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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Finiteness of results in Connes-Kreimer approach
Remark: Although this is technically a physics-related post, the content heavily relies on pure mathematics, so I deemed it more appropriate here.
When reading the papers by Connes and Kreimer (e.g. [...
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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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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):
$\...
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On the orthogonality relations for quantum Clebsch-Gordan coefficients
I am currently reading Quantum groups by Christian Kassel, more precisely, section VII.7 which is about quantum Clebsch-Gordan coefficients.
To make this question self-contained, let me introduce the ...
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$\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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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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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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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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Proving uniqueness of antipodes in Hopf algebras
Let $(H,\mu,\nu,\Delta,\epsilon)$ be a Bialgebra where H is the vector space, $\mu, \nu$ are the product and unit whilst $\Delta, \epsilon$ are the coproduct and counit. Now, for $f,g \in end(H)$ ...
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