Any braided vector space gives rise to a Nichols algebra, as first found by Nichols in

See

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These are certainly not the only references, but they are written in the way I like (of course!).

Other references are

Lest you think you never saw a Nichols algebra,
let me mention two examples: symmetric and antisymmetric ( = exterior)
algebras are Nichols. But there are a lot of examples you might have seen;
here is a list of papers in which they appeared under various names:
**[G. Lusztig,** *Introduction to Quantum Groups*, Birkhäuser, 1993.**]**
**[M. Rosso,** *Certaines formes bilineaires sur les groupes quantiques et une conjecture de Schechtman et Varchenko*, CRAS Paris, **314**, Sèrie I (1992), 5-8.**]**
**[M. Rosso,** *Groupes quantiques et algèbres de battage quantiques*, CRAS Paris, **320**, Sèrie I (1995), 145-148.**]**
**[P. Schauenburg,** *A Characterization of the Borel-like Subalgebras of Quantum Enveloping Algebras*, Comm. in Algebra **24** (1996), 9, 2811-2823**]**
**[S.L. Woronowicz,** *Differential Calculus on Compact Matrix Pseudogroups (Quantum Groups)*, Comm. Math. Phys. **122** (1989), 125-170**]**

So here are all examples I know about of finite dimensional Nichols algebras
of nonabelian group type (here by "nonabelian group type" we mean that the
braided vector space does not live in a category of modules over the Drinfeld
double of an abelian group). All of them are presented by means of quandles
and cocycles, as explained in **[AG2]**.

x_{0}^{2} = x_{1}^{2} = x_{2}^{2} = 0,

x_{0}x_{1}+x_{1}x_{2}+x_{2}x_{0} = 0,

x_{1}x_{0}+x_{2}x_{1}+x_{0}x_{2} = 0.

Besides these relations, to have a Gröbner basis we have to add the relation

x_{0}x_{1}x_{0}=x_{1}x_{0}x_{1}.

A basis is given then by multiplying one element in each row from top to bottom in all the possible ways:

- 1, x
_{0} - 1, x
_{1}, x_{1}x_{0} - 1, x
_{2}

The Hilbert polynomial is

(1+t)^{2}(1+t+t^{2})
= 1+3t+4t^{2}+3t^{3}+t^{4}

The cocycle is q = -1. The algebra has dimension 72. It is generated by x

x_{0}^{2} = x_{1}^{2}
= x_{2}^{2} = x_{3}^{2} = 0,

x_{3}x_{2}+x_{2}x_{1}+x_{1}x_{3} = 0,

x_{3}x_{1}+x_{1}x_{0}+x_{0}x_{3} = 0,

x_{3}x_{0}+x_{0}x_{2}+x_{2}x_{3} = 0,

x_{2}x_{0}+x_{0}x_{1}+x_{1}x_{2} = 0,

x_{2}x_{1}x_{0}x_{2}x_{1}x_{0}
+x_{1}x_{0}x_{2}x_{1}x_{0}x_{2}
+x_{0}x_{2}x_{1}x_{0}x_{2}x_{1} = 0.

A basis is given then by multiplying one element in each row from top to bottom in all the possible ways:

- 1, x
_{0} - 1, x
_{1}, x_{1}x_{0} - 1, x
_{2}x_{1}x_{0} - 1, x
_{2}, x_{2}x_{1} - 1, x
_{3}

The Hilbert polynomial is then

(1+t)^{2}(1+t+t^{2})^{2}(1+t^{3})
= t^{9} + 4 t^{8} + 8 t^{7} + 11 t^{6} + 12 t^{5} + 12 t^{4} + 11 t^{3} + 8 t^{2} + 4 t + 1.

x_{0}^{2} = x_{1}^{2} = x_{2}^{2} =
x_{3}^{2} = x_{4}^{2} = 0,

x_{3}x_{2}+x_{2}x_{0}
+x_{1}x_{3}+x_{0}x_{1} = 0,

x_{4}x_{0}+x_{2}x_{1}
+x_{1}x_{4}+x_{0}x_{2} = 0,

x_{4}x_{1}+x_{3}x_{4}
+x_{1}x_{0}+x_{0}x_{3} = 0,

x_{4}x_{2}+x_{3}x_{0}
+x_{2}x_{3}+x_{0}x_{4} = 0,

x_{4}x_{3}+x_{3}x_{1}
+x_{2}x_{4}+x_{1}x_{2} = 0,

x_{1}x_{0}x_{1}x_{0}
+x_{0}x_{1}x_{0}x_{1} = 0.

A basis is given then by multiplying one element in each row from top to bottom in all the possible ways:

- 1, x
_{0} - 1, x
_{1}, x_{1}x_{0}, x_{1}x_{0}x_{1} - 1, x
_{2}, x_{2}x_{1}, x_{2}x_{1}x_{2}, x_{2}x_{0}, x_{2}x_{0}x_{1}, x_{2}x_{0}x_{1}x_{2}, x_{2}x_{0}x_{1}x_{0}, x_{2}x_{0}x_{1}x_{0}x_{2}, x_{2}x_{0}x_{1}x_{0}x_{2}x_{0} - 1, x
_{3}, x_{3}x_{1}, x_{3}x_{1}x_{2}, x_{3}x_{0}, x_{3}x_{0}x_{3}, x_{3}x_{0}x_{3}x_{1}, x_{3}x_{0}x_{3}x_{1}x_{2} - 1, x
_{4}

The Hilbert polynomial is then

(1 + t)^{2} (1 + t + t^{2} + t^{3} )
(1 + t + 2 t^{2} + 2 t^{3} + 2 t^{4} + t^{5} + t^{6})
(1+t+2t^{2}+2t^{3}+t^{4}+t^{5})

= t^{16} + 5 t^{15} + 15 t^{14} + 35 t^{13}
+ 66 t^{1}2 + 105 t^{11} + 145 t^{10}
+ 175 t^{9} + 186 t^{8} + 175 t^{7}
+ 145 t^{6} + 105 t^{5} + 66 t^{4}
+ 35 t^{3} + 15 t^{2} + 5 t + 1.

x_{1}^{2} = x_{2}^{2} = x_{3}^{2}
= x_{4}^{2} = x_{5}^{2} = x_{6}^{2} = 0,

x_{4}x_{3}+x_{3}x_{4} = 0

x_{5}x_{2}+x_{2}x_{5} = 0

x_{6}x_{1}+x_{1}x_{6} = 0

x_{3}x_{2}+x_{2}x_{1}+x_{1}x_{3} = 0

x_{4}x_{1}+x_{2}x_{4}+x_{1}x_{2} = 0

x_{5}x_{1}+x_{4}x_{5}+x_{1}x_{4} = 0

x_{5}x_{3}+x_{3}x_{1}+x_{1}x_{5} = 0

x_{6}x_{2}+x_{2}x_{3}+x_{3}x_{6} = 0

x_{6}x_{3}+x_{5}x_{6}+x_{3}x_{5} = 0

x_{6}x_{4}+x_{4}x_{2}+x_{2}x_{6} = 0

x_{6}x_{5}+x_{5}x_{4}+x_{4}x_{6} = 0

A Gröbner basis is given in this file;
the variables x_{1},...,x_{6} here are called a,...,f there.

- 1, x
_{1} - 1, x
_{2}, x_{2}x_{1} - 1, x
_{3}, x_{3}x_{2}, x_{3}x_{2}x_{1} - 1, x
_{4}, x_{4}x_{3}, x_{4}x_{3}x_{2} - 1, x
_{5}, x_{5}x_{4} - 1, x
_{6}

The Hilbert polynomial is then

(1 + t)^{2} (1 + t + t^{2} )^{2} (1 + t + t^{2} + t^{3} )^{2}

=
t^{12} + 6t^{11} + 19t^{10} + 42t^{9} + 71t^{8} + 96t^{7}
+ 106t^{6} + 96t^{5} + 71t^{4} + 42t^{3} + 19t^{2} + 6t + 1.

A similar example is given by the transpositions in S_{4} with the cocycle q = -1.
The algebra then has generators
x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}
(with some identification) and has relations

x_{0}^{2} = x_{1}^{2} = x_{2}^{2}
= x_{3}^{2} = x_{4}^{2} = x_{5}^{2} = 0,

x_{3}x_{2}+x_{2}x_{3} = 0

x_{4}x_{1}+x_{1}x_{4} = 0

x_{5}x_{0}+x_{0}x_{5} = 0

x_{3}x_{0}+x_{1}x_{3}+x_{0}x_{1} = 0

x_{3}x_{1}+x_{0}x_{3}+x_{1}x_{0} = 0

x_{4}x_{0}+x_{2}x_{4}+x_{0}x_{2} = 0

x_{4}x_{2}+x_{0}x_{4}+x_{2}x_{0} = 0

x_{5}x_{1}+x_{2}x_{5}+x_{1}x_{2} = 0

x_{5}x_{2}+x_{1}x_{5}+x_{2}x_{1} = 0

x_{5}x_{3}+x_{4}x_{5}+x_{3}x_{4} = 0

x_{5}x_{4}+x_{3}x_{5}+x_{4}x_{3} = 0

A Gröbner basis is given in this file;
the variables x_{0},...,x_{5} here are called a,...,f there.

The dimension and the Hilbert polynomial are the same. Notice that both algebras are different in principle, the relations are not exactly the same though they look similar. Indeed, they are different as braided Hopf algebras.

x_{0}^{2} = x_{1}^{2} = x_{2}^{2}
= x_{3}^{2} = x_{4}^{2} = x_{5}^{2}
= x_{6}^{2} = 0

x_{3}x_{0} + x_{1}x_{3} + x_{0}x_{1} = 0

x_{3}x_{2} + x_{2}x_{0} + x_{0}x_{3} = 0

x_{4}x_{1} + x_{2}x_{4} + x_{1}x_{2} = 0

x_{4}x_{3} + x_{3}x_{1} + x_{1}x_{4} = 0

x_{5}x_{0} + x_{4}x_{5} + x_{0}x_{4} = 0

x_{5}x_{1} + x_{1}x_{0} + x_{0}x_{5} = 0

x_{5}x_{2} + x_{3}x_{5} + x_{2}x_{3} = 0

x_{5}x_{4} + x_{4}x_{2} + x_{2}x_{5} = 0

x_{6}x_{0} + x_{2}x_{6} + x_{0}x_{2} = 0

x_{6}x_{1} + x_{5}x_{6} + x_{1}x_{5} = 0

x_{6}x_{2} + x_{2}x_{1} + x_{1}x_{6} = 0

x_{6}x_{3} + x_{4}x_{6} + x_{3}x_{4} = 0

x_{6}x_{4} + x_{4}x_{0} + x_{0}x_{6} = 0

x_{6}x_{5} + x_{5}x_{3} + x_{3}x_{6} = 0

x_{2}x_{0}x_{1}x_{2}x_{0}x_{1} + x_{1}x_{2}x_{0}x_{1}x_{2}x_{0} + x_{0}x_{1}x_{2}x_{0}x_{1}x_{2} = 0

A Gröbner basis is given in this file;
the variables x_{0},...,x_{6} here are called a,...,g there.

- 1, x
_{0}. - 1, x
_{1}, x_{1}x_{0}. - 1, x
_{2}, x_{2}x_{0}, x_{2}x_{0}x_{1}, x_{2}x_{0}x_{1}x_{0}, x_{2}x_{0}x_{1}x_{0}x_{2}, x_{2}x_{0}x_{1}x_{0}x_{2}x_{0}, x_{2}x_{0}x_{1}x_{0}x_{2}x_{0}x_{1}, x_{2}x_{0}x_{1}x_{0}x_{2}x_{0}x_{1}x_{0}, x_{2}x_{0}x_{1}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}, x_{2}x_{0}x_{1}x_{0}x_{2}x_{1}, x_{2}x_{0}x_{1}x_{0}x_{2}x_{1}x_{0}, x_{2}x_{0}x_{1}x_{2}, x_{2}x_{0}x_{1}x_{2}x_{0}, x_{2}x_{1}, x_{2}x_{1}x_{0}, x_{2}x_{1}x_{0}x_{2}, x_{2}x_{1}x_{0}x_{2}x_{1}. - 1, x
_{3}, x_{3}x_{1}, x_{3}x_{1}x_{0}, x_{3}x_{1}x_{0}x_{3}, x_{3}x_{1}x_{0}x_{3}x_{1}, x_{3}x_{1}x_{0}x_{3}x_{1}x_{2}, x_{3}x_{1}x_{0}x_{3}x_{1}x_{2}x_{3}, x_{3}x_{1}x_{0}x_{3}x_{1}x_{2}x_{3}x_{1}, x_{3}x_{1}x_{0}x_{3}x_{1}x_{2}x_{3}x_{1}x_{0}, x_{3}x_{1}x_{2}, x_{3}x_{1}x_{2}x_{3}, x_{3}x_{1}x_{2}x_{3}x_{1}, x_{3}x_{1}x_{2}x_{3}x_{1}x_{0}. - 1, x
_{4}, x_{4}x_{0}, x_{4}x_{0}x_{2}, x_{4}x_{0}x_{2}x_{0}, x_{4}x_{0}x_{2}x_{0}x_{1}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{3}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{3}x_{1}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{3}x_{1}x_{2}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{3}x_{1}x_{2}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{3}x_{1}x_{2}x_{4}x_{0}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{3}x_{1}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{3}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{2}x_{4}x_{0}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{3}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{3}x_{1}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{3}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{0}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{3}, x_{4}x_{0}x_{2}x_{0}x_{1}x_{3}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{3}, x_{4}x_{0}x_{2}x_{0}x_{4}, x_{4}x_{0}x_{2}x_{0}x_{4}x_{2}, x_{4}x_{0}x_{2}x_{0}x_{4}x_{2}x_{3}, x_{4}x_{0}x_{2}x_{1}, x_{4}x_{0}x_{2}x_{1}x_{0}, x_{4}x_{0}x_{2}x_{1}x_{0}x_{3}, x_{4}x_{0}x_{2}x_{1}x_{3}, x_{4}x_{0}x_{2}x_{3}, x_{4}x_{0}x_{3}, x_{4}x_{0}x_{3}x_{4}, x_{4}x_{2}, x_{4}x_{2}x_{3}. - 1, x
_{5}, x_{5}x_{3}. - 1, x
_{6}.

Hilbert polynomial:

(1 + t)^{2} (1 + t + t^{2})^{2}
(1 + t + 2t^{2} + 3t^{3} + 4t^{4} + 5t^{5} + 4t^{6} + 5t^{7} + 4t^{8} + 3t^{9} + 2t^{10} + t^{11} + t^{12})
(1 + t + t^{2} + 2t^{3} + 2t^{4} + 2t^{5} + 2t^{6} + t^{7} + t^{8} + t^{9})
(1 + t + 2t^{2} + 2t^{3} + 3t^{4} + 3t^{5} + 2t^{6} + 2t^{7} + t^{8} + t^{9})

=1t^{36} + 7t^{35} + 28t^{34} + 84t^{33} + 210t^{32} + 462t^{31} + 918t^{30} + 1673t^{29} + 2828t^{28} + 4473t^{27} + 6664t^{26} + 9394t^{25} + 12573t^{24} + 16023t^{23} + 19488t^{22} + 22659t^{21} + 25214t^{20} + 26873t^{19} + 27448t^{18} + 26873t^{17} + 25214t^{16} + 22659t^{15} + 19488t^{14} + 16023t^{13} + 12573t^{12} + 9394t^{11} + 6664t^{10} + 4473t^{9} + 2828t^{8} + 1673t^{7} + 918t^{6} + 462t^{5} + 210t^{4} + 84t^{3} + 28t^{2} + 7t + 1

x_{0}^{2} = x_{1}^{2} = x_{2}^{2} = x_{3}^{2} =
x_{4}^{2} = x_{5}^{2} = x_{6}^{2} = x_{7}^{2} =
x_{8}^{2} = x_{9}^{2} = 0

x_{4}x_{2}+x_{2}x_{4} = 0

x_{4}x_{3}+x_{3}x_{4} = 0

x_{5}x_{1}+x_{1}x_{5} = 0

x_{5}x_{3}+x_{3}x_{5} = 0

x_{6}x_{1}+x_{1}x_{6} = 0

x_{6}x_{2}+x_{2}x_{6} = 0

x_{7}x_{0}+x_{0}x_{7} = 0

x_{7}x_{3}+x_{3}x_{7} = 0

x_{7}x_{6}+x_{6}x_{7} = 0

x_{8}x_{0}+x_{0}x_{8} = 0

x_{8}x_{2}+x_{2}x_{8} = 0

x_{8}x_{5}+x_{5}x_{8} = 0

x_{9}x_{0}+x_{0}x_{9} = 0

x_{9}x_{1}+x_{1}x_{9} = 0

x_{9}x_{4}+x_{4}x_{9} = 0

x_{4}x_{0}+x_{1}x_{4}+x_{0}x_{1} = 0

x_{4}x_{1}+x_{1}x_{0}+x_{0}x_{4} = 0

x_{5}x_{0}+x_{2}x_{5}+x_{0}x_{2} = 0

x_{5}x_{2}+x_{2}x_{0}+x_{0}x_{5} = 0

x_{6}x_{0}+x_{3}x_{6}+x_{0}x_{3} = 0

x_{6}x_{3}+x_{3}x_{0}+x_{0}x_{6} = 0

x_{7}x_{1}+x_{2}x_{7}+x_{1}x_{2} = 0

x_{7}x_{2}+x_{2}x_{1}+x_{1}x_{7} = 0

x_{7}x_{4}+x_{5}x_{7}+x_{4}x_{5} = 0

x_{7}x_{5}+x_{5}x_{4}+x_{4}x_{7} = 0

x_{8}x_{1}+x_{3}x_{8}+x_{1}x_{3} = 0

x_{8}x_{3}+x_{3}x_{1}+x_{1}x_{8} = 0

x_{8}x_{4}+x_{6}x_{8}+x_{4}x_{6} = 0

x_{8}x_{6}+x_{6}x_{4}+x_{4}x_{8} = 0

x_{9}x_{2}+x_{3}x_{9}+x_{2}x_{3} = 0

x_{9}x_{3}+x_{3}x_{2}+x_{2}x_{9} = 0

x_{9}x_{5}+x_{6}x_{9}+x_{5}x_{6} = 0

x_{9}x_{6}+x_{6}x_{5}+x_{5}x_{9} = 0

x_{9}x_{7}+x_{8}x_{9}+x_{7}x_{8} = 0

x_{9}x_{8}+x_{8}x_{7}+x_{7}x_{9} = 0

A Gröbner basis is given in this file;
the variables x_{0},...,x_{9} here are called a,...,j there.

A basis is given then by multiplying one element in each row from top to bottom in all the possible ways from this file.

Hilbert polynomial:

(1 + t)^{4} (1 + t + t^{2})^{3} (1 + t + 2t^{2} + 2t^{3} + t^{4} + t^{5})(1 + t + 3t^{2} + 6t^{3} + 9t^{4} + 13t^{5} + 19t^{6} + 24t^{7} + 27t^{8} + 31t^{9} + 32t^{10} + 31t^{11} + 27t^{12} + 24t^{13} + 19t^{14} + 13t^{15} + 9t^{16} + 6t^{17} + 3t^{18} + t^{19} + t^{20})(1 + t + 2t^{2} + 2t^{3} + t^{4} + t^{5})

= t^{40} + 10t^{39} + 55t^{38} + 220t^{37} + 711t^{36} + 1960t^{35} + 4761t^{34} + 10410t^{33} + 20796t^{32} + 38370t^{31} + 65921t^{30} + 106120t^{29} + 160857t^{28} + 230480t^{27} + 313121t^{26} + 404330t^{25} + 497207t^{24} + 583120t^{23} + 652942t^{22} + 698580t^{21} + 714456t^{20} + 698580t^{19} + 652942t^{18} + 583120t^{17} + 497207t^{16} + 404330t^{15} + 313121t^{14} + 230480t^{13} + 160857t^{12} + 106120t^{11} + 65921t^{10} + 38370t^{9} + 20796t^{8} + 10410t^{7} + 4761t^{6} + 1960t^{5} + 711t^{4} + 220t^{3} + 55t^{2} + 10t + 1