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

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Drinfeld Double definition

A while ago I was doing a reading course on Link Invariants and I came across the notion of a Drinfeld Double: given a Hopf algebra, H, the Drinfeld Double, D(H), was a quasi-triangular Hopf algebra. ...
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Endomorphism ring of an irreducible comodule

Given a Hopf algebra over $\mathbb{C}$, and a irreducible comodule $V$, I can show that Mor$(V,V) \simeq \mathbb{C}$, where Mor$(V,V)$ is the space of comodule maps from $V$ to $V$. My proof uses that ...
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37 views

How to understand the formula $\Delta(x_1 \otimes x_2) = \Delta(x_1)\Delta(x_2)$?

In the webpage, it is said that the comultiplication on the tensor algebra $TV$ is defined as follows. \begin{align} \Delta(x_1\otimes\dots\otimes x_m) = \sum_{p=0}^m ...
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Compatibility of Yetter-Drinfeld modules.

Let $H$ be a Hopf algebra. A Yetter-Drinfeld module over $H$ is a triple $(V, \cdot, \delta)$, where $\cdot : H \otimes V \to V$ , $\delta : V \to H \otimes V$ are actions and coactions respectively, ...
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68 views

Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will ...
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How to proof equivalent condition of algebra morphism and coalgebra morphism about Hopf algebra

My question is why $\mu$ is multiplication preserving iff $\mu\circ m=(m\otimes m)\circ(1\otimes \tau \otimes1)\circ(\mu \otimes \mu)$ and this holds iff $m$ is comultiplication preserving, where ...
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34 views

Multiplication map for algebras in the sense of Hopf algebra

Let $R$ be a commutative ring with unity and $A$ a ring with unity. In the definition of algebras, we require a multiplication map $\mu: A\otimes A \rightarrow A$ satisfying certain property. Indeed, ...
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Help with a statement in P.286 of Hatcher

In P.286 of Hatcher's 'Algebraic Topology', it is stated that $\alpha_{p^i}$ is primitive in $ \bigotimes_{i\geq 0} \mathbb{Z}_p[\alpha_{p^i}]/(\alpha_{p^i}^p)$, where 'primitive' means the coproduct ...
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27 views

Alternative Cotensor Definition

Let $H$ be a Hopf algebra, and $(M,\rho_r)$ and $(N,\rho_l)$ right and left $H$-comodules respectively. As usual, we define their cotensor product to be $$ M \square_H N := \text{ker}\{(\rho_r ...
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66 views

Why is Hom$_A(M,A)$ a right $\Gamma$ comodule?

I'm reading through appendix I (Hopf algebroids) of Ravenel's green book, and I came across a line I can't understand in a proof. The part of the lemma I'm interested in states: $\mathbf{Lemma ...
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81 views

Is the special orthogonal group really rationally homotopy commutative?

It is a classical result that the rational cohomology of $SO(n)$ is given by: $$H^*(SO(2m); \mathbb{Q}) = \begin{cases} S(\beta_1, \dots, \beta_{m-1}, \alpha_{2m-1}) & n = 2m \\ S(\beta_1, \dots, ...
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1answer
25 views

Why is $e_m$ a character?

Let $k$ be algebraically closed. A character on an algebraic group $G$ over $k$ is a group homomorphism $G \rightarrow k^{\ast}$. If this is also a morphism of varieties, then this is called a ...
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35 views

Examples of Algebraic quantum groups

I am reading articles about algebraic quantum groups, which are defined (see A. Van Dael) as a regular multiplier Hopf algebra $(A,\Delta)$ for which there exists a non-zero functional $\varphi$ on ...
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Why is the group $[\Sigma\Sigma X, Y]_{\ast}$ commutative?

Can anyone give a reference (or explain here), why the group $[\Sigma\Sigma X,Y]_*$ is commutative? How is it related to the fact that $\Sigma X$ is a co-H-space?
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80 views

What are the natural surjections in the proof of Hopf's classification theorem?

I am currently reading Hatcher's book, trying to understand the proof of Hopf's classification Theorem on Hopf algebras that says the following: Every Hopf algebra $A$ that is commutative and ...
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61 views

An interpretation of this construction giving an operad from a bialgebra?

Let $A$ be a cocommutative bialgebra object (or even a Hopf algebra) in a symmetric monoidal category. Define an operad $\mathtt{P}_A$ by $\mathtt{P}_A(r) = A^{\otimes r}$ (so that $\mathtt{P}_A(0) = ...
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Why do we need the condition “cocommutative” in the definition of a coPoisson Hopf algebra?

In this paper, page 5, Section 3.6, in the definition of a coPoisson Hopf algebra $H$, it is said that: a coPoisson Hopf algebra is a cocommutative Hopf algebra $A$ with a map $\delta: A \to A \otimes ...
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73 views

Relation of $G$-invariants and $g$ -invariants.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $H$ a Hopf algebra and $M$ an $H$-module. By definition, $m \in M$ is called invariant if $x.m=\epsilon(x)m$, $\forall x \in H$, where $\epsilon: H ...
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55 views

Summing over components of a basis: Coalgebra

Let $(H,m,u,\Delta,\epsilon,S)$ be a Hopf algebra with product $m$, unit $u$, coproduct $\Delta$, counit $\epsilon$ and antipode $S$. Coproduct of an element: $$\Delta: H\to H\otimes H,\quad ...
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$\mathbb{G}_m$ action on $Spec A$ is equivalent to grading… what does coassociativity do?

I can interpret the other axioms for a grading via the definition of a group scheme action appropriately (I think): Let $H = k[x,x^{-1}]$. We are given a coaction $ \rho : A \to A \otimes H$, which ...
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1answer
48 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 ...
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1answer
42 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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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 ...
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36 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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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 ...
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1answer
35 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
26 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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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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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
44 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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23 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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73 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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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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27 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 ...
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40 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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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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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
41 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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40 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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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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36 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 ...
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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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23 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 ...
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77 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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37 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 ...
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1answer
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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 ...
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1answer
88 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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25 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).
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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 $V$ ...
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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 ...