---
tag: KHMT0E4BRK
kind: concept
name: Nichols algebra
msc: [16T05, 16T20, 17B37]
sources:
  - url: https://en.wikipedia.org/wiki/Nichols_algebra
same_as:
  wikidata: Q17099388
  wikipedia: Nichols algebra
---

## Summary

The Nichols algebra $\mathfrak{B}(V)$ of a braided vector space $V$ is a graded braided Hopf algebra generated by $V$.
Its only primitive elements are those of $V$.
It is the quotient of the [[tensor algebra|P4J5JZDW0B]] $T(V)$ by the kernels of the quantum symmetrizers.
These symmetrizers are built from the action of the [[braid group|BNR4H5GCTP]] on $V^{\otimes n}$.
Over a field of characteristic $0$, for the ordinary flip it is the symmetric algebra, and for minus the flip it is the [[exterior algebra|0MDMZ965VE]].
The positive part of a [[quantum enveloping algebra|1V8H34F6N9]] is a Nichols algebra, and Nichols algebras are central in the classification of pointed Hopf algebras.

## Explanations

### L1 human

Start with a vector space and a rule $c$ for swapping two vectors, which may twist them in the process.
Form all words in the vectors, and declare a word zero whenever "symmetrizing it with the twisted swap" gives zero.
What remains is the Nichols algebra.
With the ordinary swap this gives polynomials, where $xy = yx$, and with minus the swap it gives the exterior algebra, where $xy = -yx$.
For a single vector $x$ whose swap multiplies by a root of unity $q$ of order $N$, the result is $k[x]/(x^N)$.