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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How to think of the group ring as a Hopf algebra?
Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
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What are Quantum Groups?
I am going to do a semester project (kind of a little thesis) this spring. I met a professor and asked him about some possible arguments. Among others, he proposed something related to quantum groups. ...
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Why is the Quasitriangular Hopf algebra called "Quasitriangular"?
The precise definition of a Quasitriangular Hopf algebra can be found on wikipedia.
What is the reason behind the word "Quasitriangular"?
Is it because the R-matrix is a triangular matrix, ...
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Hopf-like monoid in $(\Bbb{Set}, \times)$
I am looking for a nontrivial example of the following:
Let a monoid $A$ be given with unit $e$, and two of its distinguished disjoint submonoids $B_1$ and $B_2$ (s.t. $B_1\cap B_2=\{e\}$), endowed ...
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Good Introduction to Hopf Algebras with Examples
I want to learn more about hopf algebras but I am having trouble finding a down to earth introduction to the subject with lots of motivation and examples. My algebra knowledge ranges from Dummit and ...
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Motivation/Intuition for the Pentagon Axiom
I have just started reading a bit on monoidal categories, and there is I just can't make much sense of: the Pentagon Axiom.
To provide some context, we have a category $\mathcal{C}$ together with a ...
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Integrals of a Hopf algebra: Why that name?
1. Context: The notion of an integral
Let $H$ be a Hopf algebra over a field $\mathbb k$. We call its $\mathbb k$-linear subspace
$$
I_l(H)= \{x \in H; h \cdot x=\epsilon(h)x \quad for \>all\>h\...
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different definitions of Hopf algebras
(i). In the book Algebraic Topology, A. Hatcher, p. 283， the notion Hopf algebra is defined as follows:
(ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is ...
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Different notions of q-numbers
It seems that most of the literature dealing with q-analogs defines q-numbers according to
$$[n]_q\equiv \frac{q^n-1}{q-1}.$$
Even Mathematica uses this definition: with the built-in function QGamma ...
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Meaning of the antipode in Hopf algebras?
What I understand so far is that Hopf algebra is a vector space which is both algebra and coalgebra. In addition to this, there is a linear operation $S$, which for each element gives a so-called '...
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Exercises to help a student become accustomed to Sweedler notation
For a coassociative coalgebra $A$, we have a comultiplication map $\Delta \colon A \to A \otimes A$. An element $c \in A$ is sent to a sum of simple tensors, which can be a mess of indices, so we can ...
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Example of $V^* \otimes V^*$ not isomorphic to $(V \otimes V)^*$
There is always an injection between $V^* \otimes V^*$ and $(V \otimes V)^*$ given by
$$
f(v^* \otimes w^*)(x \otimes y)=v^*(x)w^*(y),
$$
where $x,y \in V$. I've been given to understand that in ...
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Hopf Algebras in Combinatorics
I know that many examples of Hopf algebras that come from combinatorics. But I'm interested in knowing how Hopf algebras are applied in solving combinatorial problem.
Are there examples of open ...
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Duality between universal enveloping algebra and algebras of functions
There is a well known duality (of Hopf algebras) between universal enveloping algebra $U(\mathfrak{g})$ of a complex Lie algebra $\mathfrak{g}$ of a compact group $G$ and the algebra of continuous ...
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$\mathbb{G}_m$ action on $\operatorname{Spec}A$ is equivalent to grading... what does coassociativity do?
I can interpret the other axioms for a grading via the definition of a group scheme action appropriately (I think):
Let $H = k[x,x^{-1}]$. We are given a coaction $ \rho : A \to A \otimes H$, which is ...
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