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Source: 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

U

q

(

g

)

+

{\displaystyle U_{q}({\mathfrak {g}})^{+}}

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

u

q

(

g

)

+

{\displaystyle u_{q}({\mathfrak {g}})^{+}}

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

B

(
V
)

{\displaystyle {\mathfrak {B}}(V)}

where

V

{\displaystyle V}

is a Yetter–Drinfel'd module over a finite group

G

{\displaystyle G}

, where the group is generated by the support of

V

{\displaystyle V}

. For more details on Nichols algebras see Nichols algebra.

There are two major cases:

G

{\displaystyle G}

abelian, which implies

V

{\displaystyle V}

is diagonally braided

x

i

⊗

x

j

↦

q

i
j

x

j

⊗

x

i

{\displaystyle x_{i}\otimes x_{j}\mapsto q_{ij}x_{j}\otimes x_{i}}

.

G

{\displaystyle G}

nonabelian.
The rank is the number of irreducible summands

V
=

⨁

i
∈
I

V

i

{\displaystyle V=\bigoplus _{i\in I}V_{i}}

in the semisimple Yetter–Drinfel'd module

V

{\displaystyle V}

.
The irreducible summands

V

i

=

O

[
g
]

χ

{\displaystyle V_{i}={\mathcal {O}}_{[g]}^{\chi }}

are each associated to a conjugacy class

[
g
]
⊂
G

{\displaystyle [g]\subset G}

and an irreducible representation

χ

{\displaystyle \chi }

of the centralizer

Cent
⁡
(
g
)

{\displaystyle \operatorname {Cent} (g)}

.
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

V

i

{\displaystyle V_{i}}

and edges depending on their braided commutators in the Nichols algebra.
The Hilbert series of the graded algebra

B

(
V
)

{\displaystyle {\mathfrak {B}}(V)}

is given. An observation is that it factorizes in each case into polynomials

(
n

)

t

:=
1
+
t
+

t

2

+
⋯
+

t

n
−
1

{\displaystyle (n)_{t}:=1+t+t^{2}+\cdots +t^{n-1}}

. We only give the Hilbert series and dimension of the Nichols algebra in characteristic

0

{\displaystyle 0}

.
Note that a Nichols algebra only depends on the braided vector space

V

{\displaystyle V}

and can therefore be realized over many different groups. Sometimes there are two or three Nichols algebras with different

V

{\displaystyle V}

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.