# 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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### 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 $H\subset G(k)$...
1 vote
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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 ...
1 vote
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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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### 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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### 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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### 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 ...
1 vote
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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? 32 views

### 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 ...
1 vote
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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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### 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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### 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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### 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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### 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 ...
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 $T_1(a \otimes b) = \sum a_{(1)} \otimes a_{(2)} ... 1 vote 0 answers 76 views ### Hopf "algebroid" structure of a groupoid convolution algebra? To male thinks simple as possible, lets say we have a discrete group$G.$Then the then the group algebra$\mathbb{C}[G]$(of finitely supported complex valued functions on$G$) has a convolution and ... 1 vote 0 answers 32 views ### Is every sub-Hopf algebra a Frobenius extension? Recall that a ring extension$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\...
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 $A$ ...