# Higher braidings of diagonal type

Michael Cuntz  Michael Cuntz, Leibniz Universität Hannover, Institut für
Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und
Physik, Welfengarten 1, D-30167 Hannover, Germany<EMAIL_ADDRESS>and  Tobias Ohrmann  Tobias Ohrmann, Leibniz Universität Hannover, Institut
für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik
und Physik, Welfengarten 1, D-30167 Hannover, Germany<EMAIL_ADDRESS>hannover.de

###### Abstract.

Heckenberger introduced the Weyl groupoid of a finite dimensional Nichols
algebra of diagonal type. We replace the matrix of its braiding by a higher
tensor and present a construction which yields further Weyl groupoids. Abelian
cohomology theory gives evidence for the existence of a higher braiding
associated to such a tensor.

###### Key words and phrases:

Nichols algebra, braiding, Weyl groupoid

###### 2020 Mathematics Subject Classification:

17B22, 16T30, 20F55

###### Contents

  1. 1 Nichols algebras of diagonal type
    1. 1.1 Nichols algebras
    2. 1.2 Adjoint action
    3. 1.3 The Weyl groupoid of a Nichols algebra
    4. 1.4 Definition of a Weyl groupoid
      1. 1.4.1 Cartan graphs
      2. 1.4.2 Root systems
  2. 2 Tensor powers of reflections
    1. 2.1 Additive notation
    2. 2.2 Tensor products and eigenvectors
    3. 2.3 Higher tensor powers
    4. 2.4 Higher Rosso conditions
    5. 2.5 Weyl groupoids from higher tensor powers
    6. 2.6 Examples in rank two
  3. 3 Higher tensors from abelian cohomology theory
    1. 3.1 Nichols algebras as graded vector spaces
    2. 3.2 Abelian cohomology theory
    3. 3.3 Invariants for abelian cohomology of commutativity level 1

## Introduction

To a braided vector space $(V,c)$, i.e. a vector space $V$ and a linear
isomorphism

$c:V\otimes V\rightarrow V\otimes
V\quad\text{with}\quad(c\otimes\mathrm{id})(\mathrm{id}\otimes
c)(c\otimes\mathrm{id})=(\mathrm{id}\otimes
c)(c\otimes\mathrm{id})(\mathrm{id}\otimes c)$

one can associate a Hopf algebra $\mathfrak{B}(V)=T(V)/I$, where the ideal $I$
is the kernel of the symmetrizer of the braiding. This algebra
$\mathfrak{B}(V)$ is called the Nichols algebra of $(V,c)$. For example, given
a generalized Cartan matrix $A$ of finite type and a parameter
$q\in\mathbb{C}^{\times}$, it is well-known that the positive part of the
Drinfeld-Jimbo quantum enveloping algebra $U_{q}(A)$ is a Nichols algebra.
This is a special case of a Nichols algebra of diagonal type.

The classification of finite dimensional Nichols algebras of diagonal type
[Hec09] was achieved using certain root systems. In classical Lie theory,
these are orbits under the action of a Weyl group. For Nichols algebras,
Heckenberger [Hec06] noticed that one has to consider Weyl groupoids, i.e.
categories whose morphisms are compositions of reflections acting on a
lattice.

Unfortunately, associating a Weyl groupoid to a Nichols algebra appears to be
neither a surjective nor an injective map (up to isomorphisms). At least in
the case when the braiding defining the Nichols algebra is of diagonal type,
one does not obtain all finite Weyl groupoids as classified in [CH15]. To
obtain a much larger class of Weyl groupoids, Cuntz and Lentner [CL17]
constructed Nichols algebras associated to restrictions of crystallographic
arrangements. However, these are almost never Nichols algebras of Yetter-
Drinfeld modules of finite groups. Moreover, we do not expect to obtain too
many further Weyl groupoids of rank two in this way: in rank two, there are
infinitely many finite Weyl groupoids, but only a very small finite set is
associated to Nichols algebras of diagonal type. Restrictions of Weyl
arrangements won’t produce enough algebras using the construction by Cuntz and
Lentner.

A braiding $c$ of diagonal type is completely determined by a matrix ${\bf
q}=(q_{ij})$ (when $c(x_{i}\otimes x_{j})=q_{ij}x_{j}\otimes x_{i}$ for a
basis $x_{1},\ldots,x_{n}$). To understand the action of a reflection $\sigma$
on a Nichols algebra, it is convenient to consider the bicharacter $\chi$
associated to ${\bf q}$. The map $\sigma$ acts on the exponent vectors in
$\mathbb{Z}^{n}$ of the generators of the Nichols algebra. But to obtain the
new Nichols algebra $\sigma(\mathfrak{B}(V,c))$, one has to apply $\sigma$ on
$\chi$. In an additive notation, this turns out to be the action of
$\sigma\otimes\sigma$ on $\mathbb{Z}^{n}\otimes\mathbb{Z}^{n}$, see
Subsections 2.1 and 2.2.

Conversely, given a Weyl groupoid $\mathcal{W}$, the tensor squares of its
reflections define the variety of braidings $c$ such that $\mathcal{W}$ is the
Weyl groupoid of $\mathfrak{B}(V,c)$. In additive notation it is very natural
to generalize this machinery to higher tensor powers of reflections. For
example, fourth tensor powers would produce a variety of “tensors” ${\bf
q}=(q_{i,j,k,l})$.

To compute this variety, we need a rule to determine the reflections
associated to such a tensor. In the classical theory of Nichols algebras of
diagonal type, this rule is given by a formula that we call _Rosso’s
condition_ : the Cartan entry $c_{\ell,j}$ is minus the smallest
$m\in\mathbb{N}_{0}$ such that

(1)
$(1-q_{\ell,\ell}^{m}q_{\ell,j}q_{j,\ell})\sum_{\nu=0}^{m}q_{\ell,\ell}^{\nu}=0.$

In Section 2.3, we compute in a natural way a generalized Rosso condition for
$d$-th tensor powers when $d$ is even. In general, the Cartan entry
$c_{\ell,j}$ is minus the smallest $m$ such that

(2)
$\left(1-\prod_{\nu=1}^{d}q_{\nu}^{\frac{1}{2}((m+1)^{\nu}-m^{\nu}+(-1)^{\nu+1})}\right)\sum_{\mu=0}^{m}\prod_{\nu=1}^{d}q_{d-\nu}^{\mu\frac{1}{2}\left((m+1)^{\nu-1}+\sum_{k=0}^{\nu-1}(-1)^{\nu-k}m^{k}\right)}=0$

where

$q_{\nu}:=\prod_{\begin{subarray}{c}i_{1},\ldots,i_{d}\in\\{\ell,j\\}\\\
|\\{k\>\mid\>i_{k}=j\\}|=\nu\end{subarray}}q_{i_{1},\ldots,i_{d}}.$

We also obtain a recursion for this condition (Theorem 2.2) which is similar
to the formula obtained in the classical case for the Nichols algebra. We
prove that our generalized Rosso condition produces Cartan matrices such that
all axioms of a Weyl groupoid are satisfied (Theorem 2.10). In examples, we
obtain finite Weyl groupoids which do not occur as invariants of Nichols
algebras of diagonal type. We conjecture that all finite Weyl groupoids are
attained asymptotically when the tensor power (i.e. $d$) increases.

In Section 3 we give a precise definition of higher braidings in terms of
abelian cohomology theory. For an abelian group $G$, it is well-known that
monoidal structures on the category of finite dimensional $G$-graded vector
spaces are (up to braided monoidal equivalence) classified by the third
abelian cohomology $H_{ab}^{3}(G,\mathbb{K}^{\times})$. If the associator is
trivial, the defining abelian $3$-cocycle will simply be a bicharacter on $G$.
In this situation every object naturally becomes a diagonally braided vector
space. In particular, if we set $G=\mathbb{Z}^{n}$, a braiding on
$V=\oplus_{i=1}^{n}\,\mathbb{K}\cdot x_{i}$ is uniquely determined by a matrix
${\bf q}=(q_{ij})\in(\mathbb{K}^{\times})^{n\times n}$. Two such braidings are
equivalent if the numbers $q_{ii}$ and $q_{ij}q_{ji}$ coincide. Consistently,
we define a $d$-dimensional braiding to be an abelian $(2d-1)$-cocycle. Again,
if the (higher) associative structure is trivial this will simply be a
$d$-character on $G$. For $G=\mathbb{Z}^{n}$ the $d$-dimensional braiding is
then uniquely determined by a tensor ${\bf q}=(q_{i_{1},\dots,i_{d}})$. In
analogy to the classical case we can associate the numbers
$q_{\nu}\in\mathbb{K}^{\times}$ (see above) to such a $d$-dimensional diagonal
braiding and see that these numbers only depend on the cohomology class of the
abelian $(2d-1)$-cocycle.
We also notice that, as in the classical case ($d=2$), the higher Rosso
condition as defined before only depends on the numbers
$q_{\nu}\in\mathbb{K}^{\times}$ and hence only on the cohomology class of the
$d$-dimensional braiding ${\bf q}$ (see Example 3.12). Moreover, we conjecture
that the numbers $q_{\nu}\in\mathbb{K}^{\times}$ are defined by multi-
dimensional symmetric forms $\theta_{\lambda}\in
H^{2d-1}(G,\mathbb{K}^{\times})$ indexed by partitions $\lambda$ of $d$. This
is Conjecture 3.10 and Remark 3.11. For the case $d=2$, we obtain the well-
known quadratic form $\theta_{2}$ and symmetric bilinear form
$\theta_{(1,1)}$, which define the numbers $q_{ii}$ and $q_{ij}q_{ji}$ in the
diagonal case.
The computations in Section 3 require explicit knowledge of low-dimensional
generators and boundaries of the abelian chain complex. In Table
LABEL:tab:generators_of_abelian_chain_complex we list these up to $n=6$, as we
believe this can be helpful to other people dealing with abelian chain
complexes independently from our motivation.

##  1\. Nichols algebras of diagonal type

###  1.1. Nichols algebras

Let $G$ be an abelian group and $V$ a finite dimensional representation of the
Drinfeld double $D(G)$ over the field $\mathbb{K}$. Since $G$ is abelian, $V$
decomposes into a direct sum of $1$-dimensional modules

$V=\bigoplus_{i=1}^{n}V_{g_{i},\xi_{i}}$

where $g_{1},\ldots,g_{n}\in G$,
$\xi_{1},\ldots,\xi_{n}\in\hat{G}=\operatorname{Irr}(G)$. Choose
$x_{1},\ldots,x_{n}\in V$ such that $V_{g_{i},\xi_{i}}=\langle x_{i}\rangle$
for all $i=1,\ldots,n$. Then $G$ acts on $V$ via

$g\cdot x_{i}=\xi_{i}(g)x_{i},\quad\quad g\in G,$

and $\hat{G}$ acts on $V$ via

$\xi\cdot x_{i}=\xi(g_{i})x_{i},\quad\quad\xi\in\hat{G}.$

In this section, the relevant information about $V$ is encoded in the matrix

${\bf q}=(q_{i,j})_{i,j}:=(\xi_{j}(g_{i}))_{i,j}.$

There are many equivalent definitions of a Nichols algebra. For our purpose,
the definition via skew derivations will be the most convenient (cf. [Hec06]).

######  Definition 1.1.

Define (left and right)  skew derivations $\partial^{L}_{i}:V^{\otimes
k}\rightarrow V^{\otimes k-1}$, $\partial^{R}_{i}:V^{\otimes k}\rightarrow
V^{\otimes k-1}$, $i=1,\ldots,n$, $k\in\mathbb{N}_{0}$ inductively by

$\partial^{L}_{i}(1):=0,\quad\partial^{L}_{i}(x_{j}):=\delta_{i,j},\quad\partial^{L}_{i}(xy):=\partial^{L}_{i}(x)y+(\xi_{i}\cdot
x)\partial^{L}_{i}(y).$
$\partial^{R}_{i}(1):=0,\quad\partial^{R}_{i}(x_{j}):=\delta_{i,j},\quad\partial^{R}_{i}(xy):=x\partial^{R}_{i}(y)+\partial^{R}_{i}(x)(g_{i}\cdot
y),$

for $x,y$ of degree $\geq 1$.

Let $\varepsilon:T(V)\rightarrow\mathbb{K}$ be the homomorphism of algebras
with $\varepsilon(v)=0$ for all $v\in V$.

######  Definition 1.2.

Let $\mathfrak{I}(V)$ be the largest ideal among all ideals $I$ of $T(V)$ such
that $\varepsilon(I)=0$ and $\partial^{L}_{i}(I)\subset I$ for all
$i=1,\ldots,n$. Then $\mathfrak{B}(V):=T(V)/\mathfrak{I}(V)$ is called the
Nichols algebra (of diagonal type) of $V$.

Using methods similar to [Jos77, Lemma 2.2], one can prove the following
result:

######  Proposition 1.3.

Let $i\in\\{1,\ldots,n\\}$, $\mathcal{K}_{i}:=\ker\partial^{R}_{i}$ and

$h_{i}:=\min\left\\{m\geq
1\mid\sum_{k=0}^{m-1}q_{i,i}^{k}=0\right\\}\cup\\{\infty\\}.$

Then

$\mathfrak{B}(V)\cong\begin{cases}\mathcal{K}_{i}\otimes\mathbb{K}[x_{i}]&h_{i}=\infty,\\\
\mathcal{K}_{i}\otimes\mathbb{K}[x_{i}]/(x_{i}^{h_{i}})&h_{i}\in\mathbb{N}.\end{cases}$

as $\mathbb{Z}^{n}$-graded modules.

###  1.2. Adjoint action

Following [Hec06], we will construct a (generalized) root system associated to
a Nichols algebra of diagonal type. An essential tool is the adjoint action:

######  Definition 1.4.

With the above notation, for $i=1,\ldots,n$ and $y\in T(V)$, let

$(\operatorname{ad}x_{i})(y):=x_{i}y-(g_{i}y)x_{i}.$

In [Hec06, Prop. 1] it is shown that the algebra $\mathcal{K}_{i}$ is the
subalgebra of $\mathfrak{B}(V)$ generated by the elements
$(\operatorname{ad}x_{i})^{m}(x_{j})$, $m\geq 0$, $j\neq i$. This implies:

######  Proposition 1.5.

Let $i\in\\{1,\ldots,n\\}$. Then $\mathcal{K}_{i}$ is finitely generated if
and only if: for all $j\neq i$, there exists $m_{i,j}\geq 0$ such that

$(\operatorname{ad}x_{i})^{m_{i,j}+1}(x_{j})=0.$

The following proposition will be the key for our generalization, hence we
sketch a proof:

######  Proposition 1.6.

In the situation of Prop. 1.5, we have

(3) $m_{i,j}=\min\left\\{m\geq
0\mid\sum_{k=0}^{m}q_{i,i}^{k}=0\quad\text{or}\quad
q_{i,i}^{m}q_{i,j}q_{j,i}=1\right\\}.$

###### Proof.

Let $y\in T(V)$. Since
$g_{i}\partial^{L}_{i}(y)=q_{i,i}^{-1}\partial^{L}_{i}(g_{i}y)$,

$\displaystyle\partial^{L}_{i}((\operatorname{ad}x_{i})(y))$ $\displaystyle=$
$\displaystyle
y+q_{i,i}x_{i}\partial^{L}_{i}(y)-\partial^{L}_{i}(g_{i}y)x_{i}-(\xi_{i}g_{i}y)$
$\displaystyle=$ $\displaystyle
y-(\xi_{i}g_{i}y)+q_{i,i}(\operatorname{ad}x_{i})\partial^{L}_{i}(y).$

In particular, for $y=(\operatorname{ad}x_{i})^{k}(x_{j})$, $k\geq 0$ we get

$\partial^{L}_{i}((\operatorname{ad}x_{i})^{k+1}(x_{j}))=(1-q_{i,i}^{2k}q_{i,j}q_{j,i})(\operatorname{ad}x_{i})^{k}(x_{j})+q_{i,i}(\operatorname{ad}x_{i})\partial^{L}_{i}((\operatorname{ad}x_{i})^{k}(x_{j})).$

With

(4) $R_{0}:=1-q_{i,j}q_{j,i},\quad\quad
R_{k}:=1-q_{i,i}^{2k}q_{i,j}q_{j,i}+q_{i,i}R_{k-1}$

for $k>0$, we obtain

$\partial^{L}_{i}((\operatorname{ad}x_{i})^{k}(x_{j}))=R_{k-1}(\operatorname{ad}x_{i})^{k-1}(x_{j}).$

Using induction we see that

(5) $R_{k}=(1-q_{i,i}^{k}q_{i,j}q_{j,i})\sum_{\nu=0}^{k}q_{i,i}^{\nu}.$

Thus $(\operatorname{ad}x_{i})^{k}(x_{j})\in\mathfrak{I}(V)$ if and only if
$R_{k-1}=0$. ∎

Let us call (3) and (5) the  Rosso condition because it goes back to Rosso
[Ros98].

###  1.3. The Weyl groupoid of a Nichols algebra

Recall that the Nichols algebra of diagonal type is completely determined by
the matrix ${\bf q}$.

######  Definition 1.7.

The  Cartan matrix $(c^{\bf q}_{i,j})_{1\leq i,j\leq n}$ of ${\bf q}$ is given
by the formulas $c^{\bf q}_{i,i}=2$ and (3):

(6) $c^{\bf q}_{i,j}=-\min\\{m\in\mathbb{N}_{0}\mid
1+q_{ii}+q_{ii}^{2}+\cdots+q_{ii}^{m}=0\quad\text{or}\quad
q_{ii}^{m}q_{ij}q_{ji}=1\\},\quad i\neq j.$

Notice that depending on ${\bf q}$, some of these entries may not be defined
since an $m\in\mathbb{N}_{0}$ satisfying one of the two conditions possibly
does not exist. In this case, we will say that the Nichols algebra has no
corresponding Weyl groupoid. Prop. 1.5 ensures that a Cartan matrix exists in
the special case that the Nichols algebra is finite dimensional.

######  Definition 1.8 ([Hec06]).

Let $\mathbb{Z}^{n}=\langle\alpha_{1},\ldots,\alpha_{n}\rangle$ and
$\alpha_{1},\ldots,\alpha_{n}$ be the standard basis. Let
$\ell\in\\{1,\ldots,n\\}$ and
$\sigma_{\ell}\in\operatorname{Aut}(\mathbb{Z}^{n})$ be the reflection given
by

$\sigma_{\ell}(\alpha_{j}):=\alpha_{j}-c^{\bf q}_{\ell,j}\alpha_{\ell}$

for $\ell=1,\ldots,n$. Using $\sigma_{\ell}$ we obtain a new matrix
$\sigma_{\ell}({\bf q})$ via

(7) $\sigma_{\ell}({\bf
q})_{i,j}:=\prod_{k,s=1}^{n}q_{k,s}^{\sigma_{\ell}(\alpha_{i})_{k}\sigma_{\ell}(\alpha_{j})_{s}}.$

We can compute a Cartan matrix to $\sigma_{\ell}({\bf q})$ instead of ${\bf
q}$ using Rosso’s formula and apply another reflection and so on. This way we
obtain a groupoid called the  Weyl groupoid whose objects are the matrix ${\bf
q}$ and its images and the morphisms are compositions of the reflections
between these matrices. We recall the precise definition of a Weyl groupoid in
the following subsection.

######  Example 1.9.

Let $n=3$, $\zeta$ be a primitive third root of unity, and

${\bf q}=(q_{ij})_{i,j}=\begin{pmatrix}-1&\zeta&\zeta\\\ 1&-1&\zeta\\\
1&1&-1\end{pmatrix}.$

$\zeta$
---
$\zeta$
---
$\zeta$
---
$\zeta$
---
$\zeta$
---
$\zeta$
---
$\zeta$
---
$\zeta$
---
$\zeta$
---
$-1$
---
$-1$
---
$-1$
---
$-1$
---
$-1$
---
$-1$
---
$\zeta^{-1}$
---
$\zeta^{-1}$
---
$\zeta^{-1}$
---
$\zeta^{-1}$
---
$\zeta^{-1}$
---
$\zeta^{-1}$
---
Figure 1. Dynkin diagrams in the Weyl groupoid of Example 1.9

The  Dynkin diagram of ${\bf q}$ is a graph with vertices $1,\ldots,n$ and
edges $(i,j)$ when $q_{ij}q_{ji}\neq 1$ and labeled by $q_{ij}q_{ji}$. Fig. 1
displays the Weyl groupoid obtained from ${\bf q}$: the Dynkin diagrams are
the objects, reflections are indicated by seperating lines.

###  1.4. Definition of a Weyl groupoid

####  1.4.1. Cartan graphs

The collections of Cartan matrices produced in Example 1.9 are called Cartan
graphs111In early papers these were called Cartan schemes.. To define the
corresponding Weyl groupoid, we need several notions (compare [CH09b, CH09a]).

######  Definition 1.10 ([Kac90, §1.1]).

Let $I$ be a non-empty finite set and $\\{\alpha_{i}\mid i\in I\\}$ the
standard basis of $\mathbb{Z}^{I}$. A  generalized Cartan matrix
$C=(c_{ij})_{i,j\in I}$ is a matrix in $\mathbb{Z}^{I\times I}$ such that

  1. (M1)

$c_{ii}=2$ and $c_{jk}\leq 0$ for all $i,j,k\in I$ with $j\not=k$,

  2. (M2)

if $i,j\in I$ and $c_{ij}=0$, then $c_{ji}=0$.

######  Definition 1.11.

Let $A$ be a non-empty set, $\rho_{i}:A\to A$ a map for all $i\in I$, and
$C^{a}=(c^{a}_{jk})_{j,k\in I}$ a generalized Cartan matrix in
$\mathbb{Z}^{I\times I}$ for all $a\in A$. The quadruple

$\mathcal{C}=\mathcal{C}(I,A,(\rho_{i})_{i\in I},(C^{a})_{a\in A})$

is called a  Cartan graph if

  1. (C1)

$\rho_{i}^{2}=\mathrm{id}$ for all $i\in I$,

  2. (C2)

$c^{a}_{ij}=c^{\rho_{i}(a)}_{ij}$ for all $a\in A$ and $i,j\in I$.

######  Definition 1.12.

Let $\mathcal{C}=\mathcal{C}(I,A,(\rho_{i})_{i\in I},(C^{a})_{a\in A})$ be a
Cartan graph. For all $i\in I$ and $a\in A$ define
$\sigma_{i}^{a}\in\operatorname{Aut}(\mathbb{Z}^{I})$ by

(8)
$\displaystyle\sigma_{i}^{a}(\alpha_{j})=\alpha_{j}-c_{ij}^{a}\alpha_{i}\qquad\text{for
all $j\in I$.}$

The  Weyl groupoid of $\mathcal{C}$ is the category $\mathcal{W}(\mathcal{C})$
such that $\operatorname{Obj}(\mathcal{W}(\mathcal{C}))=A$ and the morphisms
are compositions of maps $\sigma_{i}^{a}$ with $i\in I$ and $a\in A$, where
$\sigma_{i}^{a}$ is considered as an element in
$\operatorname{Hom}(a,\rho_{i}(a))$. The category $\mathcal{W}(\mathcal{C})$
is a groupoid in the sense that all morphisms are isomorphisms. The
cardinality of $I$ is the  rank of $\mathcal{W}(\mathcal{C})$.

Since most upper indices referring to elements of $A$ are determined by the
context, we will often omit them to improve readability.

####  1.4.2. Root systems

######  Definition 1.13.

Let $\mathcal{C}$ be a Cartan graph. For all $a\in A$ let

$(R^{\mathrm{re}})^{a}=\\{\mathrm{id}^{a}\sigma_{i_{1}}\cdots\sigma_{i_{k}}(\alpha_{j})\,|\,k\in\mathbb{N}_{0},\,i_{1},\dots,i_{k},j\in
I\\}\subseteq\mathbb{Z}^{I}.$

The elements of the set $(R^{\mathrm{re}})^{a}$ are called  real roots (at
$a$). The pair $(\mathcal{C},((R^{\mathrm{re}})^{a})_{a\in A})$ is denoted by
$\mathcal{R}^{\mathrm{re}}(\mathcal{C})$. A real root
$\alpha\in(R^{\mathrm{re}})^{a}$, where $a\in A$, is called positive (resp.
negative) if $\alpha\in\mathbb{N}_{0}^{I}$ (resp.
$\alpha\in-\mathbb{N}_{0}^{I}$).

The Weyl groupoids which are relevant for the study of Nichols algebras
produce sets of real roots which satisfies additional properties:

######  Definition 1.14.

Let $\mathcal{C}=\mathcal{C}(I,A,(\rho_{i})_{i\in I},(C^{a})_{a\in A})$ be a
Cartan graph. For all $a\in A$ let $R^{a}\subseteq\mathbb{Z}^{I}$, and define
$m_{i,j}^{a}=|R^{a}\cap(\mathbb{N}_{0}\alpha_{i}+\mathbb{N}_{0}\alpha_{j})|$
for all $i,j\in I$ and $a\in A$. We say that

$\mathcal{R}=\mathcal{R}(\mathcal{C},(R^{a})_{a\in A})$

is a  root system of type $\mathcal{C}$, if it satisfies the following axioms.

  1. (R1)

$R^{a}=R^{a}_{+}\cup-R^{a}_{+}$, where
$R^{a}_{+}=R^{a}\cap\mathbb{N}_{0}^{I}$, for all $a\in A$.

  2. (R2)

$R^{a}\cap\mathbb{Z}\alpha_{i}=\\{\alpha_{i},-\alpha_{i}\\}$ for all $i\in I$,
$a\in A$.

  3. (R3)

$\sigma_{i}^{a}(R^{a})=R^{\rho_{i}(a)}$ for all $i\in I$, $a\in A$.

  4. (R4)

If $i,j\in I$ and $a\in A$ such that $i\not=j$ and $m_{i,j}^{a}$ is finite,
then $(\rho_{i}\rho_{j})^{m_{i,j}^{a}}(a)=a$.

The axioms (R2) and (R3) are always fulfilled for $\mathcal{R}^{\mathrm{re}}$.
The root system $\mathcal{R}$ is called  finite if for all $a\in A$ the set
$R^{a}$ is finite. By [CH09b, Prop. 2.12], if $\mathcal{R}$ is a finite root
system of type $\mathcal{C}$, then $\mathcal{R}=\mathcal{R}^{\mathrm{re}}$,
and hence $\mathcal{R}^{\mathrm{re}}$ is a root system of type $\mathcal{C}$
in that case.

##  2\. Tensor powers of reflections

###  2.1. Additive notation

Let us take a closer look to Rosso’s formula (6). For $i\neq j$, we are
looking for a smallest $m\in\mathbb{N}_{0}$ such that

(9) $1+q_{ii}+q_{ii}^{2}+\cdots+q_{ii}^{m}=0\quad\text{or}\quad
q_{ii}^{m}q_{ij}q_{ji}=1,$

or equivalently,

(10) $(q_{ii}^{m+1}=1\quad\text{and}\quad q_{ii}\neq 1)\quad\text{or}\quad
q_{ii}^{m}q_{ij}q_{ji}=1.$

This shows that in the whole procedure computing the Weyl groupoid, only
integral powers of $q_{ij}$ are involved (remember (7)). This allows us to
write everything in an additive notation: Let $W:=\mathbb{Z}^{n\times
n}=\langle e_{ij}\mid i,j\in\\{1,\ldots,n\\}\rangle$ and $\chi_{\bf q}$ be the
group homomorphism

$\chi_{\bf q}:(W,+)\rightarrow U:=\langle q_{ij}\mid
i,j\in\\{1,\ldots,n\\}\rangle\leq\mathbb{K}^{\times},\quad e_{ij}\mapsto
q_{ij}.$

Then we obtain an isomorphism of $\mathbb{Z}$-modules

$\varepsilon:U\rightarrow M_{\bf q}:=W/\ker(\chi_{\bf q}).$

Using the map $\varepsilon$, Rosso’s condition (10) becomes

(11) $((m+1)\varepsilon(q_{ii})=0\quad\text{and}\quad\varepsilon(q_{ii})\neq
0)\quad\text{or}\quad
m\varepsilon(q_{ii})+\varepsilon(q_{ij})+\varepsilon(q_{ji})=0$

or with $\overline{e}_{ij}:=e_{ij}+\ker(\chi_{\bf q})$ for $i,j=1,\ldots,n$ in
$M_{\bf q}$,

(12) $((m+1)\overline{e}_{ii}=0\quad\text{and}\quad\overline{e}_{ii}\neq
0)\quad\text{or}\quad
m\overline{e}_{ii}+\overline{e}_{ij}+\overline{e}_{ji}=0.$

###  2.2. Tensor products and eigenvectors

The reflection of matrices defined in (7),

$\sigma_{\ell}({\bf
q})_{i,j}:=\prod_{k,s=1}^{n}q_{k,s}^{\sigma_{\ell}(\alpha_{i})_{k}\sigma_{\ell}(\alpha_{j})_{s}}$

translates to a tensor product in additive notation:

${\mathbb{Z}^{n}}$${W\cong\mathbb{Z}^{n}\otimes\mathbb{Z}^{n}}$${U\leq\mathbb{K}^{\times}}$${\mathbb{Z}^{n}}$${W\cong\mathbb{Z}^{n}\otimes\mathbb{Z}^{n}}$$\scriptstyle{\otimes}$$\scriptstyle{\chi_{\sigma_{\ell}({\bf
q})}}$$\scriptstyle{\otimes}$$\scriptstyle{\sigma_{\ell}}$$\scriptstyle{\chi_{{\bf
q}}}$$\scriptstyle{\sigma_{\ell}\otimes\sigma_{\ell}}$

To obtain a Weyl groupoid, it is necessary that Rosso’s formula satisfies the
symmetry

(13) $c^{\bf q}_{\ell,j}=c^{\sigma_{\ell}({\bf q})}_{\ell,j}$

for all $\ell,j\in\\{1,\ldots,r\\}$. This comes from the fact that
$(m+1)e_{\ell\ell}$ and $me_{\ell\ell}+e_{\ell j}+e_{j\ell}$ are eigenvectors
of $\sigma_{\ell}\otimes\sigma_{\ell}$ to the eigenvalues $\pm 1$ for
$m=-c^{\bf q}_{\ell,j}$. Because of this fact, the conditions will remain the
same after reflecting. In order to “explain” this fact, we now compare the
degrees of $(\operatorname{ad}x_{\ell})^{m+1}(x_{j})$ and
$(\operatorname{ad}x_{\ell})^{m}(x_{j})$:

$u_{m}:=((m+1)\alpha_{\ell}+\alpha_{j})^{\otimes
2}-(m\alpha_{\ell}+\alpha_{j})^{\otimes
2}=(2m+1)\alpha_{\ell}\otimes\alpha_{\ell}+\alpha_{\ell}\otimes\alpha_{j}+\alpha_{j}\otimes\alpha_{\ell}.$

The vector $u_{m}$ is a sum of eigenvectors of
$\sigma_{\ell}\otimes\sigma_{\ell}$:

$u_{m}=\underbrace{\frac{1}{2}(\sigma_{\ell}^{\otimes
2}(u_{m})+u_{m})}_{v_{m}}\underbrace{-\frac{1}{2}(\sigma_{\ell}^{\otimes
2}(u_{m})-u_{m})}_{w_{m}},$

these are

$\displaystyle v_{m}$ $\displaystyle:=$
$\displaystyle\frac{1}{2}(\sigma_{\ell}^{\otimes
2}(u_{m})+u_{m})=(m+1)\alpha_{\ell}^{\otimes 2},$ $\displaystyle w_{m}$
$\displaystyle:=$ $\displaystyle-\frac{1}{2}(\sigma_{\ell}^{\otimes
2}(u_{m})-u_{m})=m\alpha_{\ell}^{\otimes
2}+\alpha_{\ell}\otimes\alpha_{j}+\alpha_{j}\otimes\alpha_{\ell}.$

We observe that Rosso’s condition (5) becomes

(14) $(1-q_{\ell,\ell})R_{m}=(1-\chi_{\bf q}(w_{m}))(1-\chi_{\bf q}(v_{m})).$

###  2.3. Higher tensor powers

We now generalize the additive decomposition into eigenvectors (14) to higher
tensor powers. Let $d\in\mathbb{N}$ and

${\bf
q}=(q_{i_{1},\ldots,i_{d}})_{i_{1},\ldots,i_{d}}\in\mathbb{K}^{\\{1,\ldots,n\\}^{d}}.$

Let $W:=(\mathbb{Z}^{n})^{\otimes d}$ and denote by
$e_{i_{1},\ldots,i_{d}}:=\alpha_{i_{1}}\otimes\cdots\otimes\alpha_{i_{d}}$ the
elements of its standard basis. Let $\chi_{\bf q}$ be the group homomorphism

$\chi_{\bf q}:(W,+)\rightarrow U:=\langle q_{i_{1},\ldots,i_{d}}\mid
i_{1},\ldots,i_{d}\in\\{1,\ldots,n\\}\rangle\leq\mathbb{K}^{\times},\quad
e_{i_{1},\ldots,i_{d}}\mapsto q_{i_{1},\ldots,i_{d}}.$

As before, we compare the $d$-th tensor powers of presumed degrees:

$u_{m}:=((m+1)\alpha_{\ell}+\alpha_{j})^{\otimes
d}-(m\alpha_{\ell}+\alpha_{j})^{\otimes d}.$

We decompose $u_{m}$ as a sum of eigenvectors of $\sigma_{\ell}^{\otimes d}$:

$u_{m}=\underbrace{\frac{1}{2}(\sigma_{\ell}^{\otimes
d}(u_{m})+u_{m})}_{v_{m}}\underbrace{-\frac{1}{2}(\sigma_{\ell}^{\otimes
d}(u_{m})-u_{m})}_{w_{m}}.$

We do not know yet what will be the analog of $(1-q_{\ell,\ell})$ for $d>2$,
so let us first set

(15) $\tilde{R}_{m}:=(1-\chi_{\bf q}(v_{m}))(1-\chi_{\bf q}(w_{m})).$

In the following Theorem 2.2 we find a recursion for $\tilde{R}_{m}$ similar
to Equation (4). We need some notation for its formulation:

######  Definition 2.1.

Let $\ell,j\in\\{1,\ldots,n\\}$. For $k\in\\{0,\ldots,d\\}$ and
$m\in\mathbb{N}$ we set

$\gamma_{k}:=\sum_{\begin{subarray}{c}i_{1},\ldots,i_{d}\in\\{\ell,j\\}\\\
|\\{\nu\>\mid\>i_{\nu}=j\\}|=k\end{subarray}}\alpha_{i_{1}}\otimes\ldots\otimes\alpha_{i_{d}},\quad
q_{k}:=\chi_{\bf
q}(\gamma_{k})=\prod_{\begin{subarray}{c}i_{1},\ldots,i_{d}\in\\{\ell,j\\}\\\
|\\{\nu\>\mid\>i_{\nu}=j\\}|=k\end{subarray}}q_{i_{1},\ldots,i_{d}},$
$e_{m,k}:=\sum_{\begin{subarray}{c}\nu=0\\\
(-1)^{\nu}=(-1)^{k}\end{subarray}}^{k-2}\binom{k}{\nu}m^{\nu},\quad
r_{m}:=\prod_{i=0}^{d-2}q_{i}^{e_{m,d-i}},$
$f_{m,k}:=\sum_{\begin{subarray}{c}\nu=0\\\
(-1)^{\nu}=(-1)^{k+1}\end{subarray}}^{k-1}\binom{k}{\nu}m^{\nu},\quad
z_{m}:=\prod_{i=0}^{d-1}q_{i}^{f_{m,d-i}}.$

######  Theorem 2.2.

With the above notation, $\tilde{R}_{m}$, $m\in\mathbb{N}$ satisfy the
following recursions:

(16) $\displaystyle\tilde{R}_{0}$ $\displaystyle=$
$\displaystyle(1-r_{0})(1-z_{0}),$ (17) $\displaystyle\tilde{R}_{m}$
$\displaystyle=$ $\displaystyle r_{m}\tilde{R}_{m-1}+(1-r_{m})(1-z_{m}).$

###### Proof.

For Equation (16), we compute

$\displaystyle v_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\sigma_{\ell}^{\otimes
d}(u_{0})+u_{0})=\frac{1}{2}((-\alpha_{\ell}+\alpha_{j})^{\otimes
d}-\alpha_{j}^{\otimes d}+(\alpha_{\ell}+\alpha_{j})^{\otimes
d}-\alpha_{j}^{\otimes d})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\nu=0}^{d-1}(1+(-1)^{d-\nu})\gamma_{\nu}=\sum_{\nu=0}^{d-2}e_{0,d-\nu}\gamma_{\nu}$

because $e_{0,k}=\frac{1}{2}(1+(-1)^{k})$; hence $\chi_{\bf q}(v_{0})=r_{0}$.
We check in the same way that $\chi_{\bf q}(w_{0})=z_{0}$, thus
$\tilde{R}_{0}=(1-r_{0})(1-z_{0})$. For the recursions, observe first that
using

$e_{m,k}+f_{m,k}=(m+1)^{k}-m^{k}$

we obtain

(18) $e_{m,k}-f_{m,k}=(m-1)^{k}-m^{k}=-e_{m-1,k}-f_{m-1,k}.$

This implies the relations

(19) $\displaystyle\prod_{\nu=0}^{d-1}q_{\nu}^{f_{m,d-\nu}}$
$\displaystyle\stackrel{{\scriptstyle\eqref{efef}}}{{=}}$
$\displaystyle\prod_{\nu=0}^{d-2}q_{\nu}^{e_{m,d-\nu}}\prod_{\nu=0}^{d-1}q_{\nu}^{e_{m-1,d-\nu}+f_{m-1,d-\nu}},$
(20)
$\displaystyle\prod_{\nu=0}^{d-2}q_{\nu}^{e_{m,d-\nu}}\prod_{\nu=0}^{d-1}q_{\nu}^{\frac{1}{2}(e_{m-1,d-\nu}+f_{m-1,d-\nu})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{efef}}}{{=}}$
$\displaystyle\prod_{\nu=0}^{d-1}q_{\nu}^{\frac{1}{2}(e_{m,d-\nu}+f_{m,d-\nu})}.$

We now compute $v_{m}$ and $w_{m}$:

$\displaystyle v_{m}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\sigma_{\ell}^{\otimes d}(u_{m})+u_{m})$
$\displaystyle=$
$\displaystyle\frac{1}{2}((-\alpha_{\ell}+\alpha_{j})^{\otimes
d}-\alpha_{j}^{\otimes d}+((m+1)\alpha_{\ell}+\alpha_{j})^{\otimes
d}-(m\alpha_{\ell}+\alpha_{j})^{\otimes d})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\nu=0}^{d-1}(e_{m,d-\nu}+f_{m,d-\nu}+(-1)^{d-\nu})\gamma_{\nu}$

and similarly

$w_{m}=\frac{1}{2}\sum_{\nu=0}^{d-1}(e_{m,d-\nu}+f_{m,d-\nu}+(-1)^{d-\nu+1})\gamma_{\nu}.$

Thus we get

$\displaystyle\tilde{R}_{m}$ $\displaystyle=$ $\displaystyle(1-\chi_{\bf
q}(w_{m}))(1-\chi_{\bf q}(v_{m}))$ $\displaystyle=$ $\displaystyle
1-\prod_{\nu=0}^{d-1}q_{\nu}^{\frac{1}{2}(e_{m,d-\nu}+f_{m,d-\nu}+(-1)^{d-\nu})}-\prod_{\nu=0}^{d-1}q_{\nu}^{\frac{1}{2}(e_{m,d-\nu}+f_{m,d-\nu}+(-1)^{d-\nu+1})}$
$\displaystyle+\prod_{\nu=0}^{d-1}q_{\nu}^{e_{m,d-\nu}+f_{m,d-\nu}}$
$\displaystyle\stackrel{{\scriptstyle\text{\eqref{ef1}, \eqref{ef2}}}}{{=}}$
$\displaystyle\left(\prod_{\nu=0}^{d-2}q_{\nu}^{e_{m,d-\nu}}\right)\left(1-\prod_{\nu=0}^{d-1}q_{\nu}^{\frac{1}{2}(e_{m-1,d-\nu}+f_{m-1,d-\nu}+(-1)^{d-\nu})}\right)$
$\displaystyle\left(1-\prod_{\nu=0}^{d-1}q_{\nu}^{\frac{1}{2}(e_{m-1,d-\nu}+f_{m-1,d-\nu}+(-1)^{d-\nu+1})}\right)+\left(1-\prod_{\nu=0}^{d-2}q_{\nu}^{e_{m,d-\nu}}\right)$
$\displaystyle\left(1-\prod_{\nu=0}^{d-1}q_{\nu}^{f_{m,d-\nu}}\right)$
$\displaystyle=$ $\displaystyle r_{m}\tilde{R}_{m-1}+(1-r_{m})(1-z_{m}).$

∎

######  Example 2.3.

  1. (1)

If $d=2$, then we recover the recursion (4) and Rosso’s condition from Nichols
algebras (notice that then $r_{m}=q_{0}=q_{\ell,\ell}$ for all $m$).

  2. (2)

If $d=3$, then

$\displaystyle\chi_{\bf q}(v_{m})$ $\displaystyle=$ $\displaystyle
q_{\ell\ell\ell}^{3\frac{m(m+1)}{2}}(q_{\ell\ell j}q_{\ell
j\ell}q_{j\ell\ell})^{m+1}=q_{0}^{3\frac{m(m+1)}{2}}q_{1}^{m+1},$
$\displaystyle\chi_{\bf q}(w_{m})$ $\displaystyle=$ $\displaystyle
q_{\ell\ell\ell}^{3\frac{m(m+1)}{2}+1}(q_{\ell\ell j}q_{\ell
j\ell}q_{j\ell\ell})^{m}q_{\ell jj}q_{j\ell
j}q_{jj\ell}=q_{0}^{3\frac{m(m+1)}{2}+1}q_{1}^{m}q_{2}.$

  3. (3)

$d=4$: With respect to the basis $\gamma_{0},\ldots,\gamma_{4}$,

$\displaystyle v_{m}$ $\displaystyle=$
$\displaystyle\left(2m^{3}+3m^{2}+2m+1,\frac{3}{2}m^{2}+\frac{3}{2}m,m+1,0,0\right),$
$\displaystyle w_{m}$ $\displaystyle=$
$\displaystyle\left(2m^{3}+3m^{2}+2m,\frac{3}{2}m^{2}+\frac{3}{2}m+1,m,1,0\right).$

###  2.4. Higher Rosso conditions

Remember that the original formula by Rosso (6) in multiplicative notation
includes a geometric series which we replaced by the condition that
$q_{\ell,\ell}^{m+1}=1$ and $q_{\ell,\ell}\neq 1$. In other words, when $d=2$,
we have to divide $\tilde{R}_{m}$ by $1-q_{\ell,\ell}$. Moreover, this
division is necessary to ensure Axiom (M2) (see Prop. 2.8) if we aim to get a
Weyl groupoid.
So we need to find some polynomial $\tilde{g}_{m}$ in $q_{0},\ldots,q_{d}$
such that $\tilde{R}_{m}/\tilde{g}_{m}$ is a condition that produces a Weyl
groupoid. In particular, to obtain a Cartan matrix and a Cartan graph (see
Def. 2.7 and Prop. 2.9), $\tilde{R}_{m}/\tilde{g}_{m}$ should not be rational
functions with non trivial denominators.

In the classical case of Nichols algebras, $\tilde{g}_{m}$ is a divisor of
$(1-\chi_{\bf q}(v_{m}))$. For $d>2$, the polynomial $(1-\chi_{\bf q}(v_{m}))$
in $q_{0},\ldots,q_{d}$ does not factorize in a nice way. However, we notice
that the greatest common divisor of the coordinates of $v_{m}$ (as polynomials
in $m$) is $m+1$. Indeed,

$\displaystyle v_{m}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\nu=1}^{d}((m+1)^{\nu}-m^{\nu}+(-1)^{\nu})\gamma_{d-\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\nu=1}^{d}(m+1)\left((m+1)^{\nu-1}+\sum_{k=0}^{\nu-1}(-1)^{\nu-k}m^{k}\right)\gamma_{d-\nu}.$

This suggests the following definition.

######  Definition 2.4.

With the above notation and $m\in\mathbb{N}$ we set

(21) $\displaystyle s_{m}$ $\displaystyle:=$
$\displaystyle\sum_{\nu=1}^{d}\frac{1}{2}\left((m+1)^{\nu-1}+\sum_{k=0}^{\nu-1}(-1)^{\nu-k}m^{k}\right)\gamma_{d-\nu},$
(22) $\displaystyle g_{m}$ $\displaystyle:=$ $\displaystyle\chi_{\bf
q}(s_{m})=\prod_{\nu=1}^{d}q_{d-\nu}^{\frac{1}{2}\left((m+1)^{\nu-1}+\sum_{k=0}^{\nu-1}(-1)^{\nu-k}m^{k}\right)},$
(23) $\displaystyle R_{m}$ $\displaystyle:=$
$\displaystyle\frac{\tilde{R}_{m}}{1-g_{m}}=\frac{(1-\chi_{\bf
q}(v_{m}))(1-\chi_{\bf q}(w_{m}))}{1-\chi_{\bf q}(s_{m})}.$

Unfortunately, some of the coordinates of $s_{m}$ are in
$\frac{1}{2}+\mathbb{Z}$ (the exponents of $q_{i}$ for $m$ odd, $(i+d)$ odd,
and $i<d-1$) and we possibly get square roots of $q_{0},\ldots,q_{d}$ in
$g_{m}$. In order to obtain a consistent condition and definition, we may fix
square roots $\sqrt{q_{0}},\ldots,\sqrt{q_{d}}$ first and define the $q_{i}$
to be their squares. This corresponds to enlarging the $\mathbb{Z}$-module
considered in additive notation. We will see below that in most cases, the
choice of square roots in fact does not affect the condition $R_{m}=0$.

######  Corollary 2.5.

The conditions $R_{m}$, $m\in\mathbb{N}_{0}$ satisfy:

(24) $\displaystyle R_{m}$ $\displaystyle\in$
$\displaystyle\mathbb{Z}[\sqrt{q_{0}},\ldots,\sqrt{q_{d}}],$ (25)
$\displaystyle R_{0}$ $\displaystyle=$ $\displaystyle 1-z_{0},$ (26)
$\displaystyle R_{m}$ $\displaystyle=$
$\displaystyle\frac{r_{m}(1-g_{m-1})}{1-g_{m}}R_{m-1}+\frac{(1-r_{m})(1-z_{m})}{1-g_{m}}.$

###### Proof.

Since $\chi_{\bf q}(v_{m})=g_{m}^{m+1}$, the polynomial $1-\chi_{\bf
q}(v_{m})$ is divisible by $1-g_{m}$, the quotient being a geometric series.
However, these are integral polynomials in $\sqrt{q_{i}}$, $i=0,\ldots,d$ by
the definition of $g_{m}$. One can check that

$g_{0}=\sum_{\nu=1}^{d}q_{d-\nu}^{\frac{1}{2}(1+(-1)^{\nu})}=r_{0},$

thus $R_{0}=(1-r_{0})(1-z_{0})/(1-g_{0})=1-z_{0}$. The recursion is a direct
consequence of Thm. 2.2. ∎

######  Remark 2.6.

The condition $R_{m}=0$ may be written as

(27) $(\chi_{\bf q}(v_{m})=1\text{ and }\chi_{\bf q}(s_{m})\neq
1)\quad\text{or}\quad\chi_{\bf q}(w_{m})=1$

which is equivalent to

(28) $(\chi_{\bf q}(s_{m})^{m+1}=1\text{ and }\chi_{\bf q}(s_{m})\neq
1)\quad\text{or}\quad\chi_{\bf q}(w_{m})=1.$

Note that square roots of $q_{i}$ only appear in $\chi_{\bf q}(s_{m})$ for odd
$m$ since $v_{m}\in\mathbb{Z}[q_{0},\ldots,q_{d}]$ and $s_{m}=v_{m}/(m+1)$.

Now assume that $\chi_{\bf q}(s_{m})^{2}=1$. Then the first condition cannot
be satisfied for even $m$. If $m$ is odd, then $\chi_{\bf
q}(s_{m})^{m+1}=1^{(m+1)/2}=1$ and in this case the first condition is
satisfied if and only if $\chi_{\bf q}(s_{m})=-1$. Hence $R_{m}=0$ is
equivalent to

(29) $(\chi_{\bf q}(v_{m})=1\text{ and }\chi_{\bf q}(2s_{m})\neq
1)\quad\text{or}\quad(\chi_{\bf q}(s_{m})=-1\text{ and }m\text{
odd})\quad\text{or}\quad\chi_{\bf q}(w_{m})=1.$

It may happen that $\chi_{\bf q}(s_{m})=-1$ with $m$ odd never occurs in the
construction of the Weyl groupoid of a specific tensor. In this case, the
general Rosso condition would be

(30) $(\chi_{\bf q}(v_{m})=1\text{ and }\chi_{\bf q}(2s_{m})\neq
1)\quad\text{or}\quad\chi_{\bf q}(w_{m})=1,$

and it would avoid the square roots. However, in the classical case ($d=2$)
one often has $q_{\ell,\ell}=-1$, and we expect that this condition would
exclude many cases where $s_{m}\in\mathbb{Z}^{d}$ and $\chi_{\bf
q}(s_{m})=-1$.

###  2.5. Weyl groupoids from higher tensor powers

We may now use the Rosso condition $R_{m}$ for $d\geq 2$ to compute Weyl
groupoids for a given ${\bf q}$ in the same way as for $d=2$. For
$\ell,j\in\\{1,\ldots,n\\}$ we write $R^{\ell,j}_{m}:=R_{m}$.

######  Definition 2.7.

Let $d\in\mathbb{N}$ and ${\bf
q}=(q_{i_{1},\ldots,i_{d}})_{i_{1},\ldots,i_{d}}\in\mathbb{K}^{\\{1,\ldots,n\\}^{d}}$.
For each tuple $(i_{1},\ldots,i_{d})$ we fix a square root
$\sqrt{q_{i_{1},\ldots,i_{d}}}\in\mathbb{K}$ of $q_{i_{1},\ldots,i_{d}}$ and
by abuse of notation we write

$\sqrt{{\bf q}}:=(\sqrt{q_{i_{1},\ldots,i_{d}}})_{i_{1},\ldots,i_{d}}.$

Then using Equation (23) we define

(31) $c^{\bf q}_{\ell,j}=\begin{cases}-\min\\{m\in\mathbb{N}_{0}\mid
R^{\ell,j}_{m}=0\\}&\ell\neq j\\\ 2&\ell=j\end{cases}$

and obtain a matrix $c^{\bf q}=(c^{\bf q}_{\ell,j})_{\ell,j}$ if such an $m$
exists for all $\ell\neq j$. In this case, we call $c^{\bf q}$ the  Cartan
matrix of ${\bf q}$. It defines maps $\sigma_{\ell}$ via

$\sigma_{\ell}({\bf
q})_{i_{1},\ldots,i_{d}}:=\prod_{k_{1},\ldots,k_{d}=1}^{n}q_{k_{1},\ldots,k_{d}}^{\sigma_{\ell}(\alpha_{i_{1}})_{k_{1}}\cdots\sigma_{\ell}(\alpha_{i_{d}})_{k_{d}}}.$

These maps act on the square roots in a compatible way since additively, ${\bf
q}$ is just twice the vector $\sqrt{{\bf q}}$ and $\sigma_{\ell}^{\otimes d}$
is linear.

######  Proposition 2.8.

Let $d\in\mathbb{N}$ be even and ${\bf
q}=(q_{i_{1},\ldots,i_{d}})_{i_{1},\ldots,i_{d}}\in\mathbb{K}^{\\{1,\ldots,n\\}^{d}}$.
If a Cartan matrix $c^{\bf q}$ is defined, then it is a generalized Cartan
matrix.

###### Proof.

Axiom (M1) is satisfied by definition. For (M2), remember that
$R_{0}=1-z_{0}$. Since

$z_{0}=\sum_{\nu=0}^{d-1}\frac{1}{2}(1-(-1)^{\nu})\gamma_{\nu}=\sum_{\nu=0}^{d-1}\frac{1}{2}(1-(-1)^{d-\nu})\gamma_{\nu},$

the condition $R_{0}$ is the same for $c^{\bf q}_{\ell,j}$ as for $c^{\bf
q}_{j,\ell}$. ∎

######  Proposition 2.9.

Let $d\in\mathbb{N}$ be even, $I=\\{1,\ldots,n\\}$, and $A$ be a set of
tensors ${\bf q}\in\mathbb{K}^{I^{d}}$ with defined Cartan matrix $c^{\bf q}$.
Assume that for all ${\bf q}\in A$ and $\ell\in I$ we have

$\rho_{\ell}({\bf q}):=\sigma^{{\bf q}}_{\ell}({\bf q})\in A.$

Then $(I,A,(\rho_{i})_{i\in I},(c^{\bf q})_{{\bf q}\in A})$ is a Cartan graph.

###### Proof.

Let ${\bf q}\in A$ and $\ell,j\in I$. The matrices $c^{\bf q}$ are generalized
Cartan matrices by Prop. 2.8. If Axiom (C2) is satisfied, then (C1) is
satisfied as well because $\rho_{\ell}$ is defined using a reflection which
has order two and $\sigma^{{\bf q}}_{\ell}=\sigma^{\rho_{\ell}({\bf
q})}_{\ell}$ since the relevant Cartan entries are equal by (C2).
For (C2), we need to compare $c^{\bf q}_{\ell,j}$ and $c^{\rho_{\ell}({\bf
q})}_{\ell,j}$ for $\ell\neq j$. With $m:=-c^{\bf q}_{\ell,j}$, we have
$R_{m}=0$ and $R_{k}\neq 0$ for $k<m$. Now $R_{m}=(1-\chi_{\bf
q}(v_{m}))(1-\chi_{\bf q}(w_{m}))/(1-\chi_{\bf q}(s_{m}))$. Recall that
$v_{m}$ and $w_{m}$ are eigenvectors, i.e. $(\sigma^{{\bf q}}_{\ell})^{\otimes
d}(v_{m})=v_{m}$, $(\sigma^{{\bf q}}_{\ell})^{\otimes d}(w_{m})=-w_{m}$; also,
$(\sigma^{{\bf q}}_{\ell})^{\otimes d}(s_{m})=s_{m}$ because
$(m+1)s_{m}=v_{m}$. Hence replacing ${\bf q}$ by $\sigma^{{\bf q}}_{\ell}({\bf
q})$ will map $R_{m}$ to $(1-\chi_{\bf q}(v_{m}))(1-\chi_{\bf
q}(-w_{m}))/(1-\chi_{\bf q}(s_{m}))$. But $1-\chi_{\bf q}(-w_{m})=0$ if and
only if $-w_{m}=0$ or equivalently $w_{m}=0$. Therefore, via (2.7) the new
condition produces the same Cartan entry $c^{\bf
q}_{\ell,j}=c^{\rho_{\ell}({\bf q})}_{\ell,j}$ for $\ell\neq j$. ∎

We collect our results in the following theorem.

######  Theorem 2.10.

Let $d\in\mathbb{N}$ be even, $I=\\{1,\ldots,n\\}$, ${\bf
q}_{0}\in\mathbb{K}^{I^{d}}$ a tensor, and fix square roots $\sqrt{{\bf
q}_{0}}$. Let $A$ be the smallest set with ${\bf q}_{0}\in A$ and
$\rho_{\ell}({\bf q}):=\sigma^{{\bf q}}_{\ell}({\bf q})\in A$ for all ${\bf
q}\in A$, $\ell\in I$, assuming that $c^{\bf q}$ is defined for all ${\bf
q}\in A$. Then $\mathcal{C}:=(I,A,(\rho_{i})_{i\in I},(c^{\bf q})_{{\bf q}\in
A})$ is a Cartan graph.
We call $\mathcal{W}(\mathcal{C})$ the  Weyl groupoid of ${\bf q}_{0}$.

###### Proof.

This is a special case of Prop. 2.9. ∎

###  2.6. Examples in rank two

Finite Weyl groupoids of rank two are in bijection with the set of
triangulations of convex polygons by non-intersecting diagonals:

Let ${\bf q}$ be a tensor of rank two ($n=2$, $I=\\{1,2\\}$) and consider the
sequence of Cartan entries

$c:=(-c^{\bf q}_{1,2},-c^{\sigma^{\bf q}_{1}({\bf
q})}_{2,1},-c^{\sigma^{\sigma_{1}({\bf q})}_{2}(\sigma^{\bf q}_{1}({\bf
q}))}_{1,2},\ldots).$

If the Weyl groupoid is finite, then this sequence $c$ may be interpreted as
the numbers of triangles at the vertices of a triangulation of a convex
polygon by non-intersecting diagonals. It is called a  quiddity cycle.

Writing the coordinates of the sets $R^{a}$ of a root system into a matrix
yields a so called  frieze pattern222We omit a definition since we do not need
it here., see [Cun14]. Fig. 2 shows the example of the sequence
$c=(1,4,1,2,2,2)$.

$\begin{array}[]{cccccccccccc}&&\ddots&&&&&&&&&\\\ 0&1&1&3&2&1&0&&&&&\\\
&0&1&4&3&2&1&0&&&&\\\ &&0&1&1&1&1&1&0&&&\\\ &&&0&1&2&3&4&1&0&&\\\
&&&&0&1&2&3&1&1&0&\\\ &&&&&0&1&2&1&2&1&0\\\ &&&&&&&&&\ddots&&\end{array}$

214122

Figure 2. A frieze pattern and the corresponding triangulation.

We have many examples of tensors producing interesting Weyl groupoids. Since
we do not have a classification yet, we content ourselves with two examples.

######  Example 2.11.

Let $d=4$, $\zeta$ primitive $11$-th root of unity, $\mu:=-\zeta$, and

${\bf
q}=(q_{0},\ldots,q_{4})=(\zeta^{2},\zeta^{2},\zeta^{2},\zeta^{2},\zeta^{2}),\quad\sqrt{{\bf
q}}=(\sqrt{q_{0}},\ldots,\sqrt{q_{4}})=(\mu,\mu,\mu,\mu,\mu).$

We compute the values

$\begin{array}[]{l|rrrrrr}m&0&1&2&3\\\ \hline\cr\chi_{\bf
q}(v_{m})&\mu^{4}&\mu^{4}&\mu^{2}&1\\\ \chi_{\bf
q}(w_{m})&\mu^{4}&\mu^{4}&\mu^{2}&1\\\ \chi_{\bf
q}(s_{m})&\mu^{4}&\mu^{13}&\mu^{8}&\mu^{11}\end{array}$

so the smallest $m$ with $R_{m}^{1,2}=0$ is $3$ and $c^{\bf q}_{1,2}=-3$.
Reflecting produces the new “tensor”

$\sigma_{1}^{{\bf q}}(\sqrt{{\bf
q}})=(\mu^{2},\mu^{10},\mu^{21},\mu^{13},\mu^{12}).$

To compute the next Cartan entry we need to reverse this sequence because we
now want to reflect at label $2$, the new $q_{i}$ are
$(\sqrt{q_{0}},\ldots,\sqrt{q_{4}})=(\mu^{12},\mu^{13},\mu^{21},\mu^{10},\mu^{2})$:

$\begin{array}[]{l|rrrrrr}m&0&1\\\ \hline\cr\chi_{\sigma_{1}^{{\bf
q}}}(v_{m})&\mu^{18}&\mu^{20}\\\ \chi_{\sigma_{1}^{{\bf
q}}}(w_{m})&\mu^{20}&1\\\ \chi_{\sigma_{1}^{{\bf
q}}}(s_{m})&\mu^{18}&\mu^{10}\end{array}$

The next Cartan entry is $-1$. Continuing this procedure, we obtain the
complete sequence of Cartan entries. The sequence of negative Cartan entries
is the quiddity cycle of the corresponding Weyl groupoid:

$(3,1,2,3,2,1,3).$

As a triangulation of a $7$-gon, this is:

This is not a Weyl groupoid of a Nichols algebra of diagonal type.

######  Example 2.12.

Let $d=4$, $\zeta$ primitive $7$-th root of unity, $\mu:=-\zeta$, and

${\bf
q}=(q_{0},\ldots,q_{4})=(\zeta,\zeta^{2},\zeta,\zeta^{2},\zeta^{2}),\quad(\sqrt{q_{0}},\ldots,\sqrt{q_{4}})=(\mu^{4},\mu,\mu^{4},\mu,\mu).$

Then the (negative) Cartan entries ( quiddity cycle) of the corresponding Weyl
groupoid are

$(2,1,5,1,3,1,5,1,2,3).$

The triangulation is:

Again, this is a quiddity cycle that does not appear in the classical theory
of Nichols algebras of diagonal type.

######  Conjecture 2.13.

Any finite Weyl groupoid of rank two is obtained as the Weyl groupoid of some
${\bf q}\in\mathbb{C}^{\\{1,2\\}^{d}}$, $d\in\mathbb{N}$.

It is thus interesting to find a mathematical object similar to a Nichols
algebra that would yield the recursion of Theorem 2.2 for $d>2$. This question
is the subject of the next section; the idea will be to consider higher
commutativity constraints.

##  3\. Higher tensors from abelian cohomology theory

In this section, we want to give a motivation for the construction in Section
2 in terms of higher commutativity constraints. As in the classical case
($d=2$), a higher commutativity constraint (or braiding) will be an abelian
$(2d-1)$-cocycle. In the diagonal case, these are simply determined by tensors

${\bf
q}=(q_{i_{1},\ldots,i_{d}})_{i_{1},\ldots,i_{d}}\in\mathbb{K}^{\\{1,\ldots,n\\}^{d}}$

as in Section 2.3. The numbers

$\displaystyle
q_{k}:=\prod_{\begin{subarray}{c}i_{1},\ldots,i_{d}\in\\{\ell,j\\}\\\
|\\{\nu\>\mid\>i_{\nu}=j\\}|=k\end{subarray}}q_{i_{1},\ldots,i_{d}},$

from Def. 2.1 can then be reinterpreted as invariants with respect to
$(2d-1)$st abelian cohomology. Before we introduce higher commutativity
constraints, we want to revisit the classical case in order to see where the
third abelian cohomology (which we assume to be known) comes into play.

###  3.1. Nichols algebras as graded vector spaces

Let $G$ be an abelian group and let $\mathsf{Vect}_{G}$ denote the category of
finite-dimensional $G$-graded vector spaces over a field $\mathbb{K}$.

It is well-known [EGNO15, Ex. 8.4.8] that braided monoidal structures on
$\mathsf{Vect}_{G}$ are classified (up to braided monoidal equivalence) by the
third abelian cohomology $H^{3}_{1}(G,\mathbb{K}^{\times})$. The construction
goes as follows. Let $(\omega,\theta)\in Z^{3}_{1}(G,\mathbb{K}^{\times})$ be
an abelian 3-cocycle. For $g,h,k\in G$ let $V_{g}$, $V_{h}$ and $V_{k}$ be the
corresponding simple objects in $\mathsf{Vect}_{G}$. Then we can define an
associator $a$ and a braiding $c$ on $\mathsf{Vect}_{G}$ as follows:

$\displaystyle a_{g,h,k}:(V_{g}\otimes V_{h})\otimes V_{k}$
$\displaystyle\longrightarrow V_{g}\otimes(V_{h}\otimes V_{k})$
$\displaystyle\qquad e_{g}\otimes e_{h}\otimes e_{k}$
$\displaystyle\longmapsto\omega(g,h,k)\,e_{g}\otimes e_{h}\otimes e_{k}$
$\displaystyle c_{g,h}:V_{g}\otimes V_{h}$ $\displaystyle\longrightarrow
V_{h}\otimes V_{g}$ $\displaystyle e_{g}\otimes e_{h}$
$\displaystyle\longmapsto\theta(g,h)\,e_{h}\otimes e_{g}.$

Here, the tensor product of simple objects $V_{g}$ and $V_{h}$ is simply given
by $V_{g}\otimes V_{h}:=V_{gh}$. We denote the resulting braided monoidal
category by $\mathsf{Vect}_{G}^{(\omega,\theta)}$.

If the (ordinary) 3-cocycle $\omega$ is trivial, then we can read off Table
LABEL:tab:generators_of_abelian_chain_complex ($n=4$, $k=1$) that $\theta$
must be a bicharacter. Hence, we can define a $G$-action on every simple
object $V_{g}\in\mathsf{Vect}_{G}^{(1,\theta)}$ by setting
$h.e_{g}:=\theta(g,h)\,e_{g}$. For this reason, any object in
$\mathsf{Vect}_{G}^{(1,\theta)}$ can be regarded as a Yetter-Drinfeld module
over $G$, so that we are in the setting of Section 1.1. In particular, for
$g_{1},\dots,g_{n}\in G$ the object

$\displaystyle V=\oplus_{i=1}^{n}\,V_{g_{i}}\in\mathsf{Vect}_{G}^{(1,\theta)}$

is a diagonally braided vector space with braid matrix

$\displaystyle{\bf q}=(q_{i,j})_{i,j}:=(\theta(g_{i},g_{j}))_{i,j}$

and we can define the Nichols algebra $\mathfrak{B}(V)$ of $V$. However, it
seems natural to us to think of $\mathfrak{B}(V)$ as a Hopf algebra inside the
braided monoidal category $\mathsf{Vect}_{G}^{(1,\theta)}$ because its
construction (including the operators $\partial$ and $\operatorname{ad}$)
completely takes place in $\mathsf{Vect}_{G}^{(1,\theta)}$.

###  3.2. Abelian cohomology theory

In this subsection, we recapitulate the cohomology theory of abelian groups
first introduced by Samuel Eilenberg and Saunders MacLane in [EM50a], [EM50b]
and [Mac52]. They realized that the homology theory appropriate to an abelian
group333We write $\Pi$ instead of $G$ here because this is the usual symbol
for abelian cohomology. $\Pi$ is not given by the ordinary bar complex
$A_{\bullet}(\Pi)$ of the group ring $\mathbb{Z}[\Pi]$, as the proof of the
condition $\partial\partial=0$ only uses associativity of $\Pi$ but not
commutativity. In order to resolve this issue, they defined further cell
complexes $A^{k}_{\bullet}(\Pi)$, where $k\in\mathbb{Z}_{\geq 0}$ can be
thought of as the level of commutativity which is taken into account. In
particular, we have $A_{\bullet}(\Pi)=A^{0}_{\bullet}(\Pi)$.

We will reproduce the inductive definition of the cell complexes
$A^{k}_{\bullet}(\Pi)$ from [EM50b]:

Let $A_{\bullet}(\Pi)$ be the ordinary bar complex of the abelian group $\Pi$.
More precisely, $A_{n}(\Pi)$ is the free $\mathbb{Z}$-module generated by
$n$-tuples $[x_{1},\dots,x_{n}]$ of elements $x_{i}\in\Pi$. The boundary
operators $\partial_{n}:A_{n}(\Pi)\to A_{n-1}(\Pi)$ are given by

$\displaystyle\partial_{n}[x_{1},\dots,x_{n}]:=[x_{2},\dots,x_{n}]+\sum_{i=1}^{n-1}\,(-1)^{i}[x_{1},\dots,x_{i}x_{i+1},\dots,x_{n}]+(-1)^{n}[x_{1},\dots,x_{n-1}].$

Since this is the basis of our induction, we set

$\displaystyle
A^{0}_{\bullet}(\Pi)=A_{\bullet}(\Pi),\qquad\partial^{0}_{n}=\partial_{n}\quad\text{and}\quad[x_{1}\,|_{0}\dots\dots|_{0}\,x_{n}]=[x_{1},\dots,x_{n}].$

As a free $\mathbb{Z}$-module, $A^{k}_{n}(\Pi)$ is generated by $p$-tuples
$[\alpha_{1}\,|_{k}\dots|_{k}\,\alpha_{p}]$, where the $\alpha_{i}$ are
generators of $A^{k-1}_{n_{i}}(\Pi)$, such that

$\displaystyle n=\sum_{i=1}^{p}\,n_{i}+(p-1)k.$

Hence, we can think of the symbol $|_{k}$ as a placeholder for $k$ arguments.
In particular, we have a stabilizing chain

(32) $\displaystyle A^{0}_{n}(\Pi)\leq A^{1}_{n}(\Pi)\leq\dots\leq
A^{n-2}_{n}(\Pi)=A^{n-1}_{n}(\Pi)=\dots$

of free $\mathbb{Z}$-modules.
The graded $\mathbb{Z}$-module
$A^{k}_{\bullet}(\Pi)=\bigoplus_{n=0}^{\infty}\,A^{k}_{n}(\Pi)$ is endowed
with a so-called  $k$-shuffle product

$\displaystyle\ast_{k}:A^{k}_{n}(\Pi)\times A^{k}_{m}(\Pi)\to
A^{k}_{n+m+k}(\Pi).$

For $\alpha=[\alpha_{1}\,|_{k}\dots|_{k}\,\alpha_{p}]\in A^{k}_{n}(\Pi)$ and
$\beta=[\beta_{1}\,|_{k}\dots|_{k}\,\beta_{q}]\in A^{k}_{m}(\Pi)$ a shuffle of
$\alpha$ and $\beta$ is a $(p+q)$-tuple
$\gamma=[\gamma_{1}\,|_{k}\dots|_{k}\,\gamma_{p+q}]\in A^{k}_{n+m+k}(\Pi)$
obtained by permuting the $(p+q)$-tuple

$\displaystyle[\alpha_{1}\,|_{k}\dots|_{k}\,\alpha_{p}\,|_{k}\,\beta_{1}\,|_{k}\dots|_{k}\,\beta_{q}]$

in a way that preserves the order of the $\alpha_{i}$’s and $\beta_{j}$’s,
respectively. To such a shuffle we can associate a number

$\displaystyle\epsilon_{\gamma}:=\sum_{\text{$(i,j)$ such that $\alpha_{i}$ is
after $\beta_{j}$ in $\gamma$}}\,(n_{i}+k)(n_{j}+k).$

The $k$-shuffle product on $A^{k}_{\bullet}(\Pi)$ is then defined as follows:

$\displaystyle[\alpha_{1}\,|_{k}\dots|_{k}\,\alpha_{p}]\ast_{k}[\beta_{1}\,|_{k}\dots|_{k}\,\beta_{q}]:=\sum_{\text{shuffles
$\gamma$ of $\alpha$ and
$\beta$}}\,(-1)^{\epsilon_{\gamma}}[\gamma_{1}\,|_{k}\dots|_{k}\,\gamma_{p+q}].$

Having introduced the $k$-shuffle product, we are now ready to define the
boundary operators $\partial^{k}_{n}:A^{k}_{n}(\Pi)\to A^{k}_{n-1}(\Pi)$:

(33)
$\displaystyle\begin{split}\partial^{k}_{n}[\alpha_{1}\,|_{k}\dots|_{k}\,\alpha_{p}]:&=\sum_{i=1}^{p}\,(-1)^{a_{i-1}}[\alpha_{1}\,|_{k}\dots|_{k}\,\partial^{k-1}\alpha_{i}\,|_{k}\dots|_{k}\,\alpha_{p}]\\\
&+\sum_{i=1}^{p-1}\,(-1)^{a_{i}}[\alpha_{1}\,|_{k}\dots|_{k}\,\alpha_{i-1}\,|_{k}\,\alpha_{i}\ast_{k-1}\alpha_{i+1}\,|_{k}\,\alpha_{i+2}\,|_{k}\dots|_{k}\,\alpha_{p}],\end{split}$

where $a_{i}=\sum_{j=1}^{i}\,n_{j}+i\cdot k$. For the proof that
$\partial^{k}_{n-1}\partial^{k}_{n}=0$ holds, we refer to [EM50b].

We refer to the chain complex $A^{k}_{\bullet}(\Pi)$ as the  abelian complex
of $\Pi$ of commutativity level $k$.

######  Convention 3.1.

For an abelian group $A$, we will denote the cochain complex
$\operatorname{Hom}(A^{k}_{\bullet}(\Pi),A)$ by $A_{k}^{\bullet}(\Pi,A)$ with
coboundary operators $\delta_{k}^{n-1}:=\big{(}\partial_{n}^{k}\big{)}^{*}$.
As usual, for the subgroups of (co)cycles and (co)boundaries we replace the
letter $A$ by the letters $Z$ and $B$, respectively. For the homology and
cohomology groups we use the notation $H^{k}_{\bullet}(\Pi)$ and
$H_{k}^{\bullet}(\Pi,A)$, respectively.

######  Example 3.2.

In this example, we list generators of $A^{k}_{n}(\Pi)$ and their boundaries
for $n\in\\{1,2,3,4,5,6\\}$, since we are going to use them repeatedly in the
following section. Moreover, we believe that an explicit list can be helpful
for other people dealing with abelian chain complexes independently from our
motivation.

Due to the inclusions in Sequence (32) it is sufficient to only specify the
new generators appearing for every $k\in\mathbb{Z}_{\geq 0}$.

Table 1. Generators of $A^{k}_{n}(\Pi)$ for $1\leq n\leq 6$ n | k | Generator | Boundary
---|---|---|---
1 | - | $[a]$ | $\partial[a]=0$
2 | 0 | $[a,b]$ | $\partial[a,b]=[a]-[ab]+[b]$
3 | 0 | $[a,b,c]$ | $\partial[a,b,c]=[b,c]-[ab,c]+[a,bc]-[a,b]$
| 1 | $[a|b]$ | $\partial[a|b]=[a,b]-[b,a]$
4 | 0 | $[a,b,c,d]$ | $\\!\begin{aligned} \partial[a,b,c,d]&=[b,c,d]-[ab,c,d]+[a,bc,d]\\\ &-[a,b,cd]+[a,b,c]\end{aligned}$
| 1 | $[a,b|c]$ | $\\!\begin{aligned} \partial[a,b|c]&=[b|c]-[ab|c]+[a|c]\\\ &-[a,b,c]+[a,c,b]-[c,a,b]\end{aligned}$
|  | $[a|b,c]$ | $\\!\begin{aligned} \partial[a|b,c]&=[a|c]-[a|bc]+[a|b]\\\ &+[a,b,c]-[b,a,c]+[b,c,a]\end{aligned}$
| 2 | $[a||b]$ | $\partial[a||b]=-[a|b]-[b|a]$
5 | 0 | $[a,b,c,d,e]$ | $\\!\begin{aligned} \partial[a,b,c,d,e]&=[b,c,d,e]-[ab,c,d,e]+[a,bc,d,e]\\\ &-[a,b,cd,e]+[a,b,c,de]-[a,b,c,d]\end{aligned}$
| 1 | $[a,b,c|d]$ | $\\!\begin{aligned} \partial[a,b,c|d]&=[b,c|d]-[ab,c|d]+[a,bc|d]-[a,b|d]\\\ &+[a,b,c,d]-[a,b,d,c]+[a,d,b,c]-[d,a,b,c]\end{aligned}$
|  | $[a,b|c,d]$ | $\\!\begin{aligned} \partial[a,b|c,d]&=[b|c,d]-[ab|c,d]+[a|c,d]\\\ &-[a,b|d]+[a,b|cd]-[a,b|c]\\\ &-[a,b,c,d]+[a,c,b,d]-[c,a,b,d]\\\ &-[a,c,d,b]+[c,a,d,b]\\\ &-[c,d,a,b]\end{aligned}$
|  | $[a|b,c,d]$ | $\\!\begin{aligned} \partial[a|b,c,d]&=[a|c,d]-[a|bc,d]+[a|b,cd]-[a|b,c]\\\ &+[a,b,c,d]-[b,a,c,d]+[b,c,a,d]-[b,c,d,a]\end{aligned}$
|  | $[a|b|c]$ | $\\!\begin{aligned} \partial[a|b|c]&=[a,b|c]-[b,a|c]+[a|b,c]-[a|c,b]\end{aligned}$
| 2 | $[a,b||c]$ | $\\!\begin{aligned} \partial[a,b||c]&=[b||c]-[ab||c]+[a||c]\\\ &+[a,b|c]+[c|a,b]\end{aligned}$
|  | $[a||b,c]$ | $\\!\begin{aligned} \partial[a||b,c]&=-[a||c]+[a||bc]-[a||b]\\\ &-[a|b,c]-[b,c|a]\end{aligned}$
| 3 | $[a|||b]$ | $\\!\begin{aligned} \partial[a|||b]&=[a||b]-[b||a]\end{aligned}$
6 | 0 | $[a,b,c,d,e,f]$ | $\\!\begin{aligned} \partial[a,b,c,d,e,f]&=[b,c,d,e,f]-[ab,c,d,e,f]\\\ &+[a,bc,d,e,f]-[a,b,cd,e,f]\\\ &+[a,b,c,de,f]-[a,b,c,d,ef]+[a,b,c,d,e]\end{aligned}$
| 1 | $[a,b,c,d|e]$ | $\\!\begin{aligned} \partial[a,b,c,d|e]&=[b,c,d|e]-[ab,c,d|e]+[a,bc,d|e]\\\ &-[a,b,cd|e]+[a,b,c|e]\\\ &-[a,b,c,d,e]+[a,b,c,e,d]-[a,b,e,c,d]\\\ &+[a,e,b,c,d]-[e,a,b,c,d]\end{aligned}$
|  | $[a,b,c|d,e]$ | $\\!\begin{aligned} \partial[a,b,c|d,e]&=[b,c|d,e]-[ab,c|d,e]+[a,bc|d,e]-[a,b|d,e]\\\ &+[a,b,c|e]-[a,b,c|de]+[a,b,c|d]\\\ &+[a,b,c,d,e]-[a,b,d,c,e]\\\ &+[a,d,b,c,e]-[d,a,b,c,e]\\\ &+[a,b,d,e,c]-[a,d,b,e,c]+[d,a,b,e,c]\\\ &+[a,d,e,b,c]-[d,a,e,b,c]\\\ &+[d,e,a,b,c]\end{aligned}$
|  | $[a,b|c,d,e]$ | $\\!\begin{aligned} \partial[a,b|c,d,e]&=[b|c,d,e]-[ab|c,d,e]+[a|c,d,e]\\\ &-[a,b|d,e]+[a,b|cd,e]-[a,b|c,de]+[a,b|c,d]\\\ &-[a,b,c,d,e]+[a,c,b,d,e]\\\ &-[a,c,d,b,e]+[a,c,d,e,b]\\\ &-[c,a,b,d,e]+[c,a,d,b,e]-[c,a,d,e,b]\\\ &-[c,d,a,b,e]+[c,d,a,e,b]\\\ &-[c,d,e,a,b]\end{aligned}$
|  | $[a|b,c,d,e]$ | $\\!\begin{aligned} \partial[a|b,c,d,e]&=[a|c,d,e]-[a|bc,d,e]+[a|b,cd,e]\\\ &-[a|b,c,de]+[a|b,c,d]\\\ &+[a,b,c,d,e]-[b,a,c,d,e]+[b,c,a,d,e]\\\ &-[b,c,d,a,e]+[b,c,d,e,a]\end{aligned}$
|  | $[a,b|c|d]$ | $\\!\begin{aligned} \partial[a,b|c|d]&=[b|c|d]-[ab|c|d]+[a|c|d]\\\ &-[a,b,c|d]+[a,c,b|d]-[c,a,b|d]\\\ &-[a,b|c,d]+[a,b|d,c]\end{aligned}$
|  | $[a|b,c|d]$ | $\\!\begin{aligned} \partial[a|b,c|d]&=[a|c|d]-[a|bc|d]+[a|b|d]\\\ &+[a,b,c|d]-[b,a,c|d]+[b,c,a|d]\\\ &-[a|b,c,d]+[a|b,d,c]-[a|d,b,c]\end{aligned}$
|  | $[a|b|c,d]$ | $\\!\begin{aligned} \partial[a|b|c,d]&=[a|b|d]-[a|b|cd]+[a|b|c]\\\ &+[a,b|c,d]-[b,a|c,d]\\\ &+[a|b,c,d]-[a|c,b,d]+[a|c,d,b]\end{aligned}$
| 2 | $[a,b,c||d]$ | $\\!\begin{aligned} \partial[a,b,c||d]&=[b,c||d]-[ab,c||d]+[a,bc||d]-[a,b||d]\\\ &-[a,b,c|d]-[d|a,b,c]\end{aligned}$
|  | $[a,b||c,d]$ | $\\!\begin{aligned} \partial[a,b||c,d]&=[b||c,d]-[ab||c,d]+[a||c,d]\\\ &+[a,b||d]-[a,b||cd]+[a,b||c]\\\ &+[a,b|c,d]-[c,d|a,b]\end{aligned}$
|  | $[a||b,c,d]$ | $\\!\begin{aligned} \partial[a||b,c,d]&=-[a||c,d]+[a||bc,d]-[a||b,cd]+[a||b,c]\\\ &-[a|b,c,d]-[b,c,d|a]\end{aligned}$
|  | $[a|b||c]$ | $\\!\begin{aligned} \partial[a|b||c]&=[a,b||c]-[b,a||c]\\\ &-[a|b|c]-[a|c|b]-[c|a|b]\end{aligned}$
|  | $[a||b|c]$ | $\\!\begin{aligned} \partial[a||b|c]&=-[a||b,c]+[a||c,b]\\\ &-[a|b|c]-[b|a|c]-[b|c|a]\end{aligned}$
| 3 | $[a,b|||c]$ | $\\!\begin{aligned} \partial[a,b|||c]&=[b|||c]-[ab|||c]+[a|||c]\\\ &-[a,b||c]-[c||a,b]\end{aligned}$
|  | $[a|||b,c]$ | $\\!\begin{aligned} \partial[a|||b,c]&=[a|||c]-[a|||bc]+[a|||b]\\\ &[a||b,c]+[b,c||a]\end{aligned}$
| 4 | $[a||||b]$ | $\\!\begin{aligned} \partial[a||||b]=-[a||||b]-[b||||a]\end{aligned}$

###  3.3. Invariants for abelian cohomology of commutativity level 1

In this section, we study the complex $A^{1}_{\bullet}(\Pi)$ in more detail.
In particular, we study a certain family of chains $c_{\lambda}\in
A^{1}_{2d-1}(\Pi)$ indexed by compositions $\lambda$ of $d\in\mathbb{N}$.

Let $\lambda=(\lambda_{1},\dots,\lambda_{p})$ be a $p$-composition of $d$. We
define $\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}\in A_{2d-1}^{1}(\Pi)$ to
be the sum over all generators $[\beta_{1}|\dots|\beta_{d}]\in
A_{n}^{1}(\Pi)$, where the arguments $(\beta_{1},\dots,\beta_{d})$ run over
all multiset permutations of the multiset

$\displaystyle\\{\alpha_{1},\dots,\alpha_{1},\alpha_{2},\dots,\alpha_{2},\dots,\alpha_{p},\dots,\alpha_{p}\\}$

in which every element $\alpha_{i}$ appears $\lambda_{i}$ times. In
particular, $\\{a_{1};\dots;a_{p}\\}_{\lambda}\in A_{2d-1}^{1}(\Pi)$ has
$\frac{d!}{\lambda_{1}!\dots\lambda_{p}!}$ summands.

######  Remark 3.3.

Clearly, the chains $\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}\in
A_{2d-1}^{1}(\Pi)$ depend on the order of the $p$-composition
$\lambda=(\lambda_{1},\dots,\lambda_{p})$ only up to the order of the
arguments. That is, for $\sigma\in S_{p}$, we have

$\displaystyle\\{\alpha_{\sigma(1)};\dots;\alpha_{\sigma(p)}\\}_{(\lambda_{\sigma(1)},\dots,\lambda_{\sigma(p)})}=\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}.$

However, we will not restrict ourselves to proper partitions for reasons that
will later become clear.

######  Example 3.4.

For $\lambda=(2,2)$, we have

$\displaystyle\\{\alpha_{1};\alpha_{2}\\}_{\lambda}$
$\displaystyle=[\alpha_{1}|\alpha_{1}|\alpha_{2}|\alpha_{2}]+[\alpha_{1}|\alpha_{2}|\alpha_{1}|\alpha_{2}]+[\alpha_{1}|\alpha_{2}|\alpha_{2}|\alpha_{1}]$
$\displaystyle+[\alpha_{2}|\alpha_{1}|\alpha_{1}|\alpha_{2}]+[\alpha_{2}|\alpha_{1}|\alpha_{2}|\alpha_{1}]+[\alpha_{2}|\alpha_{2}|\alpha_{1}|\alpha_{1}].$

######  Lemma 3.5.

Let $\lambda$ be a $p$-composition of $d\in\mathbb{N}$ and
$\\{\alpha_{1},\dots,\alpha_{p}\\}\subseteq\Pi$.
Then, the chain $\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}\in
A_{2d-1}^{1}(\Pi)$ is a cycle, i.e.

(34) $\displaystyle\partial\\{a_{1};\dots;a_{p}\\}_{\lambda}=0.$

###### Proof.

Applying Definition 33, we obtain the general formula:

$\displaystyle\partial[\beta_{1}|\dots|\beta_{d}]$
$\displaystyle=\sum_{i=1}^{d-1}\,(-1)^{a_{i}}[\beta_{1}|\dots|\beta_{i}\ast_{0}\beta_{i+1}|\dots|\beta_{d}]$
$\displaystyle=\sum_{i=1}^{d-1}\,\Big{(}[\beta_{1}|\dots|\beta_{i},\beta_{i+1}|\dots|\beta_{d}]-[\beta_{1}|\dots|\beta_{i+1},\beta_{i}|\dots|\beta_{d}]\Big{)}.$

Note that $a_{i}=2i$ is always even in the considered case.

Looking at this formula, we notice that the $i$th summand of
$\partial[\beta_{1}|\dots|\beta_{d}]$ vanishes iff $\beta_{i}=\beta_{i+1}$. If
$\beta_{i}\neq\beta_{i+1}$, then
$[\beta_{1}|\dots|\beta_{i+1}|\beta_{i}|\dots|\beta_{d}]$ is the unique
summand of $\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}$ such that the $i$th
summand of its boundary cancels out the $i$th summand of
$\partial[\beta_{1}|\dots|\beta_{d}]$. Since this is a one-to-one
correspondence between non-vanishing summands of
$\partial\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}$, the statement follows.
∎

In the following, the elements $\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}\in
Z_{2d-1}^{1}(\Pi)$ will be referred to as  symmetrized cycles.

We will now explore how symmetrized cycles are related to each other. We start
by proving identities for small $m\in\mathbb{N}$.

######  Lemma 3.6.

For $\alpha,\beta,\gamma\in\Pi$ we have the following identities in
$H_{3}^{1}(\Pi)$:

$\displaystyle\big{[}\\{\alpha\beta\\}_{(2)}\big{]}$
$\displaystyle=\big{[}\\{\alpha\\}_{(2)}\big{]}+\big{[}\\{\beta\\}_{(2)}\big{]}+\big{[}\\{\alpha;\beta\\}_{(1,1)}\big{]},$
$\displaystyle\big{[}\\{\alpha\beta;\gamma\\}_{(1,1)}\big{]}$
$\displaystyle=\big{[}\\{\alpha;\gamma\\}_{(1,1)}\big{]}+\big{[}\\{\beta;\gamma\\}_{(1,1)}\big{]},$
$\displaystyle\big{[}\\{\alpha\\}_{(2)}\big{]}$
$\displaystyle=\big{[}\\{\alpha^{-1}\\}_{(2)}\big{]}.$

###### Proof.

Using the definition of the boundary operator, one can easily verify that the
element

$\displaystyle-\\{\alpha\beta\\}_{(2)}+\\{\alpha\\}_{(2)}+\\{\beta\\}_{(2)}+\\{\alpha;\beta\\}_{(1,1)}$

is the boundary of the following element in $A_{4}^{1}(\Pi)$:

$\displaystyle[\alpha,\beta|\alpha\beta]+[\alpha|\beta,\alpha]+[\beta|\alpha,\beta]-[\alpha,\beta,\alpha,\beta].$

Moreover, we have

$\displaystyle\partial\big{(}[\alpha,\beta|\gamma]+[\gamma|\alpha,\beta]\big{)}=-\\{\alpha\beta;\gamma\\}_{(1,1)}+\\{\alpha;\gamma\\}_{(1,1)}+\\{\beta;\gamma\\}_{(1,1)}$

and

$\displaystyle\partial\big{(}[\alpha|\alpha^{-1},\alpha]-[\alpha^{-1},\alpha|\alpha^{-1}]+[\alpha,\alpha^{-1},\alpha,\alpha^{-1}]\big{)}=\\{\alpha\\}_{(2)}-\\{\alpha^{-1}\\}_{(2)}.$

∎

Using the symmetry from Remark 3.3, the following corollary follows
immediately from the previous lemma.

######  Corollary 3.7.

For $\alpha_{1},\alpha_{2},\alpha_{3}\in\Pi$ we have:

$\displaystyle 0=$
$\displaystyle\big{[}\\{\alpha_{1}\alpha_{2}\alpha_{3}\\}_{(2)}\big{]}$
$\displaystyle-$
$\displaystyle\big{[}\\{\alpha_{1}\alpha_{2}\\}_{(2)}\big{]}-\big{[}\\{\alpha_{1}\alpha_{3}\\}_{(2)}\big{]}-\big{[}\\{\alpha_{2}\alpha_{3}\\}_{(2)}\big{]}$
$\displaystyle+$
$\displaystyle\big{[}\\{\alpha_{1}\\}_{(2)}\big{]}+\big{[}\\{\alpha_{2}\\}_{(2)}\big{]}+\big{[}\\{\alpha_{3}\\}_{(2)}\big{]}.$

######  Lemma 3.8.

For $\alpha,\beta,\gamma,\delta\in\Pi$ we have the following identities in
$H_{5}^{1}(\Pi)$:

$\displaystyle\big{[}\\{\alpha\beta\\}_{(3)}\big{]}$
$\displaystyle=\big{[}\\{\alpha\\}_{(3)}\big{]}+\big{[}\\{\beta\\}_{(3)}\big{]}+\big{[}\\{\alpha;\beta\\}_{(2,1)}\big{]}+\big{[}\\{\beta;\alpha\\}_{(2,1)}\big{]},$
$\displaystyle\big{[}\\{\alpha\beta;\gamma\\}_{(2,1)}\big{]}$
$\displaystyle=\big{[}\\{\alpha;\gamma\\}_{(2,1)}\big{]}+\big{[}\\{\beta;\gamma\\}_{(2,1)}\big{]}+\big{[}\\{\alpha;\beta;\gamma\\}_{(1,1,1)}\big{]},$
$\displaystyle\big{[}\\{\alpha;\beta\gamma\\}_{(2,1)}\big{]}$
$\displaystyle=\big{[}\\{\alpha;\beta\\}_{(2,1)}\big{]}+\big{[}\\{\alpha;\gamma\\}_{(2,1)}\big{]},$
$\displaystyle\big{[}\\{\alpha\beta;\gamma;\delta\\}_{(1,1,1)}\big{]}$
$\displaystyle=\big{[}\\{\alpha;\gamma;\delta\\}_{(1,1,1)}\big{]}+\big{[}\\{\beta;\gamma;\delta\\}_{(1,1,1)}\big{]}$
$\displaystyle\big{[}\\{\alpha\\}_{(3)}\big{]}$
$\displaystyle=-\big{[}\\{\alpha^{-1}\\}_{(3)}\big{]}$
$\displaystyle\big{[}\\{\alpha;\beta\\}_{(2,1)}\big{]}$
$\displaystyle=\big{[}\\{\alpha^{-1};\beta\\}_{(2,1)}\big{]}.$

###### Proof.

A long and tedious calculation shows that

$\displaystyle-\\{\alpha\beta\\}_{(3)}+\\{\alpha\\}_{(3)}+\\{\beta\\}_{(3)}+\\{\alpha;\beta\\}_{(2,1)}+\\{\beta;\alpha\\}_{(2,1)}$

is the boundary of the following element in $A_{6}^{1}(\Pi)$:

$\displaystyle[\alpha,\beta|\alpha\beta|\alpha\beta]+[\beta|\alpha,\beta|\alpha\beta]+[\alpha|\beta,\alpha|\beta\alpha]$
$\displaystyle+$
$\displaystyle[\beta|\alpha|\alpha,\beta]+[\alpha|\beta|\alpha,\beta]+[\alpha|\alpha|\alpha,\beta]+[\beta|\beta|\alpha,\beta]$
$\displaystyle-$
$\displaystyle[\alpha,\beta,\alpha,\beta|\alpha\beta]-[\beta|\alpha,\beta,\alpha,\beta]-[\alpha|\beta,\alpha,\beta,\alpha]$
$\displaystyle+$ $\displaystyle[\alpha,\beta,\alpha,\beta,\alpha,\beta].$

Another calculation shows that

$\displaystyle-\\{\alpha\beta;\gamma\\}_{(2,1)}+\\{\alpha;\gamma\\}_{(2,1)}+\\{\beta;\gamma\\}_{(2,1)}+\\{\alpha;\beta;\gamma\\}_{(1,1,1)}$

is the boundary of the following element in $A_{6}^{1}(\Pi)$:

$\displaystyle[\alpha,\beta|\alpha\beta|\gamma]+[\alpha,\beta|\gamma|\alpha\beta]+[\gamma|\alpha,\beta|\alpha\beta]$
$\displaystyle+$
$\displaystyle[\gamma|\beta|\alpha,\beta]+[\beta|\gamma|\alpha,\beta]+[\gamma|\alpha|\beta,\alpha]+[\alpha|\gamma|\beta,\alpha]$
$\displaystyle+$
$\displaystyle[\alpha|\beta,\alpha|\gamma]+[\beta|\alpha,\beta|\gamma]$
$\displaystyle-$
$\displaystyle[\alpha,\beta,\alpha,\beta|\gamma]-[\gamma|\alpha,\beta,\alpha,\beta].$

Moreover, we have

$\displaystyle\partial\big{(}[\alpha|\alpha|\beta,\gamma]+[\alpha|\beta,\gamma|\alpha]+[\beta,\gamma|\alpha|\alpha]\big{)}=-\\{\alpha;\beta\gamma\\}_{(2,1)}+\\{\alpha;\beta\\}_{(2,1)}+\\{\alpha;\gamma\\}_{(2,1)}.$

The fourth identity follows from the element

$\displaystyle-\\{\alpha\beta;\gamma;\delta\\}_{(1,1,1)}+\\{\alpha;\gamma;\delta\\}_{(1,1,1)}+\\{\beta;\gamma;\delta\\}_{(1,1,1)}$

being the boundary of the following element in $A_{6}^{1}(\Pi)$:

$\displaystyle[\alpha,\beta|\gamma|\delta]+[\gamma|\alpha,\beta|\delta]+[\gamma|\delta|\alpha,\beta]$
$\displaystyle+$
$\displaystyle[\alpha,\beta|\delta|\gamma]+[\delta|\alpha,\beta|\gamma]+[\delta|\gamma|\alpha,\beta].$

Furthermore, the element

$\displaystyle\\{\alpha\\}_{(3)}+\\{\alpha^{-1}\\}_{(3)}$

is the boundary of the following element in $A_{6}^{1}(\Pi)$:

$\displaystyle[\alpha^{-1}|\alpha^{-1}|\alpha^{-1},\alpha]-[\alpha^{-1}|\alpha^{-1},\alpha|\alpha]+[\alpha^{-1},\alpha|\alpha|\alpha]$
$\displaystyle-$
$\displaystyle[\alpha^{-1}|\alpha,\alpha^{-1},\alpha,\alpha^{-1}]+[\alpha,\alpha^{-1},\alpha,\alpha^{-1}|\alpha]$
$\displaystyle+$
$\displaystyle[\alpha,\alpha^{-1},\alpha,\alpha^{-1},\alpha,\alpha^{-1}].$

Finally, the element

$\displaystyle\\{\alpha;\beta\\}_{(2,1)}-\\{\alpha^{-1};\beta\\}_{(2,1)}$

is the boundary of the following element in $A_{6}^{1}(\Pi)$:

$\displaystyle[\alpha^{-1}|\alpha^{-1},\alpha|\beta]-[\alpha^{-1},\alpha|\alpha|\beta]$
$\displaystyle+$
$\displaystyle[\alpha^{-1}|\beta|\alpha^{-1},\alpha]-[\alpha^{-1},\alpha|\beta|\alpha]$
$\displaystyle+$
$\displaystyle[\beta|\alpha^{-1}|\alpha^{-1},\alpha]-[\beta|\alpha^{-1},\alpha|\alpha]$
$\displaystyle-$
$\displaystyle[\beta|\alpha,\alpha^{-1},\alpha,\alpha^{-1}]-[\alpha,\alpha^{-1},\alpha,\alpha^{-1}|\beta].$

∎

Again, using the symmetry from Remark 3.3, the following corollary follows
immediately from the previous lemma.

######  Corollary 3.9.

For $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\in\Pi$ we have:

(35)
$\displaystyle\begin{split}0=&\big{[}\\{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\\}_{(3)}\big{]}\\\
-&\big{[}\\{\alpha_{1}\alpha_{2}\alpha_{3}\\}_{(3)}\big{]}-\big{[}\\{\alpha_{1}\alpha_{2}\alpha_{4}\\}_{(3)}\big{]}-\big{[}\\{\alpha_{1}\alpha_{3}\alpha_{4}\\}_{(3)}\big{]}-\big{[}\\{\alpha_{2}\alpha_{3}\alpha_{4}\\}_{(3)}\big{]}\\\
+&\big{[}\\{\alpha_{1}\alpha_{2}\\}_{(3)}\big{]}+\big{[}\\{\alpha_{1}\alpha_{3}\\}_{(3)}\big{]}+\big{[}\\{\alpha_{1}\alpha_{4}\\}_{(3)}\big{]}\\\
+&\big{[}\\{\alpha_{2}\alpha_{3}\\}_{(3)}\big{]}+\big{[}\\{\alpha_{2}\alpha_{4}\\}_{(3)}\big{]}+\big{[}\\{\alpha_{3}\alpha_{4}\\}_{(3)}\big{]}\\\
-&\big{[}\\{\alpha_{1}\\}_{(3)}\big{]}-\big{[}\\{\alpha_{2}\\}_{(3)}\big{]}-\big{[}\\{\alpha_{3}\\}_{(3)}\big{]}-\big{[}\\{\alpha_{4}\\}_{(3)}\big{]}\end{split}$
(36)
$\displaystyle\begin{split}0=&\big{[}\\{\alpha_{1}\alpha_{2}\alpha_{3};\alpha_{4}\\}_{(2,1)}\big{]}\\\
-&\big{[}\\{\alpha_{1}\alpha_{2};\alpha_{4}\\}_{(2,1)}\big{]}-\big{[}\\{\alpha_{1}\alpha_{3};\alpha_{4}\\}_{(2,1)}\big{]}-\big{[}\\{\alpha_{2}\alpha_{3};\alpha_{4}\\}_{(2,1)}\big{]}\\\
+&\big{[}\\{\alpha_{1};\alpha_{4}\\}_{(2,1)}\big{]}+\big{[}\\{\alpha_{2};\alpha_{4}\\}_{(2,1)}\big{]}+\big{[}\\{\alpha_{3};\alpha_{4}\\}_{(2,1)}\big{]}.\end{split}$

Both Corollary 3.7 and Corollary 3.9 give evidence for the following
conjecture:

######  Conjecture 3.10.

Let $\lambda=(\lambda_{1},\dots,\lambda_{p})$ be a $p$-composition of
$d\in\mathbb{N}$ and

$\\{\alpha_{1},\dots,\alpha_{i-1},\beta_{1},\dots,\beta_{\lambda_{i}+1},\alpha_{i+1},\dots,\alpha_{p}\\}\subseteq\Pi.$

Then we have the following identity in $H_{2d-1}^{1}(\Pi)$:

$\displaystyle 0=$
$\displaystyle\big{[}\\{\alpha_{1};\dots;\alpha_{i-1};\beta_{1}\dots\beta_{\lambda_{i}+1};\alpha_{i+1};\dots;\alpha_{p}\\}_{\lambda}\big{]}$
$\displaystyle-$ $\displaystyle\sum_{1\leq
j\leq\lambda_{i}+1}\,\big{[}\\{\alpha_{1};\dots;\alpha_{i-1};\beta_{1}\dots\hat{\beta}_{j}\dots\beta_{\lambda_{i}+1};\alpha_{i+1};\dots;\alpha_{p}\\}_{\lambda}\big{]}$
$\displaystyle+$ $\displaystyle\sum_{1\leq
k<l\leq\lambda_{i}+1}\,\big{[}\\{\alpha_{1};\dots;\alpha_{i-1};\beta_{1}\dots\hat{\beta}_{k}\dots\hat{\beta}_{l}\dots\beta_{\lambda_{i}+1};\alpha_{i+1};\dots;\alpha_{p}\\}_{\lambda}\big{]}$
$\displaystyle\vdots$ $\displaystyle+$
$\displaystyle(-1)^{\lambda_{i}}\sum_{1\leq
m\leq\lambda_{i}+1}\,\big{[}\\{\alpha_{1};\dots;\alpha_{i-1};\hat{\beta}_{1}\dots\beta_{m}\dots\hat{\beta}_{\lambda_{i}+1};\alpha_{i+1};\dots;\alpha_{p}\\}_{\lambda}\big{]}.$

Moreover, we have

$\displaystyle\big{[}\\{\alpha_{1};\dots;\alpha_{i-1};\beta_{1};\alpha_{i+1};\dots;\alpha_{p}\\}_{\lambda}\big{]}=(-1)^{\lambda_{i}}\big{[}\\{\alpha_{1};\dots;\alpha_{i-1};\beta_{1}^{-1};\alpha_{i+1};\dots;\alpha_{p}\\}_{\lambda}\big{]}.$

######  Remark 3.11.

Let $A$ be an abelian group and let $\lambda=(\lambda_{1},\dots,\lambda_{p})$
be a $p$-composition of $m\in\mathbb{N}$. Moreover, let $\theta\in
A^{2d-1}_{1}(\Pi,A)$ be a cochain. We can define a map
$\theta_{\lambda}:\Pi^{\times p}\to A$ by setting

$\displaystyle\theta_{\lambda}(\alpha_{1};\dots;\alpha_{p}):=\theta\big{(}\\{\alpha_{1};\dots;\alpha_{p}\\}_{\lambda}\big{)}.$

Now, we can reinterpret Conjecture 3.10 as follows: If $\theta$ is a cocycle,
then $\theta_{\lambda}$ is a $\lambda_{i}$-form in the $i$th argument.
Moreover, from Equation (34) we know that in this case $\theta_{\lambda}$ only
depends on the cohomology class of $\theta\in Z^{2d-1}_{1}(\Pi,A)$.

######  Example 3.12.

Let $\mathbb{K}$ be a field and $d,n\in\mathbb{N}$. We set
$\Pi=\mathbb{Z}^{n}$ and $A=\mathbb{K}^{\times}$. Let
$(\alpha_{1},\dots\alpha_{n})$ be the standard basis of $\mathbb{Z}^{n}$. We
are now in the setting of Definition 2.1.

Let $\theta\in Z^{2d-1}_{1}(\mathbb{Z}^{n},\mathbb{K}^{\times})$ be a
$(2d-1)$-cocycle. For $i_{1},\dots,i_{d}\in\\{1,\dots,n\\}$ we define

$\displaystyle
q_{i_{1},\dots,i_{d}}:=\theta([\alpha_{i_{1}}|\dots|\alpha_{i_{d}}])\in\mathbb{K}^{\times}.$

Then for $\ell,j\in\\{1,\dots,n\\}$ and $k\in\\{0,\dots,d\\}$ we have

$\displaystyle q_{k}=\theta_{(d-k,k)}(\alpha_{\ell},\alpha_{j}).$

In particular, by the previous remark the Rosso condition $\tilde{R}_{m}=0$
only depends on the
cohomology class $[\theta]\in
H^{2d-1}_{1}(\mathbb{Z}^{n},\mathbb{K}^{\times})$.

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