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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.

Hopf algebras, named after Heinz Hopf, was first introduced in the theory of algebraic topology, while studying cohomology of Lie groups, but in recent years has been developed by many mathematicians and applied to other areas of mathematics such as algebraic groups, combinatorics, mathematical physics and Galois theory.

It is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism, called antipode, satisfying a certain property.

The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

There is a wide variety of variations of the notion of Hopf algebra, relaxing properties or adding structure. Examples are weak Hopf algebras, quasi-Hopf algebras, (quasi-)triangular Hopf algebras, quantum groups, hopfish algebras etc.

For more details you may find the following references:
$1.~~$ "Introduction to Hopf algebras and representations" by Kalle Kytola
$2.~~$ "Hopf Algebras in Combinatories" by Darij Grinberg & Victor Reiner
$3.~~$ "Hopf Algebra" from Wikipedia
$4.~~$ "A Very Basic Introduction to Hopf Algebras" by J.M. Selig

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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 theory of $G$ over $K$, as for instance if…
Will
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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. I am utterly ignorant of the subject, so could…
Daniel Robert-Nicoud
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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 with monoid homomorphisms $\Delta_i:A\to B_i$ such…
Berci
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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 $$ I_l(H)= \{x \in H; h \cdot x=\epsilon(h)x \quad for \>all\>h\in H\} $$ the space of left integrals. In other words,…
M.C.
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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 defined as follows: Question: why the…
Shiquan
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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 Foote, Atiyah-Macdonald commutative algebra, and…
54321user
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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, or is it related to some properties of triangles? Thanks…
yoyostein
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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 you obtain QGamma[n+1,q] / QGamma[n,q] = (q^n-1) /…
André
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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 tensor product $\otimes \colon \mathcal{C} \times…
user313212
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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 infinite dimension it is not surjective. Does anybody…
Dac0
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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 problems in combinatorics which were finally solved…
Mohan
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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 use Sweedler's notation to try to write down the…
Mike Pierce
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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 'anitpode'. Can anyone give an intuitive explanation…
mavzolej
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Monodromy element: Why that name?

Let $(H,R)$ be a quasitriangular Hopf algebra, i.e. $R$ is a choice of an universal $R$-matrix for the Hopf algebra H. (You can find a definition of the term quasitriangular Hopf algebra on wikipedia.). One calls the element $Q:=R_{21}\cdot R_{12}…
M.C.
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Cosemisimple Hopf algebra and Krull-Schmidt

A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will this decomposition obey Krull-Schmidt, by which…
Dyke Acland
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