Questions tagged [hopf-algebras]

For questions about Hopf algebras and related concepts, such as quantum groups. The study of Hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups and has wide-ranging connections to fields from theoretical physics to computer science.

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Commutative diagrams in the definition of bialgebras, what do they mean?

I am reading the definition of Bialgebras over a field $\mathbb{K}$. The definition is the following: A bialgebra over a field $\mathbb{K}$ is a vector space $B$ over $\mathbb{K}$ equipped with $\...
Saikat's user avatar
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About the regular representation of weak hopf algebra

In the group theory we know $dim_{\mathbb{K}}(\mathbb{K}G)=dim_{\mathbb{K}}(V)=\sum_i (dim_{\mathbb{K}}V_i)^2$ where $V$ is regualr representaion and $V_i$ are irreducible representations. Now ...
popo's user avatar
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Non unital Hopf relation

The following problem is an exercise in Loday-Vallette's Algebraic Operads. I hope I am understanding this correctly. Any suggestions or hints would be appreciated. Show that the restriction of the ...
Chanel Rose's user avatar
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Haar integral of a finite dimensional Hopf algebra: an explicit expression

Let $\mathcal{H}$ be a finite dimensional Hopf algebra. A nonzero element $\Omega\in \mathcal{H}$ is called an integral in $\mathcal{H}$ if $$x~\Omega=\epsilon(x)\Omega,~~\forall x\in \mathcal{H}.\tag{...
Lagrenge's user avatar
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Intuition for Coproduct of Grouplike vs Primitive elements in a Coalgebra?

I'm trying to understand Hopf Algebras as a physicist with a limited background in abstract algebra, and I might be in a little over my head. In particular I'm trying to wrap my head around the fact ...
Alex Kritchevsky's user avatar
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Are antipodes of free, finite rank Hopf algebras over general rings invertible?

It is a well-known result by Larson and Sweedler that, for finite-dimensional Hopf algebras over a field, the antipode is always a linear isomorphism. My question is whether this property still holds ...
Minkowski's user avatar
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Definition of equivalent representation of Hopf algebras

I have a question about equivalent representation of Hopf algebras because I am not unfamiliar with Hopf algebras. Here is my question: If $\rho_1$ and $\rho_2$ are two representations of a Hopf ...
fusheng's user avatar
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Shuffle product is skew-commutative?

In this old paper, Harrison develops what is today known as Harrison cohomology. On page 192 Harrison writes: Let $k$ be a field and $A$ a commutative associative algebra over $k$. Let $T$ be the ...
Peter's user avatar
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Hatcher hopf algebra sign convention

I am reading 3.C in Hatcher's algebraic topology, where I was introduced to a sign convention for the product in $A\otimes A$ $$(a\otimes b)(c\otimes d)=(-1)^{|b||c|}(ac\otimes bd)$$ Does anybody ...
DevVorb's user avatar
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Explicit 2-cocycle on a Hopf algebra

I'm studying the theory of cocycles on Hopf algebras, but I would like to see an explicity 2-coclycle, for example, in the Lauret algebra $A=\mathbb{C}[z,z^{-1}]$. Is there someone who can write one? ...
AmSa's user avatar
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Are the group-like elements of a finite dimensional Hopf algebra finite?

Let $H$ be a finite dimensional Hopf algebra over a field $k$. Let $G(H)$ be the set of group-like elements of $H$. Is $G(H)$ finite?
Mec's user avatar
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Equivalent definitions of Hopf algebras

Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two ...
ferolimen's user avatar
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Showing that the comultiplication map $E_*E\to E_*E\otimes E_*E$ is co-associative for a flat homotopy commutative ring spectrum

Let $(E,\mu,e)$ be a flat homotopy commutative ring spectrum, so we have an isomorphism $$\Phi_E:E_*E\otimes_{\pi_*E}E_*E\to E_*(E\wedge E)$$ sending homogeneous elements $x:S^n\to E\wedge E$ and $y:S^...
Isaiah Dailey's user avatar
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Are any subalgebras of the steenrod algebra isomorphic to the group algebra over for some group? [closed]

Heading says it all. Wondering if there are any subalgebras of the steenrod algebra which are isomorphic as hopf algebras to $\mathbb{F}_2{G}$ for some group $G$? In particular interest to me are the ...
categorically_stupid's user avatar
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Is the preimage of a Hopf subalgebra a Hopf subalgebra?

The following must be simple, but I have no intuition here, so excuse me. Let $F$ and $G$ be Hopf algebras over a field $k$ (in the usual sence, i.e. Hopf algebras in the category of vector spaces ...
Sergei Akbarov's user avatar
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Why the enveloping algebra of a Lie algebra is a bialgebra? [duplicate]

If $\mathfrak{g}$ is a Lie algebra (over the field $k$ ), then we can construct its universal enveloping algebra $U(\mathfrak{g})$. We can define $\Delta:U(\mathfrak{g})\rightarrow U(\mathfrak{g})\...
ckx's user avatar
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Coalgebra presentation in terms of generators and relations?

The presentation of associative algebras in terms of generators and relations are very useful as they often give a simple description of a large, potentially infinite dimensional algebra using just a ...
Lagrenge's user avatar
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What is the dual Hopf algebra of T(V)?

I am reading "Geometric versus Non-Geometric Rough Paths" by Martin Hairer and David Kelly. I don't know much about Hopf algebras but $T(V)$ is the one example I'm comfortable with so far ...
Theo Diamantakis's user avatar
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How do you define a right coadjoint action?

The left coadjoint action of a finite-dimensional Hopf algebra $H$ on $H^{*}$ is $\sum\phi_{(2)}\langle h, (S\phi_{(1)})\phi_{(3)}\rangle$, where $h\in H$ and $\phi\in H^{*}$. How do you define a ...
Fred Wealthman's user avatar
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How to show that $v$ is invertible in $H\ $?

Let $(H, \mu, \eta, \Delta, \varepsilon, S)$ be a Hopf algebra. Let $\mathcal F \in H \otimes H$ be an invertible element such that $(1)$ $(\mathcal F \otimes 1) (\Delta \otimes \text {id}) (\mathcal ...
Anacardium's user avatar
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If $H$ is a Hopf algebra deformation of $U (\mathfrak g)$ then $H/h H \simeq U (\mathfrak g)$ as a Hopf algebra.

Question $:$ If $H$ is Hopf algebra deformation of the universal enveloping algebra $U (\mathfrak g)$ for some Lie algebra $\mathfrak g$ according to the above definition then how to show that $H/h H \...
Anacardium's user avatar
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Coradical filtration and socle series $C_n=\mathrm{Soc}^{n+1}(C)$

I am reading the book Hopf Algebras and Their Actions on Rings by Susan Montgomery. In page 64, she said $C_n=\mathrm{Soc}^{n+1}(C)$, where $C$ is a coalgebra with coradical filtration $\{C_n \}$ and ...
Z.B. Zuo's user avatar
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Is there any way to show that the algebraic tensor product $A[[h]] \otimes A[[h]] \subsetneq (A \otimes A) [[h]]\ $?

Let $A$ be a Hopf algebra over $k.$ Consider the formal power series $A[[h]]$ in $h$ over $A$ endowed with the $h$-adic topology. Then how do we show that $A[[h]] \otimes_{\text {alg}} A[[h]] \...
Anil Bagchi.'s user avatar
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Compute $\operatorname{Ext}_{\mathcal{A}_2}^{s,t}(\mathcal{A}_2/ I(\mathcal A_2 . Sq^1), \mathbb F_2).$

I am trying to learn computations of the ASS by myself from "user's guide of spectral sequences" book and here is the thing I want to compute: Compute $\operatorname{Ext}_{\mathcal{A}_2}^{s,...
Brain's user avatar
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Sweedler's notation and commutators of Hopf subalgebras

Let $A$ be a cocommutative $K$-Hopf algebra, where $K$ is a field. Given Hopf subalgebras $X,Y$, one can define the commutator of $X,Y$ to be the subalgebra $[X,Y]$ of $A$ generated by the elements $$\...
Lios's user avatar
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Compute $S(E^i F^j K^l)$ in $U_q$

Here is the question I am trying to solve: Compute $S(E^i F^j K^l)$ in $U_q.$ Here is my thoughts: Definition: We define $U_q = U_q(\mathfrak{sl}(2))$ as the algebra generated by the four variables $E ...
Emptymind's user avatar
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2 answers
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Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$

Find all invariant symmetric bilinear forms of $\mathfrak {sl}(2)$ (as defined in the previous exercise; assume that the field $k$ is of characteristic zero.) Here is the exercise before it: Let $L$ ...
Emptymind's user avatar
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3 votes
2 answers
176 views

Determining the grouplike elements of a Hopf algebra

Here is the question I am trying to solve: For any Lie algebra $L$ determine the group of grouplike elements of the Hopf algebra $U(L).$ Some definitions: 1-To any Lie Algebra $L$ we assign an (...
Emptymind's user avatar
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2 votes
0 answers
31 views

Adjunction between groups and complete Hopf algebras

I'm trying to read Quillen's "Rational Homotopy Theory" and at one point he uses that there is an adjunction between the category of groups and complete Hopf algebras, given by $\hat{K}:Grp \...
categorically_stupid's user avatar
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1 answer
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Proving that a symmetric bilinear form on $L$ is invariant

Let $L$ be a Lie algebra and $\rho: L \to \mathfrak{gl}(V)$ a finite-dimensional representation of L. Define a symmetric bilinear form on $L$ by $$\langle x, y \rangle_{\rho} = tr (\rho(x) \rho(y))$$ ...
Brain's user avatar
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Problem in understanding an argument regarding tensor product of algebras.

I am going through the chapter $8$ from the Lecture Notes on Quantum Groups regarding Hopf algebras by Pavel Etingof. Before going into the definition of Hopf algebras the author discussed some ...
Anacardium's user avatar
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Why are the maps $\eta, \mu, \Delta, \varepsilon$ linear?

Here is the question I am trying to solve: (Divided powers) Consider the vector space $C = k[t]$ of polynomials in one variable. Prove that there exists a unique coalgebra structure $(C, \Delta, \...
Brain's user avatar
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1 answer
11 views

Show that $HK^+$ is a coideal where $K$ is a subHopfalgebra of $H$, $K^+=\mathrm{ker}(\epsilon)\cap K$.

Show that $HK^+$ is a coideal where $K$ is a subHopfalgebra of $H$, $K^+=\mathrm{ker}(\epsilon)\cap K$. Because $\epsilon(ha)=\epsilon(h)\epsilon(a)=0$ for $a\in K^+$, $\forall h \in H$. What we need ...
Z.B. Zuo's user avatar
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0 votes
1 answer
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Intersection of a chain of coideals in k-coalgebras

In my previous post Why can't you consider coideal generated by sets, where consequently i've shown why intersection of coideals need not to be a coideal, i said that ... the nonempty family $\{ I \...
Rafael H.'s user avatar
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1 answer
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A morphism of bialgebras between two Hopf algebras is necessarily a morphism of Hopf Algebras.

Here is the question I am trying to solve: Use the previous exercise to show that a morphism of bialgebras between two Hopf algebras is necessarily a morphism of Hopf algebras. Here is the previous ...
Brain's user avatar
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3 votes
1 answer
111 views

Proving the uniqueness of a map

Here is the question I am trying to solve: (Tensor product of coalgebras) Let $(C, \Delta, \varepsilon)$ and $(C', \Delta ', \varepsilon ')$ be coalgebras. Show that the linear maps $\pi: C \otimes C' ...
Brain's user avatar
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Action on associated graded algebra inducing action on filtered algebra

Suppose $Q$ is a filtered algebra, with associated graded algebra $\text{gr}(Q)$. If we have an action of a ring $R$ on $\text{gr}(Q)$ (i.e. $\text{gr}(Q)$ is an $R$-module) then it seems clear that, ...
Ted Jh's user avatar
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Why can't you consider coideal generated by sets.

The therm "coideal generated by a set" don't exist in literature but didn't found anything explaining why, so i formulated an example of a 6-dimensional coalgebra in wich there's a 1-...
Rafael H.'s user avatar
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2 votes
1 answer
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What is the meaning of $(\mu \otimes \mu) ({\rm id}\otimes \tau \otimes {\rm id})$?

Here is one of the figures whose commutativity express the fact that $\Delta$ is a morphism of algebra: $\require{AMScd}$ $$\begin{CD}H\otimes H @>\Delta\otimes\Delta>> (H\otimes H)\otimes (H\...
Emptymind's user avatar
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0 answers
42 views

Change of scalars for comodules as adjunctions?

Let $k$ be a commutative ring and $f: C \to C'$ be a homomorphism of $k$-coalgebras (for simplicity, we can suppose that it is surjective so that $f(C) = C'$). There is a functor: $$f_*: {}^lC-comod \...
Dat Minh Ha's user avatar
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1 answer
60 views

Why we need an orthonormal basis?

Here is the question I am trying to solve: Prove that $\lambda$ is injective. Here is the definition of the linear map $\lambda$: Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their ...
Emptymind's user avatar
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1 vote
1 answer
147 views

What happens if $(f_1\otimes g_1)(u_1 \otimes v_1) = (f_2 \otimes g_2)(u_2 \otimes v_2)$?

Prove that $\lambda$ is injective. Here is the definition of the linear map $\lambda$: Let $f: U \to U'$ and $g: V \to V'$ be linear maps. We define their tensor product $f \otimes g: U \otimes V \to ...
Emptymind's user avatar
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1 vote
1 answer
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Poincare Series of a graded algebra (revisited)

Here is the question I am trying to solve: Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
Secretly's user avatar
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2 votes
1 answer
95 views

Poincare series of a graded algebra

Here is the question I am trying to solve: Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
Secretly's user avatar
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2 votes
1 answer
50 views

If $\Delta(c) = \Delta^{\mathrm{op}}(c)$, then under which permutations is $c_{(1)} \otimes \dotsb \otimes c_{(n)}$ invariant?

Let $(C, \Delta)$ be a coalgebra and $c\in C$ an element with $\Delta(c) = \Delta^{\mathrm{op}}(c)$. For certain permutations $\sigma \in S_n$, we will have that $$c_{(\sigma(1))} \otimes \dots \...
Andromeda's user avatar
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1 vote
1 answer
66 views

Which algebraic subvarieties of a group variety have a group structure?

Let $G$ be an algebraic group. Given an algebraic subvariety $X\subseteq G$, is there any way simple criterion to determine whether $X$ has a group structure or not? For example, when $X$ and $G$ are ...
kindasorta's user avatar
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2 votes
1 answer
121 views

Grouplike elements is a group in a Hopf algebra

I have been looking at this question and two questions arose for me: (1) A simpler one: Why is $(g \otimes g)(h \otimes h)=(g h \otimes g h)$? My guess: $$ (g \otimes g)(h \otimes h) = g^2(1 \...
Martin Geller's user avatar
1 vote
1 answer
79 views

Understanding tensor product of modules over Hopf algebras

Let $A$ be an algebra over a commutative ring $k$ and $M$ and $N$ be modules over $A.$ Is there any natural way to define tensor product of $M$ and $N$ over the algebra $A\ $? My idea is that since $...
Anil Bagchi.'s user avatar
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1 vote
1 answer
360 views

Can we define tensor product of modules over an algebra?

Let $A$ be an algebra over a commutative ring $k$ and $M$ and $N$ be modules over $A.$ Is there any natural way to define tensor product of $M$ and $N$ over the algebra $A\ $? My idea is that since $...
Anil Bagchi.'s user avatar
  • 2,802
1 vote
1 answer
57 views

An isomorphism in a note by Witherspoon

Let $A$ be a Hopf algebra over a field $k$. In these notes by Witherspoon, we find the following lemma For the second part, the hint given is that $A$-intertwiners from $U$ to $V$ are the same as ...
Jo Mo's user avatar
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