Quantum groups | Hopf algebras
List of finite-dimensional Nichols algebras
In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to an object V in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of V enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases. The most well known examples for Nichols algebras are the Borel parts of the infinite-dimensional quantum groups when q is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts of the Frobenius–Lusztig kernel (small quantum group) when q is a root of unity. The following article lists all known finite-dimensional Nichols algebras where is a Yetter–Drinfel'd module over a finite group , where the group is generated by the support of . For more details on Nichols algebras see Nichols algebra. * There are two major cases: * abelian, which implies is diagonally braided . * nonabelian. * The rank is the number of irreducible summands in the semisimple Yetter–Drinfel'd module . * The irreducible summands are each associated to a conjugacy class and an irreducible representation of the centralizer . * To any Nichols algebra there is by attached * a generalized root system and a Weyl groupoid. These are classified in. * In particular several Dynkin diagrams (for inequivalent types of Weyl chambers). Each Dynkin diagram has one vertex per irreducible and edges depending on their braided commutators in the Nichols algebra. * The Hilbert series of the graded algebra is given. An observation is that it factorizes in each case into polynomials . We only give the Hilbert series and dimension of the Nichols algebra in characteristic . Note that a Nichols algebra only depends on the braided vector space and can therefore be realized over many different groups. Sometimes there are two or three Nichols algebras with different and non-isomorphic Nichols algebra, which are closely related (e.g. cocycle twists of each other). These are given by different conjugacy classes in the same column. (Wikipedia).
