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  "body": "\n## Summary\n\nThe Nichols algebra $\\mathfrak{B}(V)$ of a braided vector space $V$ is a graded braided Hopf algebra generated by $V$.\nIts only primitive elements are those of $V$.\nIt is the quotient of the [[tensor algebra|P4J5JZDW0B]] $T(V)$ by the kernels of the quantum symmetrizers.\nThese symmetrizers are built from the action of the [[braid group|BNR4H5GCTP]] on $V^{\\otimes n}$.\nOver 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]].\nThe positive part of a [[quantum enveloping algebra|1V8H34F6N9]] is a Nichols algebra, and Nichols algebras are central in the classification of pointed Hopf algebras.\n\n## Explanations\n\n### L1 human\n\nStart with a vector space and a rule $c$ for swapping two vectors, which may twist them in the process.\nForm all words in the vectors, and declare a word zero whenever \"symmetrizing it with the twisted swap\" gives zero.\nWhat remains is the Nichols algebra.\nWith the ordinary swap this gives polynomials, where $xy = yx$, and with minus the swap it gives the exterior algebra, where $xy = -yx$.\nFor a single vector $x$ whose swap multiplies by a root of unity $q$ of order $N$, the result is $k[x]/(x^N)$.\n",
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