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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1
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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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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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1
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Universal enveloping algebra as bialgebra
If $\mathfrak{g}$ is a Lie algebra (over $k$ ), then we can construct its universal enveloping algebra $U(\mathfrak{g})$. We can define $\Delta:U(\mathfrak{g})\rightarrow U(\mathfrak{g})\otimes U(\...
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1
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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 ...
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Distributions of a group scheme as differential operators.
I'm trying to understand the distributions on an affine group scheme act as differential operators. Let $X$ be a scheme and $G$ a group scheme both affine over some commutative ring $k$. Suppose $\...
5
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1
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Internal Homs in Modules over a Hopf Algebra
Given a Hopf algebra $H$, I wonder when the monoidal category of $H$-modules is left-closed, right-closed, and finally, under what circumstances right and left internal hom are isomorphic.
If I'm not ...
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4
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group-like elements of a Hopf algebra and the group algebra
Suppose we are given a $n$-dimensional Hopf algebra $H$ over field $\mathbb K$. I want to prove that
$H$ contains n distinct group-like elements if and only if there exists a group $G$ such ...
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4
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1
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The comultiplication on $\mathbb{C} S_3$ for a matrix basis?
Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra.
The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$.
For $G=S_3$, ...
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3
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3
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How to proof equivalent condition of algebra morphism and coalgebra morphism about Hopf algebra
My question is why $\mu$ is multiplication preserving iff $\mu\circ m=(m\otimes m)\circ(1\otimes \tau \otimes1)\circ(\mu \otimes \mu)$ and this holds iff $m$ is comultiplication preserving, where $\...
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Reference on correspondence between commutative Hopf Algebras and Groups
Is it true that every commutative Hopf algebra is related to a Group in such a way that the co-multiplication is originated from the multiplication of the group, the antipode from the inverse?
Making ...
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Poincare Series of a graded algebra (revisited)
Here is the question I am trying to solve:
Let $A = \bigoplus_{i \geq 1}A_i$ be a graded algebra such that the vector spaces $A_i$ are all finite-dimensional. Define the Poincare series of $A$ as the ...
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2
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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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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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2
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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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