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

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An R-matrix in a quasitriangular Hopf algebra

I am new to the theory of Hopf algebra. So I am sorry if the following question has really trivial answer. Suppose that we have a quasi triangular Hopf algebra with the universal R-matrix $R$. It ...
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28 views

Motivation behind Quasitriangular Hopf algebra

I would like to know why it is interesting to define the quasi-triangular structure on a Hop algebra. I understand that the pseudo-co-commutative (the existence of an intertwining operator between the ...
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18 views

A triangular Hopf algebra and its unitary R-matrix

Why is the R-matrix of a Hopf algebra called unitary when it satisfies the relation $$R^{-1}=R_{12},$$ I would say invertible, why then call it unitary? Is that a nomenclature that maybe comes from ...
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6 views

Cartier dual of an exact sequence

Suppose we have an exact sequence of affine finite flat commutative group schemes over an arbitrary ring $R$: \begin{equation} 0\rightarrow H\xrightarrow{i} G\xrightarrow{j} K\rightarrow 0 ...
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Defining the quantum group $U_q(\mathfrak{sl}_2)$

I've seen two defining relation for $U_q(\mathfrak{sl}_2)$ by the Serre relations $$[H,E]=E,\quad[H,F]=-F, \quad [E,F]=\frac{q^H-q^{-H}}{q-q^{-1}}, $$ or by taking $K=q^H$ $$KK^{-1}=K^{-1}K=1,\quad ...
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What does the notation $U(\frak{g})[[\hbar]]$ mean?

I'm reading the following motivation for studying quantum group but I'm unfamiliar with the double bracket notation in $$U(\frak{g})[[\hbar]].$$ Is this a special set of polynomials with coefficient ...
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35 views

Extension of homomorphism in multiplier algebra

I'm reading Timmermann's "An invitation to quantum groups and duality" and I have a problem in understanding following theorem: Every non-degenerate homomorphism $\phi : A \rightarrow M(B)$extends ...
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44 views

Example of Hopf categories and conatural transformation

Let $\mathcal{C}$ be a category. It is well known how to internalize the notion of category. Let $(C_0,C_1)$ be an internal category, with source $s$, target $t$, composition $c$ and unit $e$. One can ...
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39 views

Comultiplication in graded Hopf Algebras

Let $H$ be a graded Hopf algebra over some commutative ring $k$. I'm looking for a proof of the following result, which seems to be stated in various locations. For $x$ in $H$ of degree $n$ ...
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19 views

Reference request: product in $\mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$.

Let $a \otimes b, a'\otimes b' \in \mathbb{C}_q[X] \otimes \mathbb{C}_q[Y]$, where $X, Y$ are two algebraic varieties. Suppose that algebraic group $T$ acts on $X, Y$. Then there are coactions ...
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24 views

Is the coaction $\delta: A \to H \otimes A$ injective?

Let $A$ be an algebra and let $H$ be a bialgebra. Suppose that $A$ is an $H$-comodule. Then we have a coaction $\delta: A \to H \otimes A$. Is the coaction $\delta: A \to H \otimes A$ always ...
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Cocommutative k-Hopf algebra finite as k-vector space represents a constant group functor

I am working through some of my first exercises regarding Hopf algebras and I am kind of stuck with this one: Given an algebraically closed field $k$ and a cocommutative $k$-Hopf algebra $A$ finite ...
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31 views

Describe invariants using coaction.

Let $X$ be an algebraic variety and $G$ be an algebraic group which acts on $X$. We know that an invariant function $f$ in the coordinate ring $\mathbb{C}[X]$ is a function such that $g(f) = f$ for ...
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1answer
26 views

Coaction of a product.

Let $G$ be a group and let $X$, $Y$ be two algebraic varieties on which $G$ acts. Let $\delta_1: \mathbb{C}[X] \to \mathbb{C}[G] \otimes \mathbb{C}[X]$ be a coaction given by $\delta_1(f) = \sum ...
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Tensor product via the diagonal action of a Hopf algebra

Let $H$ be a Hopf algebra and $V$ and $W$ two left $H$-modules, then $V\otimes W$ is also a left $H$-module via the comultiplication of $H$. I now want to consider the functor $-\otimes_H (V\otimes ...
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Quotients and Hopf algebras

Let $A,B$ be $k$-Hopf algebras, for an algebraically closed field $k$. It follows that $h^A = $Hom$(A,-)$ and $h^B = $Hom$(B,-)$ are group functors. Now consider a natural transformation $\phi: h^A ...
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47 views

Frobenius map and Hopf algebras

I was wondering if I can get some help understanding a problem. Namely, consider the Frobenius map of $\mathbb{F}_p$-algebra $f:A \rightarrow A$ given by $a \mapsto a^p$. I showed that if $A$ is an ...
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57 views

Group objects in the category of rings

Are there group objects in: $\text{Ring}$ $\text{CRing}$ If so, why doesn't anyone talk about them? On the other hand, $$ \begin{align} cogroup \ objects \ in \ \text{CRing} &= co(group \ ...
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Is $\mathbb{H}P^\infty$ an H-space or not?

$\mathbb{R}P^\infty$ is H-space. $\mathbb{C}P^\infty$ is H-space. Is $\mathbb{H}P^\infty$ an H-space or not?
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64 views

Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$ Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether ...
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2answers
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Do coalgebras arise outside the study of bi/Hopf-algebras?

Hopefully the title is fairly self explanatory. I'm curious as to whether the coalgebra structure (that is, a vector space with a comultiplication and counit) comes up an any area of mathematics not ...
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How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$?

Let $H$ be a Hopf algebra and $V$ a finite dimensional $H$-module. How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$? Thank you very much.
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What is $\Delta(1)$ for $1$ in $U(\mathfrak{g})$?

Let $\mathfrak{g}$ be a semisimple Lie algebra and $U(\mathfrak{g})$ its universal enveloping algebra. Then $U(\mathfrak{g})$ is a hopf algebra. Is $\Delta(1) = 1 \otimes 1$ or $\Delta(1) = 1 \otimes ...
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Hopf Algebras Arising From Fourier Transforms?

At least for nice functions, a Fourier series is just a Laurent series on a circle (which means just substituting $z = re^{i\theta}$ into a Laurent series), so I can see Fourier analysis on Abelian ...
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34 views

The comultiplication on $\mathbb{C} Q $ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=Q$ the ...
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98 views

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
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56 views

Identities that connect antipode with multiplication and comultiplication

The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities: $$ S\otimes S\circ \varDelta=\sigma\circ\Delta\circ S $$ $$ \nabla\circ S\otimes ...
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Isomorphism of affine group schemes of rank 2

I was reading Waterhouse's book "Introduction to Affine Group Schemes", when I found, in Chapter 2, an exercise about classification of Affine Group Schemes of Rank 2. I proved essentially that every ...
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Classification of finite group schemes

I was reading an article of Tate about $p$-divisible Groups and, just at the beginning, I read this example. Consider $a,b\in R$, where $R$ is a commutative ring, such that $ab=2$. Put $A=R+Rx$, with ...
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Induced actions.

Let $U$ be the subgroup of $GL_n$ consisting of all unipotent upper triangular matrices. Suppose that there is an action of $U$ on a variety $X$. Let $\mathbb{C}[U]$ be the group algebra of $U$ and ...
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50 views

Is cocycle condition necesarry for coassociativity of coproduct and why is it called “cocycle condition”?

Let $\Delta_0$ and $\Delta$ be coproducts related by \begin{equation} \Delta h = \mathcal F \Delta_0 h \mathcal F^{-1} \end{equation} where $\mathcal F \in H\otimes H$ is Drinfeld twist and $h \in H$ ...
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Recovering a restricted Lie algebra from its restricted enveloping algebra

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $\mathfrak{g}$ be a restricted Lie algebra, with restricted enveloping algebra $u(\mathfrak{g})$. We can place a Hopf ...
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Are there always nontrivial primitive elements in a Hopf algebra?

Let $k$ be an algebraically closed field of arbitrary characteristic. Let $H$ be a Hopf algebra over $k$. We say $x\in H$ is a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$, where $\Delta$ ...
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50 views

Natural map of extension groups

Let $\Lambda$ be a cocommutative Hopf algebra over a commutative ring $R$. For two left $\Lambda$-modules $M$ and $N$, interpret $\mathrm{Ext}_{\Lambda}^n(M,N)$ as the set of equivalence classes of ...
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co-idempotents: algebraic dual of an idempotent element?

So many times you can write out the axioms for an algebraic structure (say an algebra over a ring) as commutative diagrams and then reverse all the arrows and get a new structure: say a coalgebra. ...
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Integral Homology of $BU$

We know that the integral cohomology of $BU$, $H^*(BU) = Z[c_1,c_2,...]$ where $|c_i| = 2i$ is the $i$th Chern class, with coproduct $\Delta c_i = \Sigma c_j\otimes c_{i-j}$. And at almost ...
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Comultiplication of sum

If $a$ and $b$ are elements of a Hopf algebra over a field $k$ and $\alpha, \beta \in k$, then what is $\Delta(\alpha a+\beta b)$? Is it $\alpha\Delta(a)+\beta\Delta(b)$? For example if $\Delta(x)=x ...
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Internal Homs in Modules over a Hopf Algebra

Given a Hopf algebra $H$, I wonder when the monoidal category of $H$-modules is left-closed, right-closed, and finally, under what circumstances right and left internal hom are isomorphic. If I'm not ...
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Booleanity implies commutativity, the case of a Hopf algebra.

If a group $G$ satisfies $g^2=e, \forall g\in G,$ then $gh=gh(hg)^2=ghhghg=gghg=hg, \forall g, h\in G,$ thus $G$ is commutative. Since Hopf algebras correspond to groups, one should obtain a ...
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871 views

What are Quantum Groups?

I am going to do a semester project (kind of a little thesis) this spring. I met a professor and asked him about some possible arguments. Among others, he proposed something related to quantum groups. ...
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A characterisation of the cyclic subfactors by the existence of a cyclic vector?

A cyclic subfactor is a subfactor admitting a distributive intermediate subfactors lattice. Let's start with the finite index irreducible depth 2 subfactors, i.e. the class of subfactors of the form ...
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Embedding $\mathbb{G}_a$ into $GL_2$

Let $k$ be an algebraically closed field of characteristic $p$. I'd like to find interesting examples of closed embeddings $\mathbb{G}_a(k)\hookrightarrow GL_2(k)$, where $\mathbb{G}_a(k)$ is ...
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102 views

Anti-homomorphism of Hopf Algebra

I've got a quick question regarding the anti-homomorphism property. Specifically, what does it actually mean?? For a bit of context, I have the following question. We define $U_q = U_q[o(3)]$ to be ...
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113 views

Hopf Algebra - Adjoint Representation

I've been asked to prove the following; $$ a \circ (bc) = \sum_{(a)} (a_{(1)} \circ b)(a_{(2)} \circ c)$$ Using the fact that the adjoint representation is as follows; $$ a \circ b = \sum_{(a)} ...
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The Hopf algebra structure of $GL(n, K)$.

Let $K$ be an algebraic closed field and $G=GL(n, K)$. There is a Hopf algebra structure on $K[G]=K[T_{11}, T_{12}, \ldots, T_{nn}, d^{-1}]$, $d=\det (T_{ij})$ given by $e^*(T_{ij})=\delta_{ij}$, ...
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1answer
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Question about the definition of left normal morphism of augmented algebras

On the renowned "On the Structure of Hopf Algebras" by Milnor and Moore, there is a definition of "left normal morphism of augmented algebras." It says as follows. If $A$ and $B$ are augmented ...
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Fusion rules and comultiplication for a Kac algebra

Let $\mathbb{A}$ be a Kac algebra (i.e. a Hopf C*-algebra) of finite dimension, and $\mathbb{A}^{*}$ its dual : Is there a formula revealing the fusion rules for the irreducible ...
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Applications of a theorem of Cartier and Gabriel

In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel: Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $ ...
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187 views

Hopf Algebras in Combinatorics

I know that many examples of Hopf algebras that come from combinatorics. But I'm interested in knowing how Hopf algebras are applied in solving combinatorial problem. Are there examples of open ...
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187 views

Computation of the hom-set of a comodule over a coalgebra: $Ext_{E(x)}(k, E(x)) = P(y)$.

First of all, since every other book somehow mentions that this is trivial, I apologize if it turns out that I am just misunderstanding something in the definitions. So here goes: The motivation for ...