidence algebra of a poset. Now i've read that one could start by defining the co-incidence algebra and then it is a fact that the dual vector space ...
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1answer
15 views

On proposition I.3.2 of 'Quantum groups' by Kassel.

I am reading the book Quantum groups by Kassel. In proposition I.3.2 at the very beginning the reader is asked to show that under the identifications made, the maps $\Delta,\varepsilon$ and $S$ ...
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1answer
34 views

The notation $A^{cop}$.

Let $A$ be an algebra. Then $A^{op}$ is the algebra with multiplication defined by $a \cdot b = b \circ a$, where $b \circ a$ is the multiplication in $A$. Let $A$ be a coalgebra. How to define the ...
3
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67 views

How to check if an algebra is not an Hopf Algebra

Let $k$ be a field and $A$ a $k$-algebra. Are there known techniques to check that $A$ does not admit a structure of Hopf algebra? In particular if $H$ is a Hopf $k$-algebra and $M$ an $H$-bimodule, ...
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1answer
20 views

How to show that $J$ is a coideal of $C$ if and only if $J^{\perp}$ is a subalgebra of $C^*$?

Let $C$ be a coalgebra and $J$ a subvector space of $C$. How to show that $J$ is a coideal of $C$ if and only if $J^{\perp}$ is a subalgebra of $C^*$? Here $J^{\perp} = \{f\in C^*: \langle f, v ...
0
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1answer
66 views

Understanding tensor-products in the commutative diagram of a k-Algebra

I'm having trouble getting to grips with the commutative diagram for an algebra over a field $k$. The main problem is that my understanding of the tensor product is weak. I have seen $V \otimes W$ ...
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1answer
36 views

Beginning the study of coalgebras and the sigma notation.

I'm beginning the study of coalgebras and the sigma notation using the book called Hopf Algebras of M.E. Sweedler. I'm doing the exercises and I don't know if this ideas are realy clear for me and if ...
4
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1answer
28 views

Group Ring to Hopf Algebra — I'm missing something simple

I've been working through Federico Ardila's online Hopf algebra lectures and hoped to check my understanding so far by constructing the Hopf algebra of a very small group ring from scratch. But I've ...
4
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1answer
73 views

Augmented coalgebras

Let $C, \Delta$ be a coalgebra. Assume that it is coaugmented with coaugmentation $u\: : \: k\to C$ and co unit $\epsilon\: : \ C\to k$. Since $\epsilon\circ u=id $ we get $$ C=\text{Kern}(\epsilon ...
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0answers
24 views

What is the meaning of “B is a bialgebra covariantly acting on A”?

Let $A$ be an algebra and $B$ a bialgebra. What is the meaning of "covariantly" in "B is covariantly acting on A"? Thank you very much. Edit: it is on line 13 of the abstract of the file (page 3).
3
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1answer
26 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
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1answer
25 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 ...
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0answers
29 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 ...
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0answers
35 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: ...
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0answers
22 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 ...
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23 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
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1answer
40 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 ...
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1answer
47 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
14 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
60 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
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1answer
31 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
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0answers
68 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 ...
2
votes
1answer
55 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
48 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
61 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
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1answer
54 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
31 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
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0answers
52 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
55 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
53 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 $$ ...