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

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Why not use the Cartesian product for the monoidal category of modules? Why use the tensor product?

At least for vector spaces, the Cartesian product is a direct sum (categorical product and coproduct) operation. Looking at the definition of monoidal category on Wikipedia, the Cartesian product/...
2
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1answer
28 views

Hopf gradings on complex commutative group rings

Let $G$ be a finite abelian group. The complex group ring $\mathbb{C}G$ admits a structure of Hopf algebra when the multiplication is the usual multiplication in a group ring and the co-multiplication ...
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1answer
45 views

Inverse limit of Hopfalgebras

My Question relates to Corollary 2.7 of http://www.jmilne.org/math/xnotes/tc.pdf So Let $k$ be a field and $\mathbb{G}_i$ be an projective system of affine $k$-groupschemes. I want to know if the ...
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2answers
30 views

Is the tensor product of two Yetter-Drinfeld modules a Yetter-Drinfeld module?

Let $U,V$ be two Yetter-Drinfeld modules over a bialgebra $H$. Is $U \otimes V$ a Yetter-Drinfeld modules over $H$? Thank you very much.
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0answers
20 views

When $H$ is a Yetter-Drinfeld module over itself? [closed]

Let $H$ be a bialgebra. When $H$ is a Yetter-Drinfeld module over itself? Thank you very much.
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0answers
20 views

How to show that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding?

Let $H$ be a bialgebra and ${}_H^H YD$ the category of Yetter-Drinfeld modules over $H$. It is said that Yetter-Drinfeld condition is equivalent to the condition of $H$-action commutes with braiding. ...
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1answer
12 views

Easy References for understanding Grossmann and Larson rooted trees?

I am an undergraduate student doing a project on rooted trees. I was wondering if anyone would know any easy to understand references that explains Grossman and Larson's Hopf Algebra on rooted trees? ...
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1answer
48 views

The inverse of the braiding $c: V \otimes W \to W \otimes V$.

In the article. It is said that the inverse of the map $$ {\displaystyle c_{V,W}:V\otimes W\to W\otimes V}, \\ {\displaystyle c(v\otimes w):=v_{(-1)}{\boldsymbol {.}}w\otimes v_{(0)},} $$ is $$ {\...
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0answers
15 views

Reference for the equivalence of categories between the categories of affine group schemes and commutative hopf algebras.

Where can I find the proof or a discussion that the category of affine group schemes is equivalent to the category of commutative hopf algebras? Thanks.
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1answer
46 views

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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1answer
24 views

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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0answers
47 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 \sum_{\sigma\in\mathrm{Sh}_{p,m-p}...
2
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1answer
29 views

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, $...
9
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1answer
75 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 ...
3
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3answers
61 views

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 $\...
4
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2answers
39 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, ...
2
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0answers
46 views

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 $...
3
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1answer
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 \...
3
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0answers
69 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 A1....
4
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1answer
86 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, ...
2
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1answer
27 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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0answers
36 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 $A$...
5
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1answer
66 views

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?
4
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1answer
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 ...
4
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0answers
64 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) = ...
4
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1answer
50 views

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 ...
4
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1answer
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 \...
3
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1answer
58 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 \Delta(...
4
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0answers
48 views

$\mathbb{G}_m$ 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 ...
2
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1answer
51 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 $P(...
2
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1answer
51 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) = \...
3
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0answers
51 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 ...
3
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0answers
40 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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0answers
39 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
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) = f(...
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0answers
13 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 $I=\...
4
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0answers
46 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 http://arxiv.org/pdf/...
2
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1answer
47 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 ...
2
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0answers
26 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 ...
4
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1answer
74 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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0answers
48 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
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 ...
3
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1answer
42 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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0answers
18 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
34 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
42 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 $(A,m,i,\Delta,\epsilon,...
2
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0answers
42 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
20 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 $S$ ...
0
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1answer
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 ...