+ | |
U | |
q(ak{g}) |
+ | |
u | |
q(ak{g}) |
The following article lists all known finite-dimensional Nichols algebras
ak{B}(V)
V
G
V
G
V
xi ⊗ xj\mapstoqijxj ⊗ xi
G
V=oplusi\inVi
V
Vi=l{O}
\chi | |
[g] |
[g]\subsetG
\chi
\operatorname{Cent}(g)
Vi
ak{B}(V)
2+ … | |
(n) | |
t:=1+t+t |
+tn-1
0
Note that a Nichols algebra only depends on the braided vector space
V
V
(as of 2015)
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
An\geq
Sn\geq
PSLn(Fq)
Sp2n(Fq)
Fi22
22A,22B
16C, 16D, 32A, 32B, 32C, 32D, 34A, 46A, 46B
32A, 32B, 46A, 46B, 92A, 92B, 94A, 94B
Finite-dimensional diagonal Nichols algebras over the complex numbers were classified by Heckenberger in in terms of the braiding matrix
qij
qii,qijqji
+ | |
u | |
q(ak{g}) |
qij
(\alphai,\alphaj) | |
=q |
Recently there has been progress understanding the other examples as exceptional Lie algebras and super-Lie algebras in finite characteristic.
For every finite coxeter system
(W,S)
Rank, Type of root system of ak{B}(V) | A1 | A1 | A1 | A2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dimension of V | 3 | 6 | 10 | 2+2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dimension of Nichols algebra(s) | 12 | 576 =242 | 8294400 | 64 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Hilbert series |
(3)t |
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Smallest realizing group | Symmetric group S3 | Symmetric group S4 | Symmetric group S5 | Dihedral group D4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
... and conjugacy classes |
|
,
,
|
|
,
⊕
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Source | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Comments | Kirilov–Fomin algebras | This smallest nonabelian Nichols algebra of rank 2 is the case \Gamma2 An An\cupAn |
S2\congZ2
ui(A
+ | |
1) |
Rank, Type of root system of ak{B}(V) | A1 | A1 | A1 | A1 | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dimension of V | 4 | 4 | 5 | 7 | ||||||||||||||||||||||||||||||
Dimension of Nichols algebra(s) | 72 | 5,184 | 1,280 | 326,592 | ||||||||||||||||||||||||||||||
Hilbert series |
(3)t(6)t |
|
(5)t |
(7)t | ||||||||||||||||||||||||||||||
Smallest realizing group | Special linear group SL2(3) A4 | Affine linear group Z5\rtimesZ
| Affine linear group Z7\rtimesZ
| |||||||||||||||||||||||||||||||
... and conjugacy classes | l{O}\begin{pmatrix-1&1\ 0&-1\end{pmatrix}}-1 | l{O}\begin{pmatrix1&1\ 0&
|
,
|
,
| ||||||||||||||||||||||||||||||
Source | ||||||||||||||||||||||||||||||||||
Comments | There exists a Nichols algebra of rank 2 containing this Nichols algebra | Only example with many cubic (but not many quadratic) relations. | Affine racks |
These Nichols algebras were discovered during the classification of Heckenberger and Vendramin.
only in characteristic 2 | |||||||||||||||||||||||||||||||||||
Rank, Type of root system of ak{B}(V) | B2,B2 | B2,B2 | \begin{pmatrix}2&-2\ -2&2\end{pmatrix} \begin{pmatrix}2&-2\ -1&2\end{pmatrix} \begin{pmatrix}2&-4\ -1&2\end{pmatrix} | ||||||||||||||||||||||||||||||||
Dimension of V | 3+2 | 3+1 3+2 | 3+1 3+2 | ||||||||||||||||||||||||||||||||
Dimension of Nichols algebra(s) | 2,304 | 10,368 | 2,239,488 | ||||||||||||||||||||||||||||||||
Hilbert series |
|
t ⋅
⋅
| |||||||||||||||||||||||||||||||||
Smallest realizing group and conjugacy class | S3 x Z2 | S3 x Z6 | |||||||||||||||||||||||||||||||||
... and conjugacy classes |
⊕ l{O}\{z\ |
⊕ l{O}\{z\ | |||||||||||||||||||||||||||||||||
Source | |||||||||||||||||||||||||||||||||||
Comments | Only example with a 2-dimensional irreducible representation \sigma | There exists a Nichols algebra of rank 3 extending this Nichols algebra | Only in characteristic 2. Has a non-Lie type root system with 6 roots. |
This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.
Root system | B2 | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dimension of V | 2+4 | ||||||||||||||||||||||||||||||
Dimension of Nichols algebra | 262,144 =218 | ||||||||||||||||||||||||||||||
Hilbert series |
⋅
⋅
⋅
| ||||||||||||||||||||||||||||||
Smallest realizing group | \tilde{D | ||||||||||||||||||||||||||||||
...and conjugacy class |
⊕
| ||||||||||||||||||||||||||||||
Comments | Both 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 D4 A2 | ||||||||||||||||||||||||||||||
This Nichols algebra was discovered during the classification of Heckenberger and Vendramin.
Root system | G2 | ||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dimension of V | 1+4 | ||||||||||||||||||||||||||||||||||
Dimension of Nichols algebra | 80,621,568 | ||||||||||||||||||||||||||||||||||
Hilbert series | (6)t ⋅
t ⋅
⋅
⋅
⋅
| ||||||||||||||||||||||||||||||||||
Smallest realizing group | \Z6 x SL2(3) | ||||||||||||||||||||||||||||||||||
...and conjugacy class | l{O}\{z\ | ||||||||||||||||||||||||||||||||||
Comments | The rank 1 Nichols algebra contained in this Nichols algebra is irreducible over its support SL2(3) | ||||||||||||||||||||||||||||||||||
This Nichols algebra was the last Nichols algebra discovered during the classification of Heckenberger and Vendramin.
Root system | Rank 3 Number 9 with 13 roots | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Dimension of V | 3+2+1 3+1+1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dimension of Nichols algebra | 1,671,768,834,048 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Hilbert series |
(6)t ⋅ (6)t ⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Smallest realizing group | S3 x Z6 x Z6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
...and conjugacy class |
⊕ l{O}\{z\ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Comments | The rank 2 Nichols algebra cenerated by the two leftmost node is of type \Gamma3 A2 A3 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.
only characteristic 3 | |||||
Rank, Type of root system of ak{B}(V) | An, n\geq2 | Dn, n\geq4 | E6,E7,E8 | Bn, n\geq2 | |
Constructed from this diagonal Nichol algebra with qij=\pm1 | An x An | Dn x Dn | En x En | Bn x Bn | |
Dimension of V | 2+2+ … | 2+2+ … | 2+2+ … | 2+2+ … | |
Dimension of Nichols algebra(s) | \left(2n+1\right)2 | \left(2n(n-1)\right)2 | \left(236\right)2, \left(263\right)2, \left(2120\right)2 | \left(3n(n-1)2n\right)2 | |
Hilbert series | Same as the respective diagonal Nichols algebra | ||||
Smallest realizing group | Extra special group (resp. almost extraspecial) with 2n+1 Dn, 2 | n requires a similar group with larger center of order 23 | |||
Source | |||||
Comments | Supposedly a folding of the diagonal Nichols algebra of type Bn q=\pm1 |
The following two are obtained by proper automorphisms of the connected Dynkin diagrams
2}A | |
{ | |
2n-1 |
2}E | |
,{ | |
6 |
Rank, Type of root system of ak{B}(V) | Cn, n\geq3 | F4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Constructed from this diagonal Nichol algebra with qij=\pm1 | A2n-1 | E6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dimension of V | 1+2+ … | 1+1+2+2 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dimension of Nichols algebra(s) | 2{2n | 236=68,719,476,736 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Hilbert series | Same as the respective diagonal Nichols algebra | Same as the respective diagonal Nichols algebra
⋅
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Smallest realizing group | Group of order 2n+1 22 23 n | Group of order 24+1 23 i.e. Z2 x Z2 x D4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
... and conjugacy class |
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Source | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Note that there are several more foldings, such as
2}D | |
{ | |
n |
(Simon Lentner, University Hamburg, please feel free to write comments/corrections/wishes in this matter: simon.lentner at uni-hamburg.de)
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]