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. 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 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.
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. These are given by different conjugacy classes in the same column.

State of classification

Established classification results

The case of rank 1 over a nonabelian group is still largely open, with few examples known.
Much progress has been made by Andruskiewitsch and others by finding subracks 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, 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 and representations χ. 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, more precisely the data. The small quantum groups are a special case , 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 the Nichols algebra over the conjugacy class of reflections was studied in . They discovered in this way the following first Nichols algebras over nonabelian groups :
The case is the rank 1 diagonal Nichols algebra of dimension 2.

Other Nichols algebras of rank 1

Nichols algebras of rank 2, type Gamma-3

These Nichols algebras were discovered during the classification of Heckenberger and Vendramin.
only in characteristic 2
Rank, Type of root system of
Dimension ofresp.resp.
Dimension of Nichols algebra
Hilbert series
Smallest realizing group and conjugacy class
... and conjugacy classes
Source
CommentsOnly example with a 2-dimensional irreducible representationThere exists a Nichols algebra of rank 3 extending this Nichols algebraOnly in characteristic 2. Has a non-Lie type root system with 6 roots.

The Nichols algebra of rank 2 type Gamma-4

This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.
Root system
Dimension of
Dimension of Nichols algebra
Hilbert series
Smallest realizing group
...and conjugacy class
CommentsBoth rank 1 Nichols algebra contained in this Nichols algebra decompose over their respective support: The left node to a Nichols algebra over the Coxeter group, the right node to a diagonal Nichols algebra of type.

The Nichols algebra of rank 2, type T

This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.
Root system
Dimension of
Dimension of Nichols algebra
Hilbert series
Smallest realizing group
...and conjugacy class
CommentsThe rank 1 Nichols algebra contained in this Nichols algebra is irreducible over its support and can be found above.

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.
Root systemRank 3 Number 9 with 13 roots
Dimension ofresp.
Dimension of Nichols algebra
Hilbert series
Smallest realizing group
...and conjugacy class
CommentsThe rank 2 Nichols algebra cenerated by the two leftmost node is of type and can be found above. The rank 2 Nichols algebra generated by the two rightmost nodes is either diagonal of type or .

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 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
Note that there are several more foldings, such as and also some not of Lie type, but these violate the condition that the support generates the group.

Poster with all Nichols algebras known so far