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 formula_1 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 formula_2 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 formula_3 where formula_4 is a Yetter–Drinfel'd module over a finite group formula_5, where the group is generated by the support of formula_4. For more details on Nichols algebras see Nichols algebra.
Note that a Nichols algebra only depends on the braided vector space formula_4 and can therefore be realized over many different groups. Sometimes there are two or three Nichols algebras with different formula_4 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.
State of classification.
Negative criteria.
The case of rank 1 (irreducible Yetter–Drinfel'd module) over a nonabelian group is still largely open, with few examples known.
Much progress has been made by Andruskiewitsch and others by finding subracks (for example diagonal ones) that would lead to infinite-dimensional Nichols algebras. As of 2015, known groups not admitting finite-dimensional Nichols algebras are 
Usually a large amount of conjugacy classes ae of type D ("not commutative enough"), while the others tend to possess sufficient abelian subracks and can be excluded by their consideration. Several cases have to be done by-hand. Note that the open cases tend to have very small centralizers (usually cyclic) and representations χ (usually the 1-dimensional sign representation). Significant exceptions are the conjugacy classes of order 16, 32 having as centralizers p-groups of order 2048 resp. 128 and currently no restrictions on χ.
Over abelian groups.
Finite-dimensional diagonal Nichols algebras over the complex numbers were classified by Heckenberger in in terms of the braiding matrix formula_31, more precisely the data formula_32. The small quantum groups formula_2 are a special case formula_34, but there are several exceptional examples involving the primes 2,3,4,5,7.
Recently there has been progress understanding the other examples as exceptional Lie algebras and super-Lie algebras in finite characteristic.
Over nonabelian group, rank > 1.
Nichols algebras from Coxeter groups.
For every finite coxeter system formula_35 the Nichols algebra over the conjugacy class(es) of reflections was studied in (reflections on roots of different length are not conjugate, see fourth example fellow). They discovered in this way the following first Nichols algebras over nonabelian groups :
The case formula_36 is the rank 1 diagonal Nichols algebra formula_37 of dimension 2.
Nichols algebras of rank 2, type Gamma-3.
These Nichols algebras were discovered during the classification of Heckenberger and Vendramin.
The Nichols algebra of rank 2 type Gamma-4.
 This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.
The Nichols algebra of rank 2, type T.
 This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.
The Nichols algebra of rank 3 involving Gamma-3.
 This Nichols algebra was the last Nichols algebra discovered during the classification of Heckenberger and Vendramin.
Nichols algebras from diagram folding.
The following families Nichols algebras were constructed by Lentner using diagram folding, the fourth example appearing only in characteristic 3 was discovered during the classification of Heckenberger and Vendramin.
The construction start with a known Nichols algebra (here diagonal ones related to quantum groups) and an additional automorphism of the Dynkin diagram. Hence the two major cases are whether this automorphism exchanges two disconnected copies or is a proper diagram automorphism of a connected Dynkin diagram. The resulting root system is folding / restriction of the original root system. By construction, generators and relations are known from the diagonal case.
The following two are obtained by proper automorphisms of the connected Dynkin diagrams formula_38
Note that there are several more foldings, such as formula_39 and also some not of Lie type, but these violate the condition that the support generates the group.