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

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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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23 views

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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18 views

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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1answer
35 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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1answer
51 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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84 views

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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57 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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13 views

How to show that $U(\mathfrak{n})^* \cong \mathbb{C}[U]$?

Let $U$ be the unipotent subgroup of $GL_n(\mathbb{C})$ consisting of all unipotent upper triangular matrices. Let $\mathfrak{n}$ be its Lie algebra. I heard many times that $U(\mathfrak{n})^* \cong ...
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13 views

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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2answers
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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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10 views

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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1answer
32 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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1answer
96 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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1answer
51 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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1answer
30 views

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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1answer
43 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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42 views

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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1answer
40 views

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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35 views

Actions of Hopf Algebras

i have a question about Hopf-algebras and Sweedler notation: Suppose that $A$ and $B$ both are Hopf algebras with a Hopf algebra pairing $p:A\times B\rightarrow\mathbb{C}$. I want to show that the ...
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1answer
49 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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2answers
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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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1answer
44 views

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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33 views

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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24 views

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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794 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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1answer
94 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 ...
3
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1answer
108 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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1answer
64 views

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
35 views

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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1answer
86 views

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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95 views

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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181 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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1answer
168 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 ...
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44 views

Quotient of the group schemes

When I study the quotient of the group schemes, I can not catch the key point. Many claims are similar with the prue group theory, but the prove are more long. I think that are march problems related ...
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1answer
40 views

Finitely-generated modules over a Frobenius algebra

Let $A$ be a Frobenius algebra over a field $k$ and let $P_1,\ldots, P_t$ be the principal indecomposable modules of $A_A$. If $W$ is a finitely-generated right $A$-module, then there exists ...
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1answer
75 views

A question on a Hopf algebra

Let $(H, \nu,\eta, \Delta, \epsilon, S)$ be a Hopf algebra. $S$ is the antipode. I am reading a proof of the fact $S(xy)=S(y)S(x)$. First, define maps $\nu, \rho$ in $\hom(H \otimes H, H)$ by ...
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Different definition of antipode for $SL_q(2)$?

In Majid's book (A Quantum Groups Primer) (pg11), the antipode for the Hopf algebra $SL_q(2)$ is defined as $Sd=a$, $Sa=d$, $Sb=-qb$, $Sc=-q^{-1}c$. However, in Kassel's book (Quantum Groups) (pg ...
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1answer
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Quantum Groups: prove $U_1'\cong U[K]/(K^2-1)$ and $U\cong U_1'/(K-1)$

Regarding this theorem, which is in Kassel pg 126, I have two questions. I have typed in the relevant material for reference. 1) How does $$U_1'\cong U[K]/(K^2-1)$$ imply $$U\cong U_1'/(K-1)$$? ...
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Hopf-like monoid in $(\Bbb{Set}, \times)$

I am looking for a nontrivial example of the following: Let a monoid $A$ be given with unit $e$, and two of its distinguished disjoint submonoids $B_1$ and $B_2$ (s.t. $B_1\cap B_2=\{e\}$), ...
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1answer
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Converse of Hopf Algebra Theorem

There is a theorem that if a Hopf algebra $H$ is commutative or cocommutative, then $S^2=id_H$, where $S$ denotes the antipode. May I know if the converse is true? (i.e. if $S^2=id_H$, does it ...
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Recovering Hopf Algebra from Group-Like Elements

Given the natural coalgebra structure on a group algebra $kG$, one can recover the group by taking the set of group-like elements of the coalgebra $kG$. When can you go the other way? In particular, ...
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Basis of $SL_q(2)$

While trying to show that $SL_q(2)$ is noncocommutative, I needed to prove the following fact: Show that the set $\{a^ib^jc^k\}_{i,j,k\geq 0}\cup\{b^ic^jd^k\}_{i,j\geq 0,k>0}$ is a basis of ...
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1answer
50 views

Tensor Product Question in Kassel's Quantum Groups

In Kassel's book on Quantum Groups, I am stuck on the following computation: \begin{eqnarray*} [\Delta (E), \Delta (F)] &=& \Delta (E)\Delta (F)-\Delta (F)\Delta (E)\\ &=& (1\otimes E ...
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2answers
103 views

How to show that $SL_q(2)$ and $U_q(\mathfrak{sl}(2))$ are noncommutative and noncocommutative

It is known that the two quantum groups $SL_q(2)$ and $U_q(\mathfrak{sl}(2))$ are both noncommutative and noncocommutative. May I ask how do we show that? I have attempted the following: To prove ...
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1answer
68 views

$U_q$ Quantum group and the four variables: $E, F, K, K^{-1}$

In Kassel's book on Quantum groups, it is defined that: "We define $U_q=U_q(\mathfrak{sl}(2) )$ as the algebra generated by the four variables $E$, $F$, $K$, $K^{-1}$ with the relations ...