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 $T(V)$ by the kernels of the quantum symmetrizers. These symmetrizers are built from the action of the braid group 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. The positive part of a quantum enveloping algebra is a Nichols algebra, and Nichols algebras are central in the classification of pointed Hopf algebras.