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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Regular functions on torsors

Let $X \rightarrow Y$ be a torsor for a linear algebraic group $G$ (i.e. $X$ is a principal $G$-bundle over $Y$). Assume also that both $X$, and $Y$ are affine. What can be said about $\mathbb{C}[X]$ ...
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When is an image of a Hopf algebra a Hopf algebra?

Suppose $R\subseteq S$ is a flat extension of rings and $A/R$, $B/S$ are flat Hopf algebras. Let $\varphi:A\otimes_R S\to B$ be a surjective $S$-Hopf algebra homomorphism. When is it the case that $\...
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How exactly does the coaction on the comodule X*⊗X work?

I'm struggling a bit with Sweedler notation. Let $(H,∆,ε,S,m,u)$ be a Hopf algebra over a commutative ring $k$ and let $X,Y$ be right $H$-comodules which are finitely generated projective as $k$-...
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How to construct irreducible representations of a Hopf algebra based on a finite non-Abelian group (beyond quantum double)?

I'm ultimately trying to construct a Hopf algebra based on a finite non-abelian group $G$ such that its irreducible representations: are labeled by $g\in G$ and $\rho\in\text{irr}(G)$ have the fusion ...
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Coinvariants of coaction $a\otimes b \mapsto \sum{\sigma_i(a)}\otimes \sigma_i(b)\otimes \sigma_i^*$

I've been studying Hopf-Galois Theory and currently I'm trying to understand some examples by writing all the explanations step by step by myself. The example I'm interested now is the classical ...
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Is this a valid q-deformation of $SU(2)$ via the $SU(3)$ Lie algebra?

In my physics research, I stumbled upon a rather simple deformation of the $SU(2)$ algebra, and I was wondering whether it qualifies as a `$q$-deformed Lie algebra' (a notion which is unfamiliar to me)...
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Coordinate algebra of image of homomorphism between linear algebraic groups

Let $f: G\to H$ be a homomorphism of linear algebraic groups. Let $f^*: k[H]\to k[G]$ be the corresponding Hopf algebra morphism. Then $f^*$ factors as $k[H] \twoheadrightarrow k[H]/I\hookrightarrow ...
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Subgroup of rational point of linear algebraic group

Sorry for my bad English. Let $k$ be a field, and $G=$Spec $A$ be a linear algebraic group over $k$. Then $k$-rational point $G(k)$ is abstract group, and take any a (abstract)subgroup $H\subset G(k)$...
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  • 1,010
1 vote
1 answer
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Left ideals of group algebra $K[\mathbb H]$

I've been studying Hopf-Galois theory, and while writing by myself certain example, I came up with this question: Let $K$ be a field, and $\mathbb H$ the quaternion group. Is $K[\mathbb H]$ the only ...
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Group algebra for quaternion group

I'm trying to understand Hopf Galois Theory, and I decided to try studying some example of a non commutative ring extension. The papers I've studied tell me that, for a strongly $G$-graded algebra $A$ ...
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$H$-comodule structure of $A\otimes_K A$

I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and need some help with something that the paper does not define (and that I'm not able to find anywere). Consider ...
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Prove $A$ is $K[G]$-comodule algebra

I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and there's something I'm trying to prove for what I may need some help. More precisely, is this statement from ...
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$c_{W,V}c_{V,W}(v \otimes w) = (R_{21}R)(v\otimes w)$ in Hopf Algebra

I am having trouble in seeing here (in the highlighted equation) how $$c_{W,V}c_{V,W}(v \otimes w) = (R_{21}R)(v\otimes w)$$ I can't find this identity in the book before. This is from the book "...
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  • 551
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A morphism of coalgebras related to Binomial coefficients

I am looking for a hint to prove that $$\Delta({t\choose n})= \sum\limits_{i=0}^{n} {t\choose i} \otimes {t\choose n-i} $$ where ${t\choose k}= \frac{t(t-1)...(t-k+1)}{k!}$ is a polynomial in $t$, $\...
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Does every bialgebra whose shear map is an isomorphism admit an antipode?

Let $H$ be a bialgebra in a braided (if necessary symmetric) monoidal category with unit $\eta$, multiplication $\mu$, counit $\epsilon$ and comultiplication $\Delta$. My question is: If we assume ...
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Does H satisfies the polynomial as the crossed product?

Let H be a Hopf algebra and H measures an algebra A. Let $A\sharp_{\sigma}H$ be a crossed product. $A\sharp_{\sigma}H$ satisfies a polynomial $f(x_1,...,x_m)$, that is , $\forall a_i\sharp h_i, f(a_1\...
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1 answer
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What is a cocentral homomorphism

Suppose $A$ is a bialgebra, $H$ a Hopf algebra. A map $f:A \to H$ is a cocentral bialgebra homomorphism. What does it mean? What about central homomorphisms?
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Cocommutative bimonads: Why does this diagram commute?

1. Definitions Let $(C, \otimes,I, a, l,r,c)$ be a monoidal category with braiding $c:\otimes \rightarrow\otimes ^{op}$. Let $(S,\mu,\eta,\tau,\theta)$ be a bimonad on $C$. Following Turaev and ...
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Tensor product of braided bialgebra/Hopf algebra

Braided Hopf algebra, is a Hopf algebra object in braided category. In consequence, it has an usual bialgebra structure and satisfies braided-compatibility with braided antipode map. In the case of ...
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2 votes
1 answer
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Stuck showing that the group ring $\mathbb{C}[G]$ is a bi-algebra

I am learning Hopf algebras via the following online notes on geometric representation theory, https://www.maths.ed.ac.uk/~djordan/QGpublic.pdf Exercise 1.7 of the Hopf Algebra section asks to show ...
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Is Spec $\mathbb{C}[-] $ exact?

I am struggling to find a reference to understand the following fact. Let $0 \to A \to B \to C \to 0$ be a short exact sequence of abelian groups. I first apply the functor $\mathbb{C}[-]$ taking ...
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$T(z) e^{-\partial_z} $ for Yangian is a Manin matrix

Let $T(u)$ be the generating matrix of the Yangian $Y(\mathfrak{gl}_n)$ of $\mathfrak{gl}_n$. So we have the identity $[T_{ij}(u),T_{kl}(v)]=\frac{1}{u-v}(T_{kj}(u)T_{il}(v)-T_{kj}(v)T_{il}(u))$. We ...
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1 answer
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How to prove the following results for $H$-bimodule $V$?

Suppose $(H, \cdot, \eta, \Delta,\varepsilon,S)$ be a Hopf algebra, and $V$ be a (finite-dimensional) $H$-bimodule. Then, how can we prove $\alpha\dot{}h-h.\alpha = 0$ for all $h \in H$ if we have $\...
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1 answer
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Drinfeld associators expansion to higher weights

Does anyone know where to get higher orders (for weight >10) expansion in Drinfeld associators. (the generating function for MZVs) The answer might be in the form of (4.5) in https://arxiv.org/pdf/...
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  • 99
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1 answer
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Question about hopf algebra and actions.

Let $H$ be a Hopf algebra and $M,N$ $H-$modules on the left, then $H$ define a structure of module to $Hom(M,N)$ given by $(h*f)(m) = h_1f(S(h_2)m)$, where $S$ is the antipode. My question is kind of ...
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  • 1,636
2 votes
1 answer
65 views

Question about Hopf algebra

Let $H$ be an Hopf Algebra, $M,N$ $H$-modules on the left, then we can define in $Hom(M,N)$ a natural structure of $H-$module (on the left) given by $(h*f)(m) = h_{(1)}f(S(h_{(2)})m), \forall m \in M$,...
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2 votes
0 answers
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Proof of Tannaka recognition theorem

I am trying to prove the theorem 5.12.7 in the book “Tensor categories” by Etingof, Gelaki, Nikshych and Ostrik. The statement is as follows: The assignments $(\mathcal{C},F) \mapsto H = \mathrm{End}(...
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1 vote
1 answer
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Supposse $U, V$ and $W$ are subspaces of a coalgebra $C$. Show that $\Delta(U) \subseteq V\otimes W$ implies $U\subseteq V\cap W$.

I'm new to coalgebras and this is a question from section 2.1 of the book "Hopf Algebras" from Davied E Radford. I tried to pick an element $u \in U$, so $\Delta(u) = u_1\otimes u_2 \in V\...
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1 answer
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Exercise 2.3.24 - Radford's Hopf Algebras

I am having trouble solving the following problem, which is Exercise 2.3.24 of Radford's Hopf Algebras: Let $C = C_n(k)$, where $n \geq 1$, and let $\{e_{i,j}\}_{1 \leq i,j\leq n}$ be a standard ...
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0 answers
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How to find all the regular subgroups of a permutation group of a non abelian group normalised by the left regular representation

Given a finite non abelian group $G$, and it's permutation group $Perm(G)$ how would I go about finding all of the regular subgroups $N$ of $Perm(G)$? I'm looking at applying the theorem of Greither ...
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2 votes
0 answers
72 views

Does quasi-triangularity imply invertibility of the antipode?

I have been studying quantum groups from Christian Kassel's Quantum Groups side-by-side with Majid's Foundations of Quantum Group Theory. I noticed that while Majid proves that the antipode of a quasi-...
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1 answer
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Self-duality of $U_q(b_+)$ proof (Majid, A Quantum Groups Primer, Proposition 2.5)

I am reading A Quantum Groups Primer by Shahn Majid, and I'm having trouble filling in the details for the proof of Proposition 2.5, which states that the Hopf algebra $U_q(b_+)$ is self-dual. $ \...
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0 votes
1 answer
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Invertibility of quasitriangular Hopf algebra element using Sweedler notation

The question concerns part of a theorem in the book Foundations of Quantum Group Theory, Shahn Majid (Cambridge University Press, 1995). More specifically, Theorem 2.3.4 (p.55-57) which I'll rewrite ...
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1 answer
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Steenrod's interpretation of the Hopf invariant. [closed]

Where can I read about Steenrod's interpretation of the Hopf invariant? Is there any reference?
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Hopf-Galois structures of cyclic type on a dihedral or quaternionic extension

Let $L/K$ be a dihedral or quaternionic finite field extension, that is such that $Gal(L/K)$ is either a dihedral or a quaternion group. How many Hopf-Galois structures of cyclic type are there on ...
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About the structure of a Hopf algebra on universal enveloping algebras of Lie algebras

We know that the universal enveloping algebra construction provides a functor from Lie algebras to cocommutative Hopf algebras which is left adjoint to the primitive functor. Furthermore, if we ...
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2 votes
0 answers
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For an algebraic group $G$, a $G$-module induces a $Dist(G)$-module

I'm reading Representation of Algebraic Groups of Jantzen about Distribution of Algebra. In chapter 7, page 103, 7.11, the author is stating that if $G$ is a group scheme over $k$, then any $G$-module ...
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-1 votes
1 answer
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Question about left Hopf-modules

I'm reading Radford's text on Hopf Algebras right now and I'm a bit confused about the definition of a left A-Hopf algebra. The definition given in the book is: Let $A$ be a $\Bbbk$-bialgebra. A $\...
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1 answer
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Character group of diagonalizable group scheme

Suppose $k$ is an integral domain, $\Lambda$ is a commutative group, $k[\Lambda]$ is the corresponding group ring and $\text{Diag}(\Lambda) = \text{Spec}(k[\Lambda])$ is the diagonalizable group ...
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1 vote
1 answer
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Proof that any extension that is Galois in the classical sense is also Hopf Galois

The usual action of $G$ on $L$ is by automorphisms that fix $\mathcal{K}$. Explicitly, for any $g\in G, l,m\in L, k\in \mathcal{K}$ \begin{align*} g(l+m)&=g(l)+g(m)\\ g(lm)&=g(l)g(n)\\ g(k)&...
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2 votes
1 answer
105 views

Lie operator is left exact

In page 190, part (c) of the book "Algebraic groups, the theory of group schemes of finite type over a field" of Milne, there's a part stating that: An exact sequence of algebraic groups $e ...
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6 votes
1 answer
410 views

Counterexample to "kernel determines image"

Working over a base field, there is a typical homomorphism theorem for affine algebraic groups ensuring that any two homomporphisms $G \to H_1$, $G \to H_2$ with the same kernel in $G$ have isomorphic ...
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  • 7,067
0 votes
1 answer
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Is the unit section of a finite flat commutative group scheme determined by sending group-like elements to 1?

Let $R$ be a ring and $\mathop{\mathrm{Spec}}A$ be a finite flat commutative group scheme over $\mathop{\mathrm{Spec}}R$ so that the theory of Cartier duality applies. Denote $s:R\to A,\ m:A\to A\...
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  • 901
5 votes
2 answers
225 views

Distributions of a group scheme as differential operators.

I'm trying to understand the distributions on an affine group scheme act as differential operators. Let $X$ be a scheme and $G$ a group scheme both affine over some commutative ring $k$. Suppose $\...
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3 votes
0 answers
37 views

For two modules over a Hopf algebra $H$, are the module homormphisms the same as the $H$-invariant linear maps?

Let $H$ be a Hopf algebra over a field $k$ and $V, W$ two $H$-modules. The antipode and comultiplication on $H$ allow us to turn $\mathrm{Hom}_k(V, W)$ into a $H$-module by setting $$ (h \cdot f)(v) = ...
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1 vote
1 answer
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Axioms of a coalgebra restated using Sweedler's notation

I'm struggling with understanding manipulation using Sweedler's notation at a very fundamental level. I don't understand the equivalence of the axioms of coalgebras in the standard notation [Coproduct ...
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1 answer
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Let B be a bialgebra. Show that the following are equivalent

Let B be a bialgebra. Show that the following are equivalent: B is a Hopf algebra. The maps $T_1$, $T_2$: $B \otimes B \to B \otimes B$ determined by $T_1(a \otimes b) = \sum a_{(1)} \otimes a_{(2)} ...
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Hopf "algebroid" structure of a groupoid convolution algebra?

To male thinks simple as possible, lets say we have a discrete group $G.$ Then the then the group algebra $\mathbb{C}[G]$ (of finitely supported complex valued functions on $G$) has a convolution and ...
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Is every sub-Hopf algebra a Frobenius extension?

Recall that a ring extension $S \subset R$ (i.e. fancy words for saying that $S$ is a subring of $R$) is called a Frobenius extension if the restriction functor $\operatorname{Res}:R\text{-mod} \to S\...
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1 vote
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Is the kernel of an action of a Hopf algebra on an algebra a biideal?

S.Dascalescu, C.Nastasescu and S.Raianu define the action of a Hopf algebra $H$ on an (associative) algebra $A$ as a map $H\times A\owns (h,a)\mapsto h\cdot a\in A$ which is an action of $H$ on $A$ ...
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