---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: Hopf algebra
aliases: [Hopf algebra]

---


---

- Tags 
	- #lie-theory 
- Refs:
	- #todo/add-references
- Links:
	- [universal enveloping algebra](universal%20enveloping%20algebra.md)

---

# Hopf algebra


![](attachments/Pasted%20image%2020210731183646.png)

![](attachments/Pasted%20image%2020210731191606.png)

![](attachments/Pasted%20image%2020210731191629.png)


- Comultiplication: $\Delta: H\to H\tensorpower{k}{2}$
- Counit $\eps; H\to k$
- Antipode $S: H\selfmap$


## 16:16

- See Manin's universal [quantum groups](quantum%20groups).
  - Manin defines a universal bialgebra for $A$, which coacts on $A$ in a universal way.

  - See [Unsorted/koszul duality](Unsorted/koszul%20duality.md)

  - Forgetful functor from Hopf algebras to bialgebras has a left adjoint: the [Hopf envelope](Hopf%20envelope).
  - Universal quantum group: take Hopf envelope of universal bialgebra.

- See [quadratic algebra](quadratic%20algebra)
- Twisting conditions for bialgebras: $B$ is $\ZZ\dash$graded and $\Delta(B_n) \subseteq B\tensorpower{k}{2}$.

- Zhang twist: supplies a twisted multiplication. 
  - Possibly related to [alpha twisted vector space](alpha%20twisted%20vector%20space)?

  - $\grmods{A} \iso \grmods{A^{\phi}}$ for $A^{\phi}$ a Zhang twist.

- Morita-Takeuchi equivalence: equivalence of categories of comodules.

- This talk compares cocycle twists to Zhang twists.

- For $\OO(G)$ the coordinate ring of $G\in\Alg\Grp$, elements $g\in G$ induce automorphism $r_g, \ell_g: \OO(G)\selfmap$ by left/right translation, and every twisting pair is of the form $(r_g, \ell_g\inv)$.

- Sovereign: equivalence between left and right duality functors.

- Pointed algebra: simple comodules are 1-dimensional

- Smash product of Hopf algebras: $H_1\tensor H_2$ as a vector space, with a deformed multiplication.
  - Example: $U(\lieg)\smashprod k[G]$.

- See [quantum Yang Baxter](quantum%20Yang%20Baxter) equations.
  Solutions are $R\in \Endo_k(V\tensorpower{2})$ satisfying a tensor formula corresponding to moving strands in a braid.
  
  - Can be obtained from any braiding on $\comods{H}$.

  - Use equivalence of braided monoidal cats to get new solutions: $\cmods{H} \iso \comod{A(R)\localize{g}}\iso \comods{A(R)\localize{g}^{\sigma}}$.

# Examples

The [group algebra](Unsorted/group%20algebra.md):
![](attachments/Pasted%20image%2020220316211737.png)
![](attachments/Pasted%20image%2020220316212016.png)

- $k[G]$ for $g\in \Fin\Grp$ with $g\to g\tensor g, g\to 1, g\to g\inv$
- $U(\lieg)$ with $x\to x\tensor 1 + 1\tensor x, x\to 0, x\to x\inv$.
- The coordinate ring of any $X\in \Alg\Grp$.
- Actions: $H\actson M$ where $\lambda: H\tensor M\to M$ compatible with mult
- Coaction: $\rho: M\to H\tensor M$ compatible with comult.
- Convolution algebra: $\Hom_k(H, k)$ under $fg \mapsto \mu_k \circ (f\tensor g) \circ \Delta_H$ for $\Delta_H$ the comult on $H$ and $\mu_k$ the multiplication on $k$.
- Can twist multiplication by a 2-cocycle $\sigma: H\tensorpower{2} \to k$.