# Questions tagged [hopf-algebras]

A Hopf algebra is a vector space $H$ over a field $k$ endowed with an associative product $\times:H\otimes_k H\to H$ and a coassociative coproduct $\Delta:H\to H\otimes_k H$ which is a morphism of algebras. Unit $1:k\to H$, counit $\epsilon:H\to k$ and antipode $S:H\to H$ are also required. Such a structure exists on the group algebra $k G$ of a finite group $G$.

416 questions
Filter by
Sorted by
Tagged with
118 views

190 views

### Intuitive, elementary intros to Hopf algebras/monoids

Motivation: I'm interested in understanding the role that noncrossing partitions play in Hopf algebras/monoids (HAs) as the components of the power series of the compositional inverse of formal ...
342 views

### Classification of $\operatorname{Rep} D(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models. However,...
111 views

### Submodules of $V\otimes V^*$

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$ and let $U_q(\hat{\mathfrak{g}})$ be the corresponding quantum affine algebra (here $q$ is not a root of unity). We know ...
386 views

383 views

### Comultiplication of elements of partition of unity

Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra). Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a ...
301 views

### Example of a commutative, cocommutative, $p$-torsion Hopf algebra which is dualizable but not self-dual?

Let $C$ be a symmetric monoidal category with split idempotents, and let $H$ be a Hopf algebra object in $C$. If $H$ is dualizable as an object of $C$, then $H^\vee = L \otimes H$ for some $\otimes$-...
129 views

87 views

### Invariance of Finite Dimensional Woronowicz $\mathrm{C}^*$-ideals under the Antipode

Let $(A,\Delta)=:F(G)$ be a finite dimensional $\mathrm{C}^*$-Hopf algebra, and so the algebra of functions on a quantum group $G$. Let $J$ be a closed ideal in $F(G)$ and $\pi:F(G)\rightarrow F(G)/J$...
192 views

### Milnor-Moore and the characteristic zero homology of H-spaces

In their work on Hopf-algebras: https://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf> on the last page p.263, they say that the Hurewicz map ...
This post is basically a quantum extension of Is every finite group a group of “symmetries”? Here finite quantum group means finite dimensional Hopf ${\rm C}^{\star}$-algebra. Frucht's theorem ...