Nichols algebra

In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted by ${\displaystyle {\mathfrak {B}}(V)}$ and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional truncations. Nichols algebras can immediately be used to write down new such quantum groups by using the Radford biproduct.

The classification of all such Nichols algebras and even all associated quantum groups (see Application) has been progressing rapidly, although still much is open: The case of an abelian group was solved in 2005, but otherwise this phenomenon seems to be very rare, with a handful examples known and powerful negation criteria established (see below). See also this List of finite-dimensional Nichols algebras.

The finite-dimensional theory is greatly governed by a theory of root systems and Dynkin diagrams, strikingly similar to those of semisimple Lie algebras. A comprehensive introduction is found in the lecture of Heckenberger.