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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A diagrammatic proof of antipode being antihomomorphism in a Hopf algebra
Let $(H, \mu, \eta, \Delta, \epsilon, S)$ be a Hopf algebra with $S: H \to H$ denoting the antipode. By definition, $S$ is the convolution inverse of $1: H \to H$ in $\operatorname{End}(H)$, with the ...
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On the orthogonality relations for quantum Clebsch-Gordan coefficients
I am currently reading Quantum groups by Christian Kassel, more precisely, section VII.7 which is about quantum Clebsch-Gordan coefficients.
To make this question self-contained, let me introduce the ...
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$\mathbb{C}G$-modules and $\mathbb{C}^{G}$-comodules
I know that representations of a group $G$ are essentially $\mathbb{C}G$-modules. How is it that every $\mathbb{C}G-$module is also equivalent to a $\mathbb{C}^{G}-$comodule. I have not found this ...
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For the binomial Hopf algebra, what is the group of grouplike elements of the dual algebra?
The linear space of finite polynomials over a field $\mathbb{K}$ of zero characteristic has a structure of a graded connected bialgebra (meaning the zeroth subspace is isomorphic to the base field):
$\...
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Hopf algebra related to monoidal category
Recently, I heard that
Braided rigid monoidal category corresponds to a quasi-triangular hopf algebra.
I know in braided condition gives hexagonal equations and monoidal category gives pentagon/...
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How does base change affect group schemes?
I am reading through Waterhouse's Introduction to Affine Group Schemes and am having trouble understanding how I should think of base change as it relates to the way a groups scheme "looks".
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Exercise 3.14b of Waterhouse Affine Group Schemes
I have been working on Exercise 3.14b of Waterhouse Introduction to Affine Group Schemes and want to make sure I am on the right track.
Suppose G is represented by A. Write down the map $\varphi: A \...
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Corepresentations of quantum subgroups
If $H$ is a Hopf subalgebra of the Hopf algebra $G$ and $\alpha:V\longrightarrow V\otimes H$ is right corepresentation of $H$ on the vector space $V$ (finite--dimensional), is there a way to induce a ...
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Tensor Product of Modules of Bialgebras
Lately I saw this post (in chinese) saying that the tensor product of modules relies on comultiplication, and the tensor product over a commutative algebra is a consequence of a canonical bialgebra ...
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Morphisms between modules over a Hopf algebra
Let $U$ be a Hopf algebra over a field $K$ and $M,N$ be $U$-modules. Then the set of all $K$-linear maps $\textrm{Hom}_{K}(M,N)$ has a canonical $U$-module structure given by
$$(u.f)(m) = \sum_{(u)} ...
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The 'union of factors' comultiplication in a monoid ring?
let $\mathbb{Z}[M]$ be 'the' monoid ring of $M$ over $\mathbb{Z}$; that is to say (if my understanding is right) the set of finite linear combinations of elements of $M$, with product given by $\sum_n ...
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Associated graded space as a (bi)algebra
A filtered bialgebra $(A,m,u,\delta,\epsilon)$ with $A_j,j\in\mathbb Z$ filtration is defined as a bialgebra so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$...
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Shuffle product formula for coproduct
I'm studying the coproduct $\Delta$ defined on a tensor algebra $T(V)$ and its action on tensor products of elements from a vector space $V$. The coproduct is given by $\Delta(v) = v \boxtimes 1 + 1 \...
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Comultiplication on the tensor algebra
Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal ...
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Affine group schemes
I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups.
My question
Let $k$ be an ...
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