# Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic
Hochschild cohomology

Natalia Iyudu, Maxim Kontsevich

###### Abstract

We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ over free associative algebra or
path algebra of a quiver. The ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$
complex is introduced as an appropriate tool for the definition of pre-Calabi-
Yau structure. We show that cohomologies of this complex are pure in case of
free algebras (path algebras), concentrated in degree zero. It serves as a
main ingredient for the formality proof. For any smooth algebra we choose a
small qiso subcomplex in the higher cyclic Hochschild complex, which gives
rise to a calculus of highly noncommutative monomials, we call them
$\xi\delta$-monomials. The Lie structure on this subcomplex is combinatorially
described in terms of $\xi\delta$-monomials. This subcomplex and a basis of
$\xi\delta$-monomials in combination with arguments from Gröbner bases theory
serves for the cohomology calculations of the higher cyclic Hochschild
complex. The language of $\xi\delta$-monomials in particular allows an
interpretation of pre-Calabi-Yau structure as a noncommutative Poisson
structure.

MSC: 16A22, 16S37, 16Y99, 16G99, 16W10, 17B63

Keywords: A-infinity structure, pre-Calabi-Yau structure, inner product,
cyclic invariance, graded pre-Lie algebra, Maurer-Cartan equation, Poisson
structure, double Poisson bracket, Hochschild (co)homology, L-infinity
structure, formality.

##  1 Introduction

The calculations we perform on the higher cyclic Hoshschild complex which lead
to the formality proof as well as the construction of this complex itself are
inspired by the notion of pre-Calabi-Yau algebra, which was introduced by
Kontsevich, Vlassopoulos [26] (see also the talk [25]), Seidel [31], and
Tradler, Zeinalian [36]. It turned out that this structure is present in many
different areas, including topology of compact manifolds with boundary,
algebraic geometry, symplectic geometry. For example, Fano varieties are
endowed with a pre-Calabi-Yau structure, open Calabi-Yau manifolds have this
structure, from the HMS conjecture it is expected that the Fukaya wrapped
category of an open symplectic manifold endowed with a pre-Calabi-Yau
structure. Pre-Calabi-Yau structure produces a generalization (for arbitrary
genus) of the Tamarkin-Tsygan calculus. Big class of examples comes from the
notion of algebra (category) of finite type, introduced by Toën and Vaquié
[35]. For these algebras $d$-pre-Calabi-Yau structure on $A$ produces a
$(2-d)$-shifted derived Poisson structure on the moduli stack of finite-
dimensional $A$-modules.

The philosophy presented in the Kontsevich and Soibelman paper [21] says that
it is natural to construct a formal noncommutative geometry where the role of
manifold played by $A_{\infty}$-algebra, and the role of Poisson structure, by
pre-Calabi-Yau algebra. There were numerous attempts in the past to introduce
a notion of noncommutative Poisson structure. First natural thought in this
direction is to try and define it exactly like in commutative case, as a Lie
bracket $\\{.,.\\}:A\times A\to A$ which satisfies the Leibnitz rule
$\\{a,bc\\}=\\{a,b\\}c+b\\{a,c\\}$. However this definition would give a very
restricted class of objects. As it was shown by Farkas and Letzter [12] the
only such Poisson bracket on noncommutative prime ring is the bracket
$[a,b]=ab-ba.$ So this definition turned out to be not what one would hope
for. Another notion of noncommutative Poisson bracket was suggested by Xu [40]
and by Block and Getzler [2]. It had the property that if $A$ endowed with
this noncommutative Poisson structure, then on the centre of $A$, ${\cal
Z}(A)$ there is an induced commutative Poisson structure, but the definition
did not ensure that there will be an induced Poisson structure on
representation spaces of $A$ or on their moduli. It was desirable to find a
notion of noncommutative Poisson structure, which would behave well on the
testing ground of representation spaces. This was achieved in the definition
of double Poisson bracket by Van den Bergh [39]. The bracket was ’thickened’,
that is it was defined as a map $\\{\\!\\{.,.\\}\\!\\}:A\otimes A\to A\otimes
A$ , satisfying axioms which are certain generalisation of the Leibnitz and
the Jacobi identities on $A\otimes A$. In the work of Crawley-Boevey [10] the
definition of noncommutative ($H_{0}$) Poisson structure was given as a Lie
bracket on zero Hochschild homology $H_{0}(A)=A/[A,A]$, which can be lifted to
a derivation on $A$. We put these latter developments into the context of pre-
Calabi-Yau structures (or rather associated cohomology) and embed
noncommutaive bivector fields into a calculus of ’highly noncommutative’
words. This gives a new perspective, kind of panoramic aerial view on what is
going on, how noncommutative notions involving ’thickening’ or factorizations
arise. The attempts to reach a good behaviour on representation spaces was
based on the philosophy introduced by Kontsevich and Rosenberg [24, 27], we
follow the ideas of this paper throughout the text and pursue some aspects of
those, as well as of their reincarnations, for example, in Kontsevich and
Soibelman [21].

We give several equivalent definitions of pre-CY structures, one of them in
terms of higher cyclic Hochshild complex. Amongst the advantages of this
definition is that it works not only for finite dimensional algebras, or
algebras with finite dimensional graded components. We show in section 4 the
reason why this definition contains as a particular case the double Poisson
bracket. Roughly speaking, a pre-Calabi-Yau structure is a solution of the the
Maurer-Cartan equation with respect to generalized necklace bracket in the
higher cyclic Hochschild complex (for precise definition see section 3).

We study the higher cyclic Hochschild complex
(${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$), its homologies and Lie structure.
One of the tools we use for the combinatorial description of Lie structure,
and for the proof of purity of this complex is a calculus of noncommutative
cyclic words with labels, which we introduce. We start with the free
associative algebra $A=\mathbb{K}\langle X\rangle$ with finite number of
generators $X=\\{x_{1},...,x_{r}\\}$ and ’labels’ $\delta_{1},...,\delta_{r}$,
$\xi$. Monomials from free algebra $\mathbb{K}\langle X\rangle$ are written
cyclically on the circle and separated by labels. These generalised cyclic
monomials with labels, which we call $\xi\delta$-monomials, represent
operations on tuples of monomials from $A$ and form a convenient basis in the
small subcomplex of the higher cyclic Hochschild complex. The
$\xi\delta$-monomials can be considered as highly noncommutative words, which
can be multiplied not only from the right or from the left, but from $r$
sides, where $r$ is a number of $\delta$th in the monomial. The result on
formality of the higher cyclic Hochschild complex we got provides $L_{\infty}$
quasi-isomorphism between higher cyclic Hochschild complex and its cohomology.
Thus the consideration of homology of the complex instead of the complex
itself for the notion of noncommutative Poisson structures and of
noncommutative polyvector fields becomes justified.

Whenever we are working with the higher cyclic Hochschild complex itself,
without embracing in further word combinatorics, we can speak of an arbitrary
smooth algebra $A$, as defined in [27], i.e. a finitely generated algebra $A$,
with kernel of multiplication being a projective $A$-bimodule.

To deal with the higher cyclic Hochschild complex
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}(A)=C^{(\bullet)}(A)$ we choose a
small subcomplex $\xi^{(\bullet)}$, quasi-isomorphic to $C^{(\bullet)}(A).$ We
specify a particular embedding of the subcomplex $\zeta^{(\bullet)}$ into
$C^{(\bullet)}(A)$ (section 5) by choosing a basis of $\xi\delta$-monomials in
$\xi^{(\bullet)}$ and describing an element of $C^{(\bullet)}$ (which we call
an ’operation’) corresponding to a $\xi\delta$-monomial. The element of
$\prod\limits_{r_{1},...,r_{n}\geqslant 0}{\rm
Hom}\,(\mathop{\otimes}\limits_{i=1}^{n}A^{\otimes r_{i}},A^{\otimes n})$,
corresponding to $\xi\delta$-monomial with $m=\sum r_{i}$ occurrences of
labels $\delta_{i},i=\bar{1,r}$, and $n-m$ occurrences of label $\xi$ is
schematically shown in the following picture.

0

0

The black arches are input monomials from $A$ and green arches are output
monomials, consisting of parts of inputs and parts of $\xi\delta$-monomial
defining the operation. We suppose orientation is clockwise everywhere. One
can read the output monomials following this orientation. In this particular
picture we see the $\xi\delta$-monomial which encodes an operation
$\Phi:A^{\otimes 3}\to A^{\otimes 5}$ (more precisely, $\Phi:A\otimes
A^{0}\otimes A\otimes A^{0}\otimes A\to A^{\otimes 5})$.

The particular case of operation encoded by $\xi\delta$-monomial with two
$\delta$th and no $\xi$th correspond to the Poisson double bracket. Indeed,
the operation $A\otimes A\to A\otimes A$, obtained from such a
$\xi\delta$-monomial automatically satisfies the double Leibnitz rule. The
double Jacobi identity comes from the Maurer-Cartan equation on the elements
of small subcomplex $\zeta^{(\bullet)}$ of the higher Hochschild complex. This
will be explained more precisely in section 4. We show there how the double
Poisson bracket invented by Van den Bergh [39] as a structure which induce a
Poisson bracket on representation space of algebra, appear as a particular
pre-Calabi-Yau structure. In [17] we gave a detailed proof of the following
fact. Any pre-Calabi-Yau structure with $m_{4}=0$ on arbitrary associative
algebra gives rise to a double Poisson bracket according to the formula [17]:

$(*)\quad\quad\quad\langle g\otimes f,\\{\\!\\{b,a\\}\\!\\}\rangle:=\langle
m_{3}(a,f,b),g\rangle,$

Moreover, an arbitrary double Poisson bracket can be obtained from pre-Calabi-
Yau structure of special type, with only second and third multiplications
$m_{2}$ and $m_{3}$ present. We comment here on the main idea behind this
earlier direct proof from the point of view of the definition of pre-Calabi-
Yau structure via higher cyclic Hochschild complex.

Coming back to the general situation, we describe the generalised necklace
bracket which endows the higher cyclic Hochschild complex with a graded Lie
algebra structure. In section 6 we show how this bracket works in terms of
$\xi\delta$-monomials. By this we not only prove that the small subcomplex
$\zeta^{(\bullet)}_{A}$ is a Lie subalgebra in ${\bf
g}=(C^{(\bullet)}_{A}(A),[,]_{g.n})$, but also give a concrete combinatorial
formula for this bracket on $\xi\delta$-monomials. We prove that the bracket
$[A,B]$ of two $\xi\delta$-monomials $A,B\in\zeta^{(\bullet)}_{A}$ is
$[A,B]=A\circ B-B\circ A$, where $A\circ B$ is a linear combination of
$\xi\delta$-monomials obtained from $\xi\delta$-monomials $A$ and $B$ by all
possible gluings of $\delta_{j}$ of $A$ and $x_{j}$ of $B$, as shown in the
following picture.

0

0

Namely, we glue all $\delta_{j}$ from $\xi\delta$-monomial $A$ to a
corresponding $x_{j}$ from $B$, then cut at the place of gluing, and open up
to obtain one new $\xi\delta$-monomial ($x_{j}$ and $\delta_{j}$ disappear).

The choice of the basis of $\xi\delta$-monomials in the subcomplex
$\xi^{(\bullet)}$ of the higher cyclic Hochschild complex allows, among other
things, an easy interpretation of pre-Calabi-Yau structure as a noncommutative
Poisson structure. Namely, the $\xi\delta$-monomial produce an obvious formal
analogue of polyvector field, which in turn create a Poisson structure on the
representation space of $A$, via kind of Schouten bracket.

Let us note also the following consequence of our result. In view of quite
obvious connection between antisymmetric solutions of Yang-Baxter equation and
double Poisson brackets, which was noticed first in [32], this our description
of all double Poisson brackets in terms of $\xi\delta$-words, provides at the
same time a description of all antisymmetric solutions of the Yang-Baxter
equation on the vector space $X$.

In section 7 we concentrate on homological properties of the higher cyclic
Hochschild complex and prove its homological purity. We again use the small
quasi-isomorphic subcomplex $\zeta^{(\bullet)}_{A}$ introduced in section 5.
From the expression of the differential in the whole dualised bar complex
$C^{(\bullet)}_{A}(A)$, which we spell out in section 5, we get a differential
in $\zeta=\zeta^{(\bullet)}_{\mathbb{K}}$.

Note that while the elements of higher cyclic Hochschild complex are defined
as elements of ${\rm Hom}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes
r_{i}},A^{\otimes N})^{{\mathbb{Z}}_{N}}$, invariant under
${\mathbb{Z}}_{N}$-acton, our homology calculations are reduced to a related
non-${\mathbb{Z}}_{N}$-invariant complex $\tilde{\zeta}$, corresponding to
operations with fixed point. This is possible since the differential commutes
with the cyclisation procedure (see Lemma 7.21).

The complex with the fixed point $\tilde{\zeta}=\oplus\tilde{\zeta}^{n}_{N}$,
where $\tilde{\zeta}^{n}=\\{$ monomials
$u\in\mathbb{K}\langle\xi,x_{i},\delta_{i}\rangle$, starting from $\xi$ or
$\delta_{i}$, such that $\hbox{\tt\rm deg}\,_{\delta}u=n,\hbox{\tt\rm
deg}\,_{\delta,\xi}u=N\\}_{\mathbb{K}}$ has natural bigrading by
$\delta$-degree, and by degree with respect to $\xi$ and $\delta_{i}$th,
$i=\bar{1,r}$, the latter we call weight. Essential for our considerations is
the cohomological grading by $\xi$-degree: $\zeta=\oplus\zeta(l),$ where
$\zeta(l)=\mathop{\oplus}\limits_{N-n=l}\zeta^{n}_{N}$.

######  Theorem 1.1.

Let $A$ be a free algebra with at least two generators, $A=\mathbb{K}\langle
x_{1},...,x_{r}\rangle,r\geqslant 2$, or a path algebra of a quiver with at
least two vertices, $A=P{\mathcal{Q}},|{\mathcal{Q}}_{0}|\geqslant 2$. Then
the homology of the complex $\tilde{\zeta}(A)=\oplus\tilde{\zeta}^{n}_{N}$ is
sitting in the diagonal $n=N$. Consequently, the complex
$\tilde{\zeta}=\oplus\tilde{\zeta}(l)$,
$\,\tilde{\zeta}(l)=\mathop{\oplus}\limits_{N-n=l}\tilde{\zeta}_{N}^{n}$ is
pure, that is its homology is sitting only in the last place of the complex
$\tilde{\zeta}$ with respect to cohomological grading by $\xi$-degree.
Homological purity hence holds for the higher cyclic Hochschild complex
$C^{(\bullet)}$.

In other words, Theorem1.1 holds for path algebra $P{\mathcal{Q}}$ of an
arbitrary quiver ${\mathcal{Q}}$, except for the quiver with one vertex and
one loop.

This purity result is obtained via use of $\xi\delta-monomials$ as a basis of
qiso subcomplex and of the Gröbner bases theory in the ideals of free algebra
generated by the element defined by the differential of this complex. Similar
techniques and the Gröbner bases theory in the ideals of path algebras are
used for the case when $A$ is the path algebra of a quiver $A=P{\mathcal{Q}}.$

As a consequence of purity result we are able to deduce $L_{\infty}$-formality
for this complex over $A=\mathbb{K}\langle x_{1},...,x_{r}\rangle,r\geqslant
2$ or $A=P{\mathcal{Q}},|{\mathcal{Q}}_{0}|\geqslant 2$, using standard
arguments related to pertrubation theory, similar to the ones appeared in
[11].

######  Definition 1.2.

The DGLA $(C,d)$ is called formal if it is quasi-isomorphic to its
cohomologies $H^{\bullet}C.$

######  Definition 1.3.

The complex $(C,d)$ is $L_{\infty}-formal$ if it is $L_{\infty}$ quasi-
isomorphic to its cohomologies $(H^{\bullet}C,0)$, considered with zero
differential, that is there exists an $L_{\infty}$-morphism, which is a quasi-
isomorphism of complexes.

Since there are more $L_{\infty}$-morphisms between given DGLAs, than just
DGLA morphisms, the notion of $L_{\infty}$-formality is weaker than formality.
But it is exactly what is needed for deformation theory. One of the main
points in [23] emphasise that what really determines the deformation functor,
is not just qiso class of a DGLA, but its qiso class as $L_{\infty}$-algebra.
Thus the best thing one can achieve in understanding the deformation theory is
to prove $L_{\infty}$-formality of corresponding DGLA.

######  Theorem 1.4.

Let $A$ be a free algebra with at least two generators, $A=\mathbb{K}\langle
x_{1},...,x_{r}\rangle,r\geqslant 2$, or a path algebra of a quiver with at
least two vertices, $A=P{\mathcal{Q}},|{\mathcal{Q}}_{0}|\geqslant 2$. Then
the higher cyclic Hochschild complex
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}(A)=C^{(\bullet)}(A)=\prod\limits_{N}C_{cycl}^{(N)}(A)$
is $L_{\infty}$-formal.

Thus the line of our study related to the deformation theory got its best
possible outcome in case of free algebras. The $L_{\infty}$-formality holds
also for such smooth (quasi-free) algebras, as path algebras of quivers, but
certainly not for an arbitrary associative algebras concentrated in degree
zero, or for ${\mathbb{Z}}$-graded free algebras.

##  2 Definitions

The typical example of an algebra in this paper is a free associative algebra
$A=\langle x_{1},...,x_{r}\rangle$, the most noncommutative algebra possible.
We develop elements of noncommutative geometry based on this algebra following
the spirit of [27, 24]. For example, we adopt the ideology introduced and
developed in these papers, which says that noncommutative structure should
manifest as a corresponding commutative structure on representation spaces. We
develop the ideas of these papers further and introduce a calculus of ’highly
noncommutative monomials’: monomials which can be multiplied not only from the
right and from the left, but from any number of specified directions, we call
them $\xi\delta$-monomials.

Throughout the text $A$ will be an associative unital algebra over the field
$\mathbb{K}$ of characteristic zero, if not specified otherwise. Denote by
$A-mod-A$ the category of all $A$-bimodules, which is the same as
$A^{e}$-modules, i.e. modules over the enveloping $A^{e}=A\otimes A^{op}$. We
consider mainly Homs of $A$-bimodules or $A^{\otimes N}$-bimodules which we
denote ${\rm Hom}\,_{A-mod-A}$ or ${\rm Hom}\,_{A^{\otimes N}-mod-A^{\otimes
N}}$ respectively.

To give a definition of pre-Calabi-Yau structure as it was originally defined
in [25], [26], [31] we start with reminding the definition of
$A_{\infty}$-algebra, or strong homotopy associative algebra introduced by
Stasheff [34].

First note, that there are two accepted conventions on grading of an
$A_{\infty}$-algebra. They differ by a shift in numeration of graded
components. In one convention, we call it shifted convention, each operation
has degree 1. While the other, which we call a naive convention is determined
by making the binary operation to have degree 0, hence the degrees of
operations $m_{n}$ of arity $n$ become $2-n$. If the degree of element $x$ in
naive convention is ${\rm deg}x=|x|$, then shifted degree in $A^{sh}=A[1]$,
which fall into shifted convention, will be ${\rm deg^{sh}}x=|x|^{\prime}$,
where $|x|^{\prime}=|x|-1$, since $x\in A^{i}=A[1]^{i+1}$.

The formulae for the graded Lie bracket, Maurer-Cartan equations and cyclic
invariance of the inner form are somewhat different in different conventions.
We mainly use the shifted convention, but sometimes need the naive convention
as well.

Let $A$ be a ${\mathbb{Z}}$ graded vector space
$A=\mathop{\oplus}\limits_{n\in{\mathbb{Z}}}A_{n}$, and $C^{l}(A,A)$ be
Hochschild cochains $C^{(}A,A)=\underline{{\rm Hom}\,}(A^{\otimes l},A)$, for
$l\geqslant 0$, $C^{\bullet}(A,A)=\prod\limits_{k\geqslant 1}C^{l}(A,A).$

On $C^{\bullet}(A,A)[1]$ there is a natural structure of graded pre-Lie
algebra, defined via composition:

$\circ:C^{l_{1}}(A,A)\otimes C^{l_{2}}(A,A)\to C^{l_{1}+l_{2}-1}(A,A):$
$f\circ g(a_{1}\otimes...\otimes a_{l_{1}+l_{2}-1})=$
$\sum(-1)^{|g|\sum\limits_{j=1}^{i-1}|a_{j}|}f(a_{1}\otimes...\otimes
a_{i-1}\otimes g(a_{i}\otimes...\otimes a_{i+l_{2}+1})\otimes...\otimes
a_{l_{1}+l_{2}-1})$

The operation $\circ$ defined in this way does satisfy the graded right-
symmetric identity:

$(f,g,h)=(-1)^{|g||h|}(f,h,g)$

where

$(f,g,h)=(f\circ g)\circ h-f\circ(g\circ h).$

As it was shown in [14] the graded commutator on a graded pre-Lie algebra
defines a graded Lie algebra structure.

Thus the Gerstenhaber bracket $[-,-]_{G}$:

$[f,g]_{G}=f\circ g-(-1)^{|f||g|}g\circ f$

makes $C^{\bullet}(A)$ into a graded Lie algebra. Equipped with the derivation
$d={\rm ad}\,\,m_{2}$, $\,\,\,(C^{\bullet}(A),m_{2})$ becomes a DGLA, which is
a Hochschild cochain complex.

Graphically the corresponding composition can be depicted as follows.

0 0

With respect to the Gerstenhaber bracket $[-,-]_{G}$ we have the Maurer-Cartan
equation

$[m^{(1)},m^{(1)}]_{G}=\sum\limits_{p+q=k+1}\sum\limits_{i=1}^{p-1}(-1)^{\varepsilon}m_{p}(x_{1},\dots,x_{i-1},m_{q}(x_{j},\dots,x_{i+q-1}),\dots,x_{k})=0,$
(2.1)

where

$\varepsilon=|x_{1}|^{\prime}+{\dots}+|x_{i-1}|^{\prime},\qquad|x_{i}|^{\prime}=|x_{i}|-1={\rm
deg}x_{i}-1$

The Maurer-Cartan equation in the naive convention looks like:

$[m^{(1)},m^{(1)}]=\sum\limits_{p+q=k+1}\sum\limits_{i=1}^{p-1}(-1)^{\varepsilon}m_{p}(x_{1},\dots,x_{i-1},m_{q}(x_{j},\dots,x_{i+q-1}),\dots,x_{k})=0,$
(2.2)

where

$\varepsilon=i(q+1)+q(|x_{1}|+{\dots}+|x_{i-1}|),$

######  Definition 2.1.

An element $m^{(1)}\in C^{\bullet}(A,A)[1]$ which satisfies the Maurer-Cartan
equation $[m^{(1)},m^{(1)}]_{G}=0$ with respect to the Gerstenhaber bracket
$[-,-]_{G}$ is called an $A_{\infty}$-structure on $A$.

Equivalently, it can be formulated in a more compact way as a coderivation on
the coalgebra of the bar complex of $A$.

In particular, for example, associative algebra with zero derivation
$(A,m=m_{2}^{(1)})$ is an $A_{\infty}$-algebra. The component of the Maurer-
Cartan equation of arity 3 says that the binary operation from this structure,
the multiplication $m_{2}$ is associative:

$(ab)c-a(bc)=dm_{3}(a,b,c)+(-1)^{\sigma}m_{3}(da,b,c)+(-1)^{\sigma}m_{3}(a,db,c)+(-1)^{\sigma}m_{3}(a,b,dc)$

We can give now one of definitions of pre-Calabi-Yau structure (in the shifted
convention).

######  Definition 2.2.

A d-pre-Calabi-Yau structure on a finite dimensional $A_{\infty}$-algebra $A$
is

(I). an $A_{\infty}$-structure on $A\oplus A^{*}[1-d]$,

(II). cyclic invariant with respect to natural non-degenerate pairing on
$A\oplus A^{*}[1-d]$, meaning:

$\langle
m_{n}(\alpha_{1},...,\alpha_{n}),\alpha_{n+1}\rangle=(-1)^{|\alpha_{1}|^{\prime}(|\alpha_{2}|^{\prime}+...+|\alpha_{n+1}|^{\prime})}\langle
m_{n}(\alpha_{2},...\alpha_{n+1}),\alpha_{1})\rangle$

where the inner form $\langle,\rangle$ on $A\oplus A^{*}$ is defined naturally
as $\langle(a,f),(b,g)\rangle=f(b)+(-1)^{|g|^{\prime}|a|^{\prime}}g(a)$ for
$a,b\in A,f,g\in A^{*}$

(III) and such that $A$ is an $A_{\infty}$-subalgebra in $A\oplus A^{*}[1-d]$.

The signs in this definition, written in shifted convention, are assigned
according to the Koszul rule. Note, by the way, that in the naive convention,
the cyclic invariance condition with respect to the natural non-degenerate
pairing on $A\oplus A^{*}[1-d]$ from (II) sounds:

$\langle
m_{n}(\alpha_{1},...,\alpha_{n}),\alpha_{n+1}\rangle=(-1)^{n+|\alpha_{1}|^{\prime}(|\alpha_{2}|^{\prime}+...+|\alpha_{n+1}|^{\prime})}\langle
m_{n}(\alpha_{2},...\alpha_{n+1}),\alpha_{1}\rangle.$

The appearance of the arity $n$, which influence the sign in this formula,
does not really follow the Koszul rule, this is the feature of the naive
convention, and this is why the shifted one is preferable.

The cyclic invariance and inner form symmetricity in the shifted convention
look like:

$\langle
m_{n}(\alpha_{1},...,\alpha_{n}),\alpha_{n+1}\rangle=(-1)^{|\alpha_{1}|^{\prime}(|\alpha_{2}|^{\prime}+...+|\alpha_{n+1}|^{\prime})}\langle
m_{n}(\alpha_{2},...\alpha_{n+1}),\alpha_{1})\rangle.$ (2.3)

$\langle x,y\rangle=-(-1)^{|x|^{\prime}\,|y|^{\prime}}\langle y,x\rangle$
(2.4)

Since the bilinear form on $A\oplus A^{*}$, which gives natural pairing, has
degree zero (in non-shifted convention), the $A^{*}$ in the above definition
is shifted by $1-d$ in order the corresponding cyclic Calabi-Yau structure on
$A\oplus A^{*}$ is of degree $d$.

The most simple example of pre-Calabi-Yau structure demonstrates that this
structure does exist on any associative algebra. Namely, the structure of
associative algebra on $A$ can be extended to the associative structure on
$A\oplus A^{*}[1-d]$ in such a way, that the natural inner form is
(graded)cyclic with respect to this multiplication. This amounts to the
following fact: for any $A$-bimodule $M$ the associative multiplication on
$A\oplus M$ is given by $(a+f)(b+g)=ab+af+gb.$ In this simplest situation both
structures on $A$ and on $A\oplus A^{*}$ are in fact associative algebras.
More examples one can find, for example, in [16], [8].

Note that the notion of pre-Calabi-Yau algebra introduced in [26], [31], [36],
as an $A_{\infty}$-atructure on $A\oplus A^{*}$, uses the fact that $A$ is
finite dimensional, since there is no natural grading on the dual algebra
$A^{*}={\rm Hom}(A,\mathbb{K})$, induced form the grading on $A$ in infinite
dimensional case. One can reformulate it to give the general definition via
the higher cyclic Hochschild complex (see [26], [25]), not requiring any
finiteness conditions.

This reformulation is based on the fact that due to the cyclic invariance of
the natural (evaluation) pairing on $A\oplus A^{*}$, any tensor of the type
$C_{1}\otimes...\otimes C_{k}\otimes A$, where $C_{i}=A$ or $A^{*}$, can be
considered as element of ${\rm
Hom}\,_{\mathbb{K}}(C_{1}^{*}\otimes...\otimes{\mathbb{C}}_{k}^{*},A)$.
Generally, in linear map from one tensor $C_{1}\otimes...\otimes C_{l}$ to
another $C_{l+1}\otimes...\otimes C_{k+1}$ input from $A$ can be made into
output from $A^{*}$ and vice versa. The obtained in this way new definition
will be given in the next section. It is equivalent to the definition above,
where the ${\rm Hom}(A,\mathbb{K})$ considered as graded Hom:
$A^{*}=\oplus(A_{n})^{*}=\underline{{\rm Hom}}(A,\mathbb{K})$, in case the
graded components of $A$ are finite dimensional.

We also should remark here that the theorem saying that pre-Calabi-Yau
algebras give rise to TQFTs, analogous to the one proved in [36], for the
definition 2.2, holds also for the definition via the higher cyclic Hochschild
complex.

##  3 Higher cyclic Hochschild complex

We start with the definition of the higher cyclic Hochschild cochains and
generalised necklace bracket.

First let us consider higher Hochschild cochain complex
${\mathcal{H}}{\mathcal{H}}$:

$C^{(N),n}(A):=\bigoplus\limits_{{r_{1},...,r_{N}\geqslant
0}\atop{r_{1}+...+r_{N}=n}}{\rm Hom}_{A^{\otimes N}-mod-A^{\otimes
N}}(A^{\otimes i_{1}}\otimes...\otimes A^{\otimes i_{N}},A^{\otimes
N}_{cycl})$

The complex $C^{(N)}=\prod\limits_{n}C^{(N),n}$ is defined as $A^{\otimes
N}$-bimodule ${\rm Hom}\,$ from the $N$th power of the bar complex
${\mathcal{B}}(A)$:

$C^{(N)}={\rm Hom}_{A^{\otimes N}-mod-A^{\otimes N}}({\mathcal{B}}^{\otimes
N},A^{\otimes N}_{cycl})$

to the ${A}^{\otimes N}$-bimodule $A^{\otimes N}_{cycl}$ with the following
bimodule structure. For any $x_{1}\otimes...\otimes x_{N}\in{A}^{\otimes
N}_{cycl}$ and elements $a_{1}\otimes...\otimes a_{N},b_{1}\otimes...\otimes
b_{N}\in{A}^{\otimes N}$,

$(a_{1}\otimes...\otimes a_{N})\bullet(x_{1}\otimes...\otimes
x_{N})\bullet(b_{1}\otimes...\otimes b_{N})=a_{1}x_{1}b_{2}\otimes...\otimes
a_{N}x_{N}b_{1}.$

Now we define the higher cyclic Hochshild complex
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$.

######  Definition 3.1.

For $N\geqslant 1$ the space of N-higher cyclic Hochschild cochains is defined
as

$C^{(N)}_{cycl}(A):=\prod_{r_{1},...,r_{N}\geqslant 0}{\rm Hom}_{A^{\otimes
N}-mod-A^{\otimes N}}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes
r_{i}},A^{\otimes N}_{cycl})^{{\mathbb{Z}}_{N}},$

The differential is coming from the bar complex of $A^{\otimes N}$-bimodules,
after it is dualised by
${\rm Hom}_{A^{\otimes N}-mod-A^{\otimes N}}(-,A_{cycl}^{\otimes N})$. It will
be written precisely in section 5.

Elements of ${\rm Hom}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes
r_{i}},A^{\otimes N}_{cycl})$ can be obviously interpreted as collections of
$N$ operations with one output each. This interpretation is obtained if we
first pass to the isomorphic complex over $\mathbb{K}$. Then elements of ${\rm
Hom}_{\mathbb{K}}\prod\limits_{r_{1},...,r_{N}\geqslant 0}(A^{\otimes
r_{i}-2},A_{cycl}^{\otimes N})$ are interpreted as a collection of $N$
operations.

In this interpretation it is easy to see that there is a natural
${\mathbb{Z}}_{N}$ group action on ${{\rm
Hom}\,}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes
N}_{cycl})$, which cyclically permutes operations and assigns a sign
$(-1)^{(d-1)(N-1)}$, according to the Koszul rule (and taking into account
degree of operation).

The difference between ${\mathcal{H}}{\mathcal{H}}$ and
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ is that the latter consists only of
elements invariant under this ${\mathbb{Z}}_{N}$ action, which we denote ${\rm
Hom}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes
N}_{cycl})^{Z_{N}}.$ Let us introduce new notation:
$C^{(N,d)}=C^{(N)}_{cycl}(A)\subset C^{(N)}(A)$,
$\,\,C^{(N,d)}=C^{(N)}_{(d-1)(N-1)({\rm mod}2)},$ where $C^{(N)}_{0}$ are
cochains symmetric under cyclic permutation of operations, and $C^{(N)}_{1}$
are antisymmetric cochains. By this we stress that
${\mathbb{Z}}_{N}$-invariant elements of ${\mathcal{H}}{\mathcal{H}}$,
constituting ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$, consist of either
symmetric or antisymmetric cochains, depending on $N$ and $d$. The fact that
we should take only ${\mathbb{Z}}_{N}$-invariant elements comes from the
condition of cyclic symmetry of the $A_{\infty}$-structure on $A\oplus
A^{*}[1-d]$ w.r.t. the natural pairing.

We can point out at the difference between ${\mathcal{H}}{\mathcal{H}}$ and
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ also by saying that elements of
${{\rm Hom}\,}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes r_{i}},A^{\otimes
N}_{cycl})$ have a fixed point, corresponding to the operation from which we
start, however in ${{\rm Hom}\,}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes
r_{i}},A^{\otimes N}_{cycl})^{{\mathbb{Z}}_{N}}$, after it is symmetrized,
that is invariants are taken, the fixed (starting) point does not exist any
more.

Denote by
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}=C^{(\bullet)}_{cycl}(A)=\mathop{\prod}\limits_{N\geqslant
1}C^{(N)}_{cycl}(A)$ the space of all higher cyclic Hochschild cochains.
Further throughout the paper we frequently omit the subscript $cycl$, if it
does not produce any confusion.

The space of all higher cyclic Hochschild cochains is denoted by
$C^{(\bullet)}_{A}(A)$ or $C^{(\bullet)}_{\mathbb{K}}(A)$, depending on
whether we are dealing with $A^{\otimes N}$ \- bimodule Homs, or consider
corresponding $\mathbb{K}$-module Homs. Sometimes we omit the $\mathbb{K}$
when it is clear from the context.

Note, that $C^{(1)}_{\mathbb{K}}(A)$ is the space of usual Hochschild
cochains. To set up notations remind that the usual Hochschild cochains are:

$C^{(1)}(A):=\prod_{r}{\rm Hom}_{A-mod-A}(A^{\otimes r},A):$

$A\to{\rm Hom}\,_{A-mod-A}(A,A)\mathop{\to}\limits^{D}{\rm
Hom}\,_{A-mod-A}(A^{\otimes 2},A)\to...,$

where $A$ is sitting in degree $0$, ${\rm Hom}\,(A,A)$ in degree $1$, etc. and

$(D_{\mathbb{K}}h)(v_{1}\otimes...\otimes
v_{n-1})=v_{1}h(v_{2}\otimes...\otimes v_{n-1})-h(v_{1}v_{2}\otimes
v_{2}...\otimes v_{n-1})+...$

$(-1)^{n-2}h(v_{1}\otimes...\otimes
v_{n-2}v_{n-1})+(-1)^{n-1}h(v_{1}\otimes...\otimes v_{n-2})v_{n-1}.$

Now, when we start to define a Lie bracket on the higher cyclic Hochschild
complex $C^{(\bullet)}_{A}(A)$, it becomes important which shifts of the
grading on $A$ we chose, so we consider the higher cyclic Hochschild complex
with the following shifts:

${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}[1]=C^{(N)}_{cycl}(A)[1]:=\prod_{r_{1},...,r_{N}\geqslant
0}{{\rm Hom}\,}_{A^{\otimes N}-mod-A^{\otimes
N}}(\mathop{\otimes}\limits_{i=1}^{N}A[1]^{\otimes r_{i}},A[1]^{\otimes
N}_{cycl})^{{\mathbb{Z}}_{N}}[1].$

######  Definition 3.2.

The generalized necklace bracket between two elements $f,g\in
C_{cycl}^{(N)}(A)$ is given as $[f,g]_{g.n}=f\circ g-(-1)^{\sigma}g\circ f,$
where composition $f\circ g$ consists of inserting all outputs of $g$ to all
inputs from $f$ with signs assigned according to the Koszul rule.

The composition for the generalised necklace bracket can be graphically
depicted as follows:

0

0

0

Note that the (starting) point is not fixed in the elements of our complex
(operations), thus generalized necklace bracket should produce also operations
without a fixed point. Thus we need to clarify what means ’insertion’ of one
operation into another in the cyclic situation of definition above: we should
think of insertion of operations with fixed point according to the above rule,
and then symmetrizing the result, by taking each resulting operation with all
possible fixed points to the output.

Since the defined above composition $f\circ g$ makes
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}[1]=C_{cycl}^{(\bullet)}[1]$ into a
graded pre-Lie algebra, the generalized necklace bracket obtained from it as a
graded commutator, makes $C_{cycl}^{(\bullet)}[1]$ into a graded Lie algebra.
We denote it by ${\bf g}=(C_{cycl}^{(\bullet)}(A)[1],[,]_{g.n}).$

######  Definition 3.3.

The pre-Calabi-Yau structure on $A$ is an element
$m=\mathop{\sum}\limits_{N\geqslant 0}m^{(N)}$, $m^{(N)}\in C^{(N,d)}(A)[1]$
from the space of the ${\mathbb{Z}}_{N}$-invariant higher cyclic Hochschild
cochains

${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}[1]=C^{(N,d)}[1]=\prod\limits_{r_{1},...,r_{N}\geqslant
0}{{\rm Hom}\,}_{A^{\otimes N}-mod-A^{\otimes
N}}(\mathop{\otimes}\limits_{i=1}^{N}A[1]^{\otimes r_{i}},A[1]^{\otimes
N}_{cycl})^{{\mathbb{Z}}_{N}}[1],$

which is a solution to the Maurer-Cartan equation $[m,m]_{g.n}=0$ with respect
to generalised necklace bracket.

Since in shifted by 1 situation operations have degree one, those elements of
the ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}[1]$ which correspond to $d$-pre-
Calabi-Yau structure should have degree $(d-1)(N-1)$. While reformulating the
definition we take into account that in the definition 2.2 $A^{*}$ is shifted
by $1-d$.

For the sake of clarity, simplicity, and since the main formality result holds
only in this situation, we mainly consider here the grading, where $A$ is
sitting in degree zero: $A_{0}=A$. This prompts us to deal with 2-pre-Calabi-
Yau structures.

In the case $n=2,N=2$, which will be interesting for us in the next section,
the definition says that we should take antisymmetric elements of
${\mathcal{H}}{\mathcal{H}}$ into ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$
complex.

##  4 Double Poisson bracket and the Maurer-Cartan equation

In this section we discuss a correspondence between particular part of pre-
Calabi-Yau structure and the structure of double Poisson bracket invented by
Van den Bergh [39] as a structure which produces the Poisson bracket on
representation spaces.

There were many efforts to construct a reasonable notion of noncommutative
Poisson bracket, which would according to the ideology of [24, 27] induce a
kind of Poisson bracket on representation spaces or their moduli. First, the
noncommutative Poisson bracket was defined in an obvious way: the same way as
it is done in the commutative case, as a bracket on $A$, $\\{-,-\\}:A\to A$
which satisfy the Leibnitz rule: $\\{a,bc\\}=b\\{a,c\\}+\\{a,b\\}c$. But this
notion turns out to be too restrictive, in [12] it was shown that defined this
way bracket on noncommutative prime rings can be only commutator bracket
$[a,b]=ab-ba$. There was a notion of ’noncommutative Poisson structure’
introduced in [40, 2], but it is only known that this bracket on the center of
$A$ produces the usual commutative Poisson structure, it is unclear what it
gives on moduli of representations. Then the attempts to introduce the notion
of noncommutative Poisson bracket lead to a good definition in [39, 10]. In
[39] the notion of the double Poisson bracket was defined as a map
$\\{\\!\\{\cdot,\cdot\\}\\!\\}:A\otimes A\to A\otimes A$, satisfying the
axioms which are certain generalization (thickening) of the usual Poisson
axioms of anti-symmetry, Leibnitz and Jacobi identities. In [10] the
noncommutative Poisson structure, called $H_{0}$-Poisson structure was defined
as a Lie bracket on zero Hochschild homology of $A$: $H_{0}=A/[A,A]$, such
that the map $\\{\bar{a},-\\}:A/[A,A]\to A/[A,A]$ is induced by a derivation
$d_{a}:A\to A$. These explained many effects, for example, clarified the study
of quasi-poisson structures [3]. There were further developments like [5], but
we are trying here to continue the line of initial ideas from [24, 27].

We will put the earlier approaches in a more general framework, which explains
a pattern of this generalisation (thickening) process from the perspective of
the whole pre-Calabi-Yau structure. For example, the twisted structure of the
diagonal bimodule $A^{\otimes N}$ will show how multiple derivations
associated to a noncommutative polyvector fields interfere. We will see also
in this section how the ’double’ definition comes as a particular part of
general situation, coming from pre-Calabi-Yau structure. Namely, we will
demonstrate that double bracket is defined by $\xi\delta$-words with two
$\delta$th and no $\xi$th.

Remind, that double Poisson bracket is defined as a map
$\\{\\!\\{\cdot,\cdot\\}\\!\\}:A\otimes A\to A\otimes A$ satisfying the
following axioms:

Anti-symmetry:

$\\{\\!\\{a,b\\}\\!\\}=-\\{\\!\\{b,a\\}\\!\\}^{op}$ (4.1)

Here $\\{\\!\\{b,a\\}\\!\\}^{op}$ means the twist in the tensor product, i.e.
if $\\{\\!\\{b,a\\}\\!\\}=\sum\limits_{i}b_{i}\otimes c_{i}$, then
$\\{\\!\\{b,a\\}\\!\\}^{op}=\sum\limits_{i}c_{i}\otimes b_{i}.$

Double Leibniz:

$\\{\\!\\{a,bc\\}\\!\\}=b\\{\\!\\{a,c\\}\\!\\}+\\{\\!\\{a,b\\}\\!\\}c$ (4.2)

(here we use an outer bimodule structure on $A\otimes A:\quad a(b\otimes
c)=ab\otimes c,(a\otimes b)c=a\otimes bc),$ and double Jacobi identity:

$\\{\\!\\{a,\\{\\!\\{b,c\\}\\!\\}\\}\\!\\}_{L}+\tau_{(123)}\\{\\!\\{b,\\{\\!\\{c,a\\}\\!\\}\\}\\!\\}_{L}+\tau_{(132)}\\{\\!\\{c,\\{\\!\\{a,b\\}\\!\\}\\}\\!\\}_{L}$
(4.3)

Here for $a\in A\otimes A\otimes A,\,\,$ and $\sigma\in S_{3}\,\,$

$\tau_{\sigma}(a)=a_{\sigma^{-1}(1)}\otimes a_{\sigma^{-1}(2)}\otimes
a_{\sigma^{-1}(3)}.$

The $\\{\\!\\{\,\\}\\!\\}_{L}$ defined as

$\\{\\!\\{b,a_{1}\otimes...\otimes
a_{n}\\}\\!\\}_{L}=\\{\\!\\{b,a_{1}\\}\\!\\}\otimes a_{1}\otimes...\otimes
a_{n}$

The connection between the two structures is described by the following
theorem.

######  Theorem 4.1.

Let we have $A_{\infty}$-structure on $(A\oplus A^{*},m=\sum\limits_{i=2,i\neq
4}^{\infty}m_{i}^{(1)})$. Define the bracket by the formula

$(*)\quad\quad\langle g\otimes f,\\{\\!\\{b,a\\}\\!\\}\rangle:=\langle
m_{3}(a,f,b),g\rangle,$

where $a,b\in A$, $f,g\in A^{*}$ and $m_{3}(a,f,b)=c\in A$ corresponds to the
component of solution to the Maurer-Cartan $m_{3}$: $A\times A^{*}\times A\to
A$ corresponding to the cyclic tensor $A\otimes A^{*}\otimes A\otimes A^{*}$.
Then this bracket does satisfy all axioms of the double Poisson algebra.

Moreover, pre-Calabi-Yau structures corresponding to the cyclic tensor
$A\otimes A^{*}\otimes A\otimes A^{*}$ with $m_{i}=0,i\geqslant 4$ are in the
bijective correspondence defined by $(*)$ with the double Poisson brackets for
an arbitrary associative algebra $A$.

The detailed proof of this theorem, taking into account signs and other
details, was given in [17] (see also [19]) in terms of definition 2.2 of pre-
Calabi-Yau structure.

We present here main idea of this proof, using definition 3.3 via higher
cyclic Hochschild complex. It looks very transparent this way, which
emphasises another advantage of this definition.

In terms of definition 3.3 the Maurer-Cartan equation on ’invariant’ with
respect to the action of cyclic group elements from the higher cyclic
Hochschild complex of particular kind, described in the theorem looks like:

0

0

Hence the Maurer-Cartan (in appropriate arity) is equivalent to the following
equations:

0

0

0

0

These two are clearly exactly the double Leibnitz and double Jacobi identities
respectively. The thing to be checked now is related to the following fact.
The element of the higher cyclic Hochschild complex with two inputs and two
outputs canonically corresponds (via the pairing on $A\oplus A^{*}$) to the
maps $D:A\times A\to A\times A$ and $M:A\times A^{*}\times A\to A$. This
correspondence is defined however only up to an arbitrary permutation of terms
$A$. To establish an isomorphism between the two structures, we then need to
choose appropriately this correspondence, which is done by formula (*) in
[17]. After that axioms of double Poisson bracket can be checked, taking into
account signs. Moreover we ensure that no other axioms appear from the Maurer-
Cartan equation in case of the structure $(A\oplus A^{*},m_{2}+m_{3})$, hence
all double Poisson brackets can be obtained from these structures, that is the
map defined by (*) is a surjection. This means that structures of mentioned
type are indeed in a bijective correspondence with the double Poisson
brackets.

The question on better understanding of the structure needed to produce
Poisson bracket on representation spaces recently received much attention. It
is discussed, apart from mentioned paper [19], for example, in [4, 41]. Our
work [17], [19] was also extended to the case when $A\neq A_{0}$, by means of
the same correspondence given by formula (*), with signs added in [13].

##  5 Small subcomplex of the higher cyclic Hochshild complex

Here we work with the smooth algebras [27], which are finitely generated
2-formally smooth algebras. By 2-formally smooth algebra we mean a formally
smooth algebra in the sense of J. Cuntz and D. Quillen [9]:

######  Definition 5.1.

An algebra A is 2-formally smooth (=quasi-free) if and only if it satisfies
one of the following equivalent properties:

(1) (Lifting property for nilpotent extensions) for any algebra $B$, a two-
sided nilpotent ideal $I\in B(I=BIB,I^{n}=0$ for $n\gg 0)$, and for any
algebra homomorphism $f:A\to B/I$, there exists an algebra homomorphism
$\tilde{f}:A\to B$ such that $f=pr_{B\to B/I}\circ\tilde{f}$ is a natural
projection.

(2) $Ext^{2}_{A-mod-A}(A,M)=0$ for any bimodule $M\in A-mod-A$.

(3) The A-bimodule $\Omega_{A}^{1}={\rm Ker}(m_{A}:A\otimes A\to A)$ is
projective.

First, consider a subcomplex $\zeta$ of the higher cyclic Hochschild complex,
which we define as follows. Take a quotient complex

$R_{\rm min}=[0\to\Omega\to A\otimes A]\twoheadrightarrow A$

of the bar complex (considered as a complex of $A$-bimodules). Namely, $R_{\rm
min}={\cal B}/{\cal F}$, where ${\cal B}$ is the bar complex

${\cal B}=[\dots A\otimes A\otimes A\mathop{\to}\limits^{D_{3}}A\otimes
A]\twoheadrightarrow A.$

Denote its usual differential by $D_{A}$ or just $D$, when it is clear that we
are talking about complex of $A$-bimodules. Obviously, ${\mathcal{B}}$ as well
as ${\mathcal{R}}_{min}$ provide a free $A$-bimodule resolution for the
diagonal bimodule $A$.

Let $\cal F$ be the subcomplex generated by $A^{\otimes k}$ with $k\geqslant
4$ and $\text{\rm ker}\,D_{3}$, i.e. ${\cal F}=\oplus A^{\otimes\geqslant
4}\oplus{\rm Ker}D_{3}$. Note that $\Omega=A^{\otimes 3}/\text{\rm
ker}\,D_{3}$ is isomorphic to the kernel of the multiplication map
$\mu:A\otimes A\to A$. We equip $R_{\rm min}$ with the grading for which
$R_{-1}=\Omega$ and $R_{0}=A\otimes A$. Thus we have a resolution $R_{\rm
min}\in{\rm Compl}(A^{\rm e}-mod)$ of a diagonal bimodule $A$.

Then consider $N$th tensor power of $R_{\rm min}$:

$R_{\rm min}^{\otimes N}\in{\rm Compl}((A^{\rm e})^{\otimes N}-{\rm mod}),$

and dualise it by taking Hom to an $A^{\otimes N}$-bimodule $A^{\otimes
N}_{cycl}$ with the defined above structure:

${\rm Hom}_{(A^{\rm e})^{\otimes N}}(R_{\rm min}^{\otimes N},A^{\otimes
N}_{{\rm cycl}})=:\zeta^{(N)}.$

For $N=1$ applying the functor ${\rm Hom}_{A^{\rm e}}(-,A)$ to $R_{\rm
min}\in{\rm Compl}(A^{e}-{\rm mod})$ we get a subcomplex $\zeta={\rm
Ann}({\cal F})={\rm Hom}_{A^{\rm e}}(R_{min},A)$ of the usual Hochschild
complex $C^{\bullet}(A,A)=C^{(1)}(A)={\rm Hom}_{A^{\rm e}}({\cal B},A)$:

$C^{\bullet}(A,A)\supset{\rm Hom}_{A^{\rm e}}(R_{\rm min},A),$

where

${\zeta}={\rm Ann}({\cal F})=\\{\Phi\in C^{\bullet}(A,A)[1]:\Phi(h)=0\
\text{for}\ h\in{\cal F}\\}.$

Thus

$\zeta={\rm Ann}({\cal F})=\\{\text{chains in}\ {\rm Hom}_{A^{e}}({\cal
B},A),\ \text{turning}\ {\cal F}\ \text{into $0$}\\}$

$=\\{\Phi(a_{1}\otimes\dots\otimes a_{n})$, s.t.
$\Phi(a_{1}\otimes\dots\otimes a_{n})=0$, $n>3$ and

$\Phi(a_{1}\otimes a_{2}\otimes a_{3})=0$ iff $a_{1}\otimes a_{2}\otimes
a_{3}\in\text{\rm ker}\,D_{3}={\rm Im}D_{4}$}.

Thus, $\Phi\in{\rm Hom}\,_{A^{e}}(A^{\otimes 3},A)$ is in $\zeta$ if and only
if it is an $A$-bimodule derivation, that is satisfies the Leibnitz rule:

$\Phi(a_{1}\otimes a_{2}a_{3}\otimes a_{4})=\Phi(a_{1}\otimes a_{2}\otimes
a_{3}a_{4})-\Phi(a_{1}a_{2}\otimes a_{3}\otimes a_{4}).$

Note that ${{\rm Hom}\,}_{A^{\rm e}}(A\otimes A,A)$ is naturally isomorphic to
$A$, while ${{\rm Hom}\,}_{A^{\rm e}}(\Omega,A)$ is naturally identified with
${\rm Der}_{A^{e}}(A^{\otimes 3},A)$, which interprets the complex
$\zeta_{A}={\rm Hom}\,_{A^{e}}(R_{min},A)$

$0\leftarrow{\rm Hom}\,_{A^{e}}(\Omega,A)\mathop{\leftarrow}^{D_{3}^{*}}{\rm
Hom}\,_{A^{e}}(A\otimes A,A)\mathop{\leftarrow}^{D_{2}^{*}}{\rm
Hom}\,_{A^{e}}(A,A)$

as

$0\leftarrow{\rm Der}_{A^{e}}(A^{\otimes
3},A)\mathop{\leftarrow}^{D_{3}^{*}}A\mathop{\leftarrow}^{D_{2}^{*}}\mathbb{K}.$

We can pass from ${\rm{\rm Hom}\,}_{A^{\rm e}}$ to ${\rm Hom}\,_{\mathbb{K}}$,
and since

${\rm Hom}\,_{A^{\rm e}}(A^{\otimes n+2})\simeq{\rm
Hom}\,_{\mathbb{K}}(A^{\otimes n},A),\quad{\rm Hom}\,_{A^{\rm e}}(A\otimes
A,A)\simeq A,\quad{\rm Hom}\,_{A^{\rm e}}(\Omega,A)\simeq{\rm
Der}_{\mathbb{K}}(A)\,\subset{\rm Hom}\,_{\mathbb{K}}(A,A)$

we have an isomorphic complex over $\mathbb{K}$:

$\zeta_{\mathbb{K}}:\quad 0\leftarrow{\rm
Der}_{\mathbb{K}}\,A\mathop{\leftarrow}^{d_{3}^{*}}A$

where ${\rm Der}_{\mathbb{K}}\,A$ is the space of derivations of $A$ from
${\rm Hom}\,_{\mathbb{K}}(A,A)$.

Cohomologies of the usual Hochschild cochain complex:

$H^{\bullet}(C^{(1)})(A)=Ext_{A-mod-A}^{\bullet}(A,A)$

are the Hochschild cohomologies.

Analogously, denote cohomologies of $N$th slice the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex $C^{(N)}$ by

$H^{\bullet}(C^{(N)}(A))=Ext_{A^{\otimes N}-mod-A^{\otimes
N}}^{\bullet}(A^{\otimes N},A^{\otimes N}_{cycl}),$

and cohomologies of the whole ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$
complex $C^{(\bullet)}$ by $H^{\bullet}C^{(\bullet)}.$

We explain here in more details what we do in the case of arbitrary $N$. The
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex (before taking invariants)
is the $N$th tenzor power of the bar complex ${\mathcal{B}}^{\otimes N}$
dualised by ${\rm Hom}\,_{A^{\otimes N}-mod-A^{\otimes N}}(-,A^{\otimes
N}_{cycl}):$

$C^{(N)}(A)={\rm Hom}\,_{A^{\otimes N}-mod-A^{\otimes
N}}({\mathcal{B}}^{\otimes N},A^{\otimes N}_{cycl}).$

The complex ${\mathcal{R}}_{min}$ is obviously a quotient complex of
${\mathcal{B}}^{\otimes N}$. Indeed, $R_{min}^{\otimes N}=({\cal B}/{\cal
F})^{\otimes N}={\cal B}^{\otimes N}/{\cal J}$ where ${\cal
J}={\mathcal{F}}\otimes{\mathcal{B}}^{\otimes(N-1)}+{\mathcal{B}}\otimes{\mathcal{F}}\otimes{\mathcal{B}}^{\otimes(N-2)}+...+{\mathcal{B}}^{\otimes(N-1)}\otimes{\mathcal{F}}$.
Here we need to check of course that ${\mathcal{J}}$ is a submodule in
$A^{\otimes N}$-bimodule ${\mathcal{B}}^{\otimes N}.$

To obtain the complex $\zeta^{(N)}$ we take $N$th tensor power of small
complex $R_{min}$ and dualize it by ${\rm Hom}_{A^{\otimes N}-mod-A^{\otimes
N}}(-,A^{\otimes N}_{cycl})$. Then we take invariants under ${\mathbb{Z}}_{N}$
action.

The structure of $A^{\otimes N}$ bimodule on ${\cal B}^{\otimes N}$ is
natural, on $A^{\otimes N}_{cycl}$ as defined above.

Thus $\zeta^{(N)}_{A}={\rm Ann}\,{\mathcal{J}}={\rm Hom}\,_{A^{\otimes
N}-mod-A^{\otimes N}}(R_{min}^{\otimes N},A^{\otimes
N}_{cycl})^{{\mathbb{Z}}_{N}}\subset{\rm Hom}\,_{A^{\otimes N}-mod-A^{\otimes
N}}({\mathcal{B}}^{\otimes N},A^{\otimes N}_{cycl})^{{\mathbb{Z}}_{N}}.$

We can describe this annihilator as

$\zeta^{(N)}={\rm Ann}\,{\mathcal{J}}=\\{\Phi\in{\rm Hom}\,_{A^{\otimes
N}-mod-A^{\otimes N}}({\mathcal{B}}^{\otimes N},A^{\otimes
N}_{cycl}))^{{\mathbb{Z}}_{N}}\,|\,$

$\Phi({{\mathcal{B}}}^{\otimes
r}\otimes{{\mathcal{F}}}\otimes{{\mathcal{B}}}^{\otimes s})=0,\quad\forall
r+s=N-1\\}.$

This means $\zeta^{(N)}={\rm Ann}\,{\mathcal{J}}$ formed by those element of
vector space

$E^{\otimes N}=\bigcap\limits_{s+r=N-1}\\{{{\mathcal{B}}}^{\otimes
r}\otimes{\rm Ann}\,{{\mathcal{F}}}\otimes{{\mathcal{B}}}^{\otimes s}\\},$

which are $A^{\otimes N}$-bimodule morphisms.

This leads us to the description of the small subcomplex $\zeta^{(N)}$ of the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex $C^{(N)}(A)$ in terms of the
appropriately chosen basis.

Starting from this place, when we choose a basis, we will deal with a free
algebra $A$, in stead of general smooth algebra. The obvious free basis in
$\Omega_{A-mod-A}$ consists of $dx_{i}=1\otimes x_{i}-x_{i}\otimes 1$. Denote
the basis of free $A$-bimodule $A\otimes A$ by $\xi$. Elements of dual bases
in ${\rm Hom}\,_{A-mod-A}(\Omega,A)$ and ${\rm Hom}\,_{A-mod-A}(A\otimes A,A)$
denote by $\delta_{i}$ and $\xi^{*}$:
$\delta_{i}(dx_{j})=\delta_{ij}1,\,\xi^{*}(\xi)=1$ respectively. Corresponding
bases of ${\rm Hom}\,_{\mathbb{K}}(\Omega,A)$ and ${\rm
Hom}\,_{\mathbb{K}}(A\otimes A,A)$ are $\\{\delta_{i}u,u\in\langle
X\rangle\\}$ and $\\{\xi u,u\in\langle X\rangle\\}$ respectively.

The basis of the complex $\zeta^{(N)}_{\mathbb{K}}={\rm
Hom}\,_{\mathbb{K}}(R_{\min}^{\otimes N},A^{\otimes
N}_{cycl})^{{\mathbb{Z}}_{N}}$ will consist of cyclic monomials on $\xi^{*}$,
$\delta_{i}$ and $x_{i}$ which we depict as follows (we will further write
just $\xi$ in stead of $\xi^{*}$).

0

0

0

pic.1

Now the important for further considerations point is to embed the subcomplex
$\zeta^{(N)}$ into the higher cyclic Hochschild complex $C^{(N)}$. We define
this embedding by specifying the operation, i.e. an element of the higher
cyclic Hochschild complex $\Phi_{\omega}\in\mathop{{\rm
Hom}\,}_{\mathbb{K}}(\mathop{\otimes}\limits_{i=1}^{N}A^{\otimes
n_{i}},A^{\otimes N})^{{\mathbb{Z}}_{N}}$, which corresponds to a given
$\xi\delta$-monomial $\omega$. Here $n=\sum n_{i}$ is the $\delta$-degree of
$\xi\delta$-monomial $w$ and $N$ is its $\delta,\xi$-degree.

Let $X$-monomials $u_{1},\dots,u_{n}$ be an input of $\Phi_{\omega}$. The
output will be a linear combination of tuples of monomials from $A$ colored
green in the following picture. All circles and arcs in the picture are
oriented clockwise, so one can read outputs following the orientation of the
circles. The sum in the linear combination is over all ’gluings’ of variables
$x_{i}$ from the input monomials (black) with $\delta_{i}$ (with the same
index $i$) in the $\xi\delta$-monomial $w$.

The $\xi\delta$-monomial depicted below represent an operation
$\Phi:A^{\otimes 3}\to A^{\otimes 5},$ (or more precisely: $A\otimes
A^{0}\otimes A\otimes A\otimes A^{0}\to A^{\otimes 5}$) from the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex $C^{(N)}$ for $N=5$.

0

0

pic.2

In terms of the above $\xi\delta$-basis we now describe differentials
$D^{*}_{A}=D^{*}$ and $D^{*}_{\mathbb{K}}=d^{*}$ in dualized complexes.

Let us spell out first the usual differential $D_{A}$ on one copy of the bar
complex ${\mathcal{B}}$:

$D_{A}(u_{1}\otimes...\otimes u_{n})=u_{1}u_{2}\otimes...\otimes
u_{n}-u_{1}\otimes u_{2}u_{3}\otimes...\otimes
u_{n}+...+(-1)^{n}u_{1}\otimes...\otimes u_{n-1}u_{n}.$

After we dualise this complex by ${\rm Hom}\,_{A-mod-A}(-,A)$, we get a usual
dual differential $D^{*}_{A}$, $D^{*}f(u)=f(Du)$ on the Hochschild complex :

$(D^{*}_{A}f)(u_{1}\otimes...\otimes u_{n+1})=f(u_{1}u_{2}\otimes...\otimes
u_{n+1})-f(u_{1}\otimes u_{2}u_{3}\otimes...\otimes
u_{n+1})+...+(-1)^{n+1}f(u_{1}\otimes...\otimes u_{n}u_{n+1}),$

where $f\in{\rm Hom}\,_{A-mod-A}({\mathcal{B}},A).$

When we pass to ${\rm Hom}\,_{\mathbb{K}}({\mathcal{B}},A)$, since $f$ is an
$A$-bimodule morphism, an element $h\in{\rm
Hom}\,_{\mathbb{K}}({\mathcal{B}},A)$ is defined by $h(v_{1}\otimes...\otimes
v_{n})=f(1\otimes v_{1}\otimes...\otimes v_{n}\otimes 1),$ and

$(D^{*}_{\mathbb{K}}h)(v_{1}\otimes...\otimes
v_{n-1})=v_{1}h(v_{2}\otimes...\otimes v_{n-1})-h(v_{1}v_{2}\otimes
v_{2}...\otimes v_{n-1})+...$

$(-1)^{n-2}h(v_{1}\otimes...\otimes
v_{n-2}v_{n-1})+(-1)^{n-1}h(v_{1}\otimes...\otimes v_{n-2})v_{n-1},$

where $h\in{\rm Hom}\,_{\mathbb{K}}({\mathcal{B}},A).$

Doing the same for the tensor product of $N$ copies of the bar complex
${\mathcal{B}}^{\otimes N}$ and dualising it by ${\rm Hom}\,_{A^{\otimes
N}-mod-A^{\otimes N}}(-,A^{\otimes N}_{cycl})$, we obtain the expression for
the differential in the higher cyclic Hochschild complex:

$D^{*}h(v_{1},...,v_{N})=\sum\limits_{\alpha=1}^{n}(-1)^{s_{1}+...+s_{\alpha-1}}D^{*\alpha}h(v_{1},...,v_{N}),$

where $v_{\alpha}=x_{1}^{\alpha}...x_{s_{\alpha}}^{\alpha}\in A^{\otimes
s_{\alpha}}\subset{\mathcal{B}}$, and

$D^{*\alpha}_{\mathbb{K}}h(v_{1},...,v_{N})\sum\limits_{j=1}^{s_{\alpha}-1}(-1)^{j}h(v_{1}\otimes...\otimes
v_{\alpha-1}\otimes x_{1}^{\alpha}\otimes...\otimes
x_{j}^{\alpha}x_{j+1}^{\alpha}\otimes...\otimes x_{s_{\alpha}}^{\alpha}\otimes
v_{\alpha+1}\otimes...\otimes v_{N})+$

$(1\otimes...\otimes 1\otimes x_{1}^{\alpha}\otimes...\otimes 1)\bullet
h(v_{1}\otimes...\otimes v_{\alpha-1}\otimes x_{2}^{\alpha}\otimes...\otimes
x_{s_{\alpha}}^{\alpha}\otimes v_{\alpha+1}\otimes...\otimes v_{N})+$

$(-1)^{s_{\alpha}}h(v_{1}\otimes...\otimes v_{\alpha-1}\otimes
x_{1}^{\alpha}\otimes...\otimes x_{s_{\alpha-1}}^{\alpha}\otimes
v_{\alpha+1}\otimes...\otimes v_{N})\bullet(1\otimes...\otimes 1\otimes
x_{s_{\alpha}}^{\alpha}\otimes...\otimes 1).$

Here the element $(1\otimes...\otimes 1\otimes x_{1}^{\alpha}\otimes...\otimes
1)$ has $x_{1}^{\alpha}$ in the place $\alpha$, the element
($1\otimes...\otimes 1\otimes x_{s_{\alpha}}^{\alpha}\otimes...\otimes 1)$ has
$x^{\alpha}_{s_{\alpha}}$ in place $\alpha+1({\rm mod}N)$, and $\bullet$ stays
for the multiplication in $A^{\otimes N}$-bimodule $A^{\otimes N}_{cycl}$, as
described in section 3.

We seen tat $R_{min}^{\otimes N}={\mathcal{B}}^{\otimes N}/{\cal J}$, thus
$\zeta={\rm Hom}\,(R_{min}^{\otimes N},A^{\otimes N}_{cycl})$ is a subcomplex
of the ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex ${\rm
Hom}\,({\mathcal{B}}^{\otimes N},A^{\otimes N}_{cycl}).$ More precisely, in
both complexex we should take ${\mathbb{Z}}_{N}$-invariant elements.

The subcomplex $\zeta$ is quasi-isomorphic to the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex since $A$ is smooth.

Remind that we denoted $H^{\bullet}C^{(N)}(A)=Ext_{A^{\otimes
N}-mod-A^{\otimes N}}(A^{\otimes N},A^{\otimes N}_{cycl})$ homologies of $N$th
slice of ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex $C^{(N)}(A)$.

######  Proposition 5.2.

Let $A$ be a smooth algebra (in particular, finitely generated free
associative algebra). Then $H^{\bullet}C^{(N)}(A,A)=H^{\bullet}\zeta^{(N)}$.

###### Proof.

The statement follows from the fact that our algebra is smooth and both
complexes comes from projective resolutions of a diagonal bimodule $A^{\otimes
N}$.

∎

One of our main goals will be to prove that these homologies are pure, namely
concentrated in the diagonal of the defined above bigrading on the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}=C^{(N),n}$, and the complex is
$L_{\infty}$-formal. We believe that concrete calculations in [30] (check of
the Jacobi identity, etc.) for noncommutative bivector fields (in sense of
[10]) are easily explained, and perhaps inspired by our results [18] and are
in fact performed in our basis of the small complex $\zeta$. The case of
noncommutative projective space corresponds to the free path algebra of the
Kronecker quiver ${\mathcal{K}}_{n}$, for which we prove the formality in
section 7.2.

In the next section we consider the Lie structure on $C^{(\bullet)}[1]$,
described in section 3, and prove that the subcomplex $\zeta^{(\bullet)}$ of
the complex ${\bf g}=C^{(\bullet)}[1]$ is closed under the Lie bracket in
${\bf g}$. Thus it forms a Lie subalgebra ${\bf g_{0}}\subset{\bf g}$.
Moreover, we show how this bracket is combinatorially described in terms of
the basis in $\bf g_{0}=\zeta^{(\bullet)}$, consisting of
$\xi\delta$-monomials.

##  6 Lie bracket on $\zeta^{(\bullet)}$

We give here a constructive description of the bracket in the small subcomplex
$\zeta_{\mathbb{K}}^{(N)}$ of the Hochschield complex in terms of $\xi\delta$
calculus which makes it into a Lie subalgebra of $C_{\mathbb{K}}^{(N)}$.

######  Theorem 6.1.

I. The above described embedding $\zeta_{\mathbb{K}}\to C^{(\bullet)}(A,A)[1]$
is an embedding of complexes, whose image is a Lie subalgebra of ${\bf
g}=C^{(\bullet)}(A,A)[1]$ equipped with the generalised necklace bracket.

II. Precise combinatorial description of this bracket is given by (*) (pic.
3).

###### Proof.

To prove this we need to show that the bracket of the Lie algebra ${\bf
g}=C^{\bullet}(A,A)[1]$ applied to $\xi\delta$-monomials yields a member of
$\zeta$, that is, a linear combination of $\xi\delta$-monomials again. Here we
demonstrate how it works.

Let $A$ and $B$ be two $\xi\delta$-monomials. We perform composition of
corresponding operations $U\circ W$ from the Hochschild complex according to
the necklace bracket rule. We will see that we can not express the resulting
operation $U\circ W$ via $\xi\delta$ monomials, but we can do it for the
operation $[U,V]=U\circ W-W\circ U$. Perform first $U\circ W$. This
composition of operations from $C^{(\bullet)}(A)$ is realised as application
of $\xi\delta$-monomial $A$ to the input (according to the procedure described
by pic.2), and then application of $\xi\delta$-monomial $B$ to the output of
the first operation. As an output of this composition we will get linear
combination of monomials from $A^{\otimes i}.$ We call such a monomial non-
essential if it is obtained as a result of gluing some letter $x_{i}$ from the
input of operation $A$ to some $\delta_{i}.$ Gluing letters $x_{j}$ from the
operation A itself (red arcs in pic.2) to the $\delta_{j}$ will result in
obtaining essential monomials in the output of composition $U\circ W$. Note
that the copy of the same monomial present in composition $U\circ W$ can be
essential or not, depending on how it is obtained. Thus to be essential is not
a property of the monomial, but it just characterises the way it got into the
output of the composition of these operations.

We claim that non-essential output monomials for the operations $U\circ W$ and
$W\circ U$ will be the same (with the same coefficients) and therefore they
will cancel out in the bracket $[U,W].$ This bracket is formed only by
essential outputs meaning exactly that it is described by the operation on
$\xi\delta$-monomials, described in picture 3. Namely, letters $x_{i}$ from
the monomial $A$ (red in pic. 3) are getting inserted in $\delta_{i}$ of the
second $\xi\delta$ word $B$ (and the other way around for $W\circ U$).
Insertions of this kind alone define $[A,B]$.

0

0

0

0

pic.3

The complete proof of this claim consists of consideration of 8 cases
depending on which combinations of $\xi$ and $\delta_{j}$ surrounds the place
of insertion as well as whether the two letters of the input involved in the
two compositions come from the same input or from different ones. ∎

We illustrate he proof by the following example.

Example

Consider the following two operations:

0

0

pic.4

Namely, let $A=\xi u\delta_{i}v\delta_{j}w\xi$ and $B=\delta_{3}y\delta_{4}z$.
As an input consider three monomials $w_{1}$, $w_{2}$ and $w_{3}$ of the form

$w_{1}=ax_{i}b,\ \ \ w_{2}=cx_{j}dx_{t}e,\ \ \ w_{3}=fx_{s}g.$

We fixed certain points in them, to construct coupling monomials which will
cancel in $[U,W]$. Consider first operation corresponding to a $\xi\delta$
monomial $A$. After inserting $x_{i}$ of $w_{1}$ and $x_{j}$ of $w_{2}$ into
$\delta_{i}$ and $\delta_{j}$ of $A$, we get three outputs $aw$, $udx_{t}e$
and $cvb$. After inserting $x_{t}$ from one of those outputs $udx_{t}e$, and
$x_{s}$ of $w_{3}$ into the operation corresponding to $B$, we get four
outputs for the composition $aw$, $cvb$, $udzg$ and $fye$. These are non-
essential monomials since we started from letter $x_{i}$ in the input $w_{1}$
of operation $U$, not from the internal letter in the $\xi\delta$-presentation
of operation $U$.

0

0

pic.5

Next, consider composition of operations $W\circ U$. If we insert $x_{s}$ of
$w_{3}$ and $x_{t}$ of $w_{2}$ into the operation corresponding to $B$, we get
two outputs $fye$ and $ucx_{j}dzg$. Now inserting $x_{i}$ of $w_{1}$, and
$x_{j}$ of the second output into the operation defined by $A$, we get the
same four outputs for the composition: $aw$, $cvb$, $udzg$ and $fye$.

0

0

pic.6

This example demonstrate that compositions in different order exhibit the same
non-essential output monomials. As a matter of fact, the same argument works
for every non-essential output monomial, so we can see that all non-essential
outputs cancel in $[U,W]=U\circ W-W\circ U$.

##  7 Homological purity and formality

###  7.1 Homological purity of the higher Hochschild complex

The goal of this section will be to prove homological purity of the complex
$\zeta=\zeta^{(N)}_{\mathbb{K}}={\rm Hom}\,_{\mathbb{K}}(R_{min}^{\otimes
N},A_{cycl}^{\otimes N})^{{\mathbb{Z}}_{N}}$ considered in previous sections.
It is a subcomplex of a complex $\tilde{\zeta}={\rm
Hom}\,_{\mathbb{K}}(R_{min}^{\otimes N},A_{cycl}^{\otimes N})$, which is the
version where we do not take invariants under ${\mathbb{Z}}_{N}$-action, or in
other words operations do have a fixed point. That is we have

$\begin{array}[]{ccc}\zeta&\subset&{\tilde{\zeta}}\\\ \cap&&\cap\\\
{\mathcal{H}}{\mathcal{C}}{\mathcal{H}}&\subset&{\mathcal{H}}{\mathcal{H}}.\\\
\end{array}$

The underlying set of this complex can be described as
$\tilde{\zeta}=\\{$monomials
$u\in\mathbb{K}\langle\xi,x_{i},\delta_{i}\rangle$, $i=\bar{1,r},$ starting
from $\xi$ or $\delta_{i}\\}_{\mathbb{K}}$. Whereas the underlying set of
$\zeta$, consists of certain linear combinations of arbitrary monomials
$u\in\mathbb{K}\langle\xi,x_{i},\delta_{i}\rangle$.

It will be instrumental in the proof of purity for $\zeta$. Our proof, which
is using Gröbner bases techniques in the ideals of free algebras, namely, for
an ideal generated by one element associated to a differential, can be
considered in a way as a construction of corresponding homotopy map.

We naturally have a bigrading on $\zeta=\oplus\zeta^{k}_{m}$: the grading by
$\delta$-degree, and by degree with respect to $\xi$ and $\delta_{i}$th,
$i=\bar{1,r}$, which we call weight. That is, $u\in\zeta^{k}_{m}$, if
$\hbox{\tt\rm deg}\,_{\xi,\delta}u=:w(u)=m,$ and $\hbox{\tt\rm
deg}\,_{\delta}u=:g(u)=k$. Essential for our considerations will be the
cohomological grading by $\xi$-degree: $\zeta=\oplus\zeta(l)$, where
$\zeta(l)=\mathop{\oplus}\limits_{m-k=l}\zeta^{k}_{m}.$

$\begin{array}[]{ccccccccc}&&\delta&&\delta\delta&&\delta\delta\delta&&\delta\delta\delta\delta\\\
&\nearrow&|&\nearrow&|&\nearrow&&\nearrow&\\\
\xi&&\xi\delta&&\xi\delta\delta&&\xi\delta\delta\delta&&\\\
&\nearrow&|&\nearrow&|&\nearrow&&&\\\
\xi\xi&&\xi\xi\delta&&\xi\xi\delta\delta&&&&\\\ &\nearrow&|&\nearrow&|&&&&\\\
\xi\xi\xi&&\xi\xi\xi\delta&&&&&&\\\ &\nearrow&|&&&&&&\\\
\xi\xi\xi\xi&&&&&&&&\end{array}$

If $\zeta_{m}$ is a subcomplex of $\zeta$, namely a slice consisting of
elements of weight $m$, then we will use also splitting
$\zeta_{m}=\mathop{\oplus}\limits_{s}\zeta_{m,s}$, where $s$ is an $x$-degree:
$u\in\zeta_{m,s},$ if $w(u)=m,$ and $\hbox{\tt\rm
deg}\,_{x_{1},...,x_{r}}(u)=s.$

The differential inherited from the complex ${\rm Hom}\,_{A^{\otimes
N}-mod-A^{\otimes N}}(\otimes_{i=1}^{N}A^{r_{i}},A^{\otimes N}_{cycl}),$
explained in section 5, in terms of $\xi\delta$-monomials boils down to the
following differential on $\tilde{\zeta}_{\mathbb{K}}$ (and on
$\zeta_{\mathbb{K}}$):

$d(u_{1}\xi u_{2}\xi\dots u_{n})=\sum(-1)^{g(u_{1}\xi
u_{2}\xi...u_{i})}u_{1}\xi u_{2}\xi\dots u_{i}\Delta u_{i+1}\dots u_{n}$

if $u_{1}\neq\varnothing$ ($u_{1}$ starting with $\delta_{i}$), here
$\Delta=\sum\limits_{i=1}^{r}\delta_{i}x_{i}-x_{i}\delta_{i}$, and

$d(\xi u_{1}\xi u_{2}\xi\dots u_{n})=\xi d(u_{1}\xi u_{2}\xi\dots
u_{n})+\sum_{i=1}^{r}[\delta_{i}x_{i}u_{1}\xi u_{2}\xi\dots
u_{n}-\delta_{i}u_{1}\xi u_{2}\xi\dots u_{n}x_{i}]$

if the monomial starts with $\xi.$ Here $u_{i}\in\mathbb{K}\langle
x_{i},\delta_{i}\rangle.$

######  Theorem 7.1.

The homology of the complex $\zeta(A)=(\zeta,d)$, $\zeta=\oplus\zeta^{k}_{m}$,
for $A=\mathbb{K}\langle x_{1},...,x_{r}\rangle,r\geqslant 2$ is sitting in
the diagonal $k=m$, consequently, the complex $\zeta=\oplus\zeta(l)$,
$\zeta(l)=\mathop{\oplus}\limits_{m-k=l}\zeta^{k}_{m}$ is pure, that is the
homology is sitting only in the last place of this complex with respect to
cohomological grading by $\xi$-degree. Thus the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex is pure as well.

###### Proof.

To prove this, we consider related complex $\widehat{\zeta}$ with the
following differential:

$d_{\widehat{\zeta}}(u_{1}\xi u_{2}\xi\dots u_{n})=\sum(-1)^{g(u_{1}\xi
u_{2}\xi...u_{i})}u_{1}\xi u_{2}\xi\dots u_{i}\Delta u_{i+1}\dots u_{n},$

where $\Delta=\sum\limits_{i=1}^{r}\delta_{i}x_{i}-x_{i}\delta_{i}$.

We first prove that homologies are sitting in one place in the complex
$(\widehat{\zeta},d_{\widehat{\zeta}})$ and then reduce the considerations for
$(\tilde{\zeta},d)$ to this. After that argument of Lemma 7.21 shows that for
the subcomplex $(\zeta,d)\subset(\tilde{\zeta},d)$ the homology is also
sitting in one place, if it is the case for $(\tilde{\zeta},d)$. Since $\zeta$
is quasi-isomorphis to the ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex
$C^{(\bullet)}(A)$ for smooth algebras, we get that $C^{(\bullet)}(A)$ is also
pure.

######  Theorem 7.2.

The $m$-th slice of the complex $(\widehat{\zeta},d_{\widehat{\zeta}})$ over
$A=\mathbb{K}\langle x_{1},...,x_{r}\rangle,r\geqslant 2,$

$\widehat{\zeta}_{m}=\\{u\in{\mathbb{K}}\langle\xi,x_{i},\delta_{i}\rangle:w(u)={\rm
deg}_{\xi,\delta}u=m\\}$

has non-trivial homology only in the last place with respect to cohomological
grading by $\xi$-degree.

###### Proof.

Induction by $m$. For arbitrary $m$ we will first need to consider the case of
cohomological degree one, that is the case of one $\xi$.

######  Lemma 7.3.

Consider the place in $(\widehat{\zeta}_{m},d_{\widehat{\zeta}})$, where ${\rm
deg}_{\xi}u=1$, $u\in\widehat{\zeta}_{m}$ (one but last place in the complexes
$\widehat{\zeta}_{m}$). Then the homology in this place is trivial.

###### Proof.

Let $d_{\widehat{\zeta}}(u)=0$ for $u\in\widehat{\zeta}$ with ${\rm
deg}_{\xi}u=1$. We show that $u\in{\rm Im}\,d_{\widehat{\zeta}}$. Since ${\rm
deg}_{\xi}u=1$, $u$ has the shape

$u=\sum a_{i}\xi b_{i},\ \ a_{i},b_{i}\in{\mathbb{K}}\langle
x_{1}\dots,x_{r},\delta_{1}\dots\delta_{r}\rangle.$

Then

$d_{\widehat{\zeta}}u=\sum(-1)^{g(a_{j})}a_{j}\Delta b_{j}=0.$

Consider the ideal $I$ in ${\mathbb{K}}\langle
x_{1}\dots,x_{r},\delta_{1}\dots\delta_{r}\rangle$ generated by $\Delta$:
$I={\rm Id}(\Delta)$.

We will use notions of Gröbner bases theory and the following lemma (version
of the Diamond lemma) to describe when this above equality might happen. We
remind them here.

######  Definition 7.4.

Monomials $u,v\in{\mathbb{K}}\langle Y\rangle$ form an ambiguity $(u,v),$ if
for some $w\in{\mathbb{K}}\langle Y\rangle$, $uw=wv$.

Suppose in ${\mathbb{K}}\langle Y\rangle$ we have fixed some admissible well-
ordering (that is, ordering compatible with multiplication, which satisfies
d.c.c.), for example, (left-to-right) degree-lexicographical ordering: we fix
an order on variables, say $y_{1}<...<y_{n}$, and compare monomials on $Y$
lexicographically (from left to right). Polynomials are compared by their
highest terms.

######  Definition 7.5.

Let $u,v$ be two monomials $u,v$, which are highest terms of the elements
$U,V$ from the ideal $I\in{\mathbb{K}}\langle
Y\rangle:U=u+\tilde{u},V=v+\tilde{v}$, where
$\tilde{u},\tilde{v}\in{\mathbb{K}}\langle Y\rangle$, smaller than
$u,v\in\langle Y\rangle$ respectively: $\tilde{u}<u,\tilde{v}<v$. Then the
resolution of the ambiguity $(u,v)$ formed by monomials $u,v$ is a polynomial
$Uw-wV=\tilde{u}w-w\tilde{v}$, which is reducible to zero modulo generators of
an ideal.

######  Definition 7.6.

A reduction on $\mathbb{K}\langle Y\rangle$ modulo generators of an ideal
$f_{i}=\bar{f}_{i}+\tilde{f}_{i}$, where $\bar{f}_{i}$ is a highest term of
$f_{i}$, is a collection of linear maps defined on monomials as follows:
$r_{u\bar{f}_{i}v}(w)=u\tilde{f}_{i}v$, if $w=u\bar{f}_{i}v$, and $w$
otherwise.

The polynomial is called reducible to zero if there exists a sequence of
reductions modulo generators of an ideal, which results in zero.

######  Lemma 7.7.

(Version of Diamond Lemma [29]) Let $A={\mathbb{K}}\langle
y_{1}\dots,y_{n}\rangle/{\rm Id}(r_{1},\dots,r_{m})$. Let $M$ be the syzygy
module of the relations $r_{1},\dots,r_{m}$, that is $M$ is the submodule of
the free ${\mathbb{K}}\langle y_{1}\dots,y_{n}\rangle$-bimodule generated by
the symbols $\widehat{r_{1}},\dots\widehat{r_{m}}$ consisting of $\sum
f_{i}\widehat{r_{s_{i}}}g_{i}$ such that $\sum f_{i}r_{s_{i}}g_{i}=0$.

Then $M$ is generated by trivial syzygies
$\widehat{r_{i}}ur_{j}-r_{i}u\widehat{r_{j}}$ and the syzygies obtained by
resolutions of ambiguities between highest terms of relations (with respect to
some ordering).

Let us fix the ordering $\delta_{1}>\delta_{2}>\dots>x_{1}>x_{2}>\dots$. Then
the leading term of the polynomial $\Delta$ is $\delta_{1}x_{1}$. It does not
produce any ambiguities. Hence by Lemma 7.7 (version of Diamond Lemma), the
corresponding syzygy module $M$ is generated by trivial syzygies, and
therefore

$(*)\quad\quad\sum a_{j}\widehat{\Delta}b_{j}=\sum
u_{k}(\widehat{\Delta}v_{k}\Delta-\Delta v_{k}\widehat{\Delta})w_{k}$

After we know this we can construct an element

$g=\sum{\gamma_{k}}u_{k}\xi v_{k}\xi w_{k}$

where $\gamma_{k}\in{\mathbb{C}}$ are chosen in such a way that in the
following sum all summands have positive signs

$d_{\widehat{\zeta}}(g)=\sum(u_{k}\xi v_{k}\Delta w_{k}-u_{k}\Delta v_{k}\xi
w_{k})$

We can see then that the latter expression is just the same as the above
formula $(*)$ with $\widehat{\Delta}$ substituted by $\xi$, hence

$\sum(u_{k}\xi v_{k}\Delta w_{k}-u_{k}\Delta v_{k}\xi w_{k})=\sum a_{j}\xi
b_{j}=u.$

And we finally have

$d_{\widehat{\zeta}}(g)=\sum a_{j}\xi b_{j}=u.$

∎

To continue the proof of Theorem 7.2, we need a basis of induction. So we
prove that in the complex $(\widehat{\zeta}_{2},d_{\widehat{\zeta}})$

$0\to\dots\xi\dots\xi\dots\to\dots\xi\dots\delta\dots\to\dots\delta\dots\delta\dots\to
0$

the homology is sitting in the last place.

Since we already have Lemma 7.3, which deals with the case of one $\xi$ it is
equivalent to proving exactness only in one term, where $\xi$-degree is equal
to two. That is, we need to show that if ${\rm deg}_{\xi}u=2$, ${\rm
deg}_{\delta_{i}}u=0$, and $d_{\widehat{\zeta}}(u)=0$, then $u=0$. Write
$u=\xi u_{0}+v$, where $v$ does not have $\xi$ on the first position. Then we
have

$0=d_{\widehat{\zeta}}(u)=\xi d_{\widehat{\zeta}}(u_{0})+\Delta
u_{0}+d_{\widehat{\zeta}}(v).$

The only term starting with $\xi$ is $\xi d_{\widehat{\zeta}}(u_{0})$, so
$d_{\widehat{\zeta}}(u_{0})=0$. Since ${\rm deg}_{\xi}u_{0}=1$, we are in
situation of Lemma 7.3 and $u_{0}=d_{\widehat{\zeta}}(v_{0})$. Since $u_{0}$
is free from $\delta_{i}$, we have $u_{0}=0$. Thus $u=v=\sum x_{i}u_{i}$ and
$0=d_{\widehat{\zeta}}(u)=\sum x_{i}d_{\widehat{\zeta}}(u_{i})$ implies
$d_{\widehat{\zeta}}(u_{i})=0$. Applying the same argument to $u_{i}$
repeatedly, we arrive at $u=0$, as required.

Step of induction in the proof of Theorem 7.2. Let ${\rm
deg}_{\xi,\delta_{i}}u=m$ and $u$ is homogeneous with respect to $\xi$ as well
as with respect to $x_{i},\xi,\delta_{i}$ and $u$ is not in the last term of
the complex: ${\rm deg}_{\xi}u\geqslant 1$. We need to show that $u\in{\rm
Im}(d_{\widehat{\zeta}})$ provided $d_{\widehat{\zeta}}(u)=0$. We present
$u=\xi u_{0}+v$, where $v$ is not starting from $\xi$. Then

$d_{\widehat{\zeta}}(u)=\Delta u_{0}+\xi
d_{\widehat{\zeta}}(u_{0})+d_{\widehat{\zeta}}(v)=0.$

The only term starting with $\xi$ can not cancel with anything, so
$d_{\widehat{\zeta}}(u_{0})=0$. Now ${\rm deg}_{\xi,\delta_{i}}(u_{0})=m-1$.
By induction hypothesis and Lemma 7.3 (if $m=2$),
$u_{0}=d_{\widehat{\zeta}}(w)$. Consider $d_{\widehat{\zeta}}(\xi w)=\Delta
w+\xi d_{\widehat{\zeta}}(w)$. Then

$u^{\prime}=u-d_{\widehat{\zeta}}(\xi w)=\xi u_{0}+v-\Delta w-\xi
d_{\widehat{\zeta}}(w)=v-\Delta w.$

Thus $u^{\prime}$ equals $u$ modulo ${\rm Im}\,d_{\widehat{\zeta}}$ and does
not have $\xi$ in the first position:

$u^{\prime}=\sum x_{j}\xi u_{j}+\sum\delta_{j}\xi v_{j}+v,$

where $\xi$ is absent from $v$ in the first two positions. Then

$0=d_{\widehat{\zeta}}(u^{\prime})=\sum x_{j}\Delta u_{j}-\sum\delta_{j}\Delta
v_{j}+\sum x_{j}\xi d_{\widehat{\zeta}}(u_{j})+\sum\delta_{j}\xi
d_{\widehat{\zeta}}(v_{j})+d_{\widehat{\zeta}}(v).$

Considering terms with $\xi$ in the second position, we deduce
$d_{\widehat{\zeta}}(u_{j})=d_{\widehat{\zeta}}(v_{j})=0$ for all $j$. By the
induction hypothesis $u_{j}=d_{\widehat{\zeta}}(w_{j})$ and
$v_{j}=d_{\widehat{\zeta}}(s_{j})$. Now $u^{\prime\prime}$ equals $u^{\prime}$
modulo ${\rm Imd_{\widehat{\zeta}}}$ and $u^{\prime\prime}$ has no $\xi$ in
the first two positions, where

$u^{\prime\prime}=u-d_{\widehat{\zeta}}(\sum x_{j}\xi w_{j}+\sum\delta_{j}\xi
s_{j}).$

After repeating this procedure, at the end we get $u=t\xi^{m}$ modulo ${\rm
Im}\,d_{\widehat{\zeta}}$, where $\hbox{\tt\rm deg}\,_{\xi}t=0$. Now
$0=d_{\widehat{\zeta}}(t\xi^{m})=td_{\widehat{\zeta}}(\xi^{m})$ and therefore
$t=0$ since
$d_{\widehat{\zeta}}(\xi^{m})=\Delta\xi^{m-1}+\dots+\xi^{m-1}\Delta\neq 0$.
Hence $u\in{\rm Im}\,d_{\widehat{\zeta}}$.

∎

Now we prove the theorem for $(\tilde{\zeta},d)$.

######  Theorem 7.8.

The $m$-th slice of the complex $(\bar{\zeta},d)$,

$\bar{\zeta}_{m}=\\{u\in{\mathbb{K}}\langle\xi,x_{i},\delta_{i}\rangle:w(u)={\rm
deg}_{\xi,\delta}u=m\\}$

for each $m\geqslant 2$ has non-trivial homology only in the last place with
respect to cohomological grading by $\xi$-degree.

###### Proof.

First we need preliminary exactness result for the case of one $\xi$.

######  Lemma 7.9.

Consider the place in $(\bar{\zeta}_{m},d)$ for any $m\geqslant 2$, where
${\rm deg}_{\xi}u=1$, $u\in\bar{\zeta}_{m}$ (one but last place in the
complex). Then the homology in this place is trivial.

###### Proof.

Let $u\in\bar{\zeta}$ be such that ${\rm deg}_{\xi}u=1$, ${\rm
deg}_{\xi,\delta_{i}}u\geqslant 2$ and $d_{\tilde{\zeta}}(u)=0$. We have to
show that $u\in{\rm Im}(d)$. Write

$u=\xi u_{0}+\sum a_{i}\xi b_{i},\ \ \text{$a_{i}\neq$const}.$

Then

$0=d(u)=\sum\delta_{i}[x_{i},u_{0}]+\sum(-1)^{\sigma}a_{i}\Delta b_{i}.$

Thus the following equality holds in $B={\mathbb{K}}\langle
x_{1}\dots,x_{r},\delta_{1}\dots\delta_{r}\rangle/{\rm Id}(\Delta)$:

$0=d(u)=\sum\delta_{i}[x_{i},u_{0}].$ (7.1)

######  Lemma 7.10.

The equality $\sum\delta_{i}[x_{i},u]=0$ in $B$ implies $[x_{i},u]=0$ in $B$
for any $i$.

###### Proof.

Let us consider ordering $x_{1}>x_{2}>...>\delta_{1}>\delta_{2}>...$, then
$\Delta$ forms a Gröbner basis. Take a normal form
$N([x_{i},u])\in{\mathbb{K}}\langle
x_{1}\dots,x_{r},\delta_{1}\dots\delta_{r}\rangle=\mathbb{K}\langle XD\rangle$
with respect to the Gröbner basis of the ideal ${\rm Id}(\Delta)$. In other
words, we present the element $[x_{i},u]\in{\mathbb{K}}\langle
x_{1}\dots,x_{r},\delta_{1}\dots\delta_{r}\rangle$ as a sum of monomials which
does not contain $x_{1}\delta_{1}$ as a submonomial. Then element
$N(\sum\delta_{i}[x_{i},u])=\sum\delta_{i}N([x_{i},u])=0$ in
$\mathbb{K}\langle XD\rangle$, hence $N[x_{i},u]=0$ in $\mathbb{K}\langle
XD\rangle$, which means $[x_{i},u]=0$ in $B$. ∎

######  Lemma 7.11.

(Centralizer) If in $B={\mathbb{K}}\langle
x_{1}\dots,x_{r},\delta_{1}\dots\delta_{r}\rangle/{\rm Id}(\Delta)$,
$r\geqslant 2$, $[u,x_{i}]=0$ for all $i$, then $u\in{\mathbb{K}}$.

###### Proof.

Fix the ordering $\delta_{1}>\delta_{2}>\dots>x_{1}>x_{2}>\dots$ The highest
term of $\Delta$ is $\delta_{1}x_{1}$. Then the set $N$ of corresponding
normal words (those which do not contain $\delta_{1}x_{1}$) is closed under
multiplication by $x_{2}$ on either side: $x_{2}N\subset N$ and $Nx_{2}\subset
N$.

Let $u\in A$ and $[u,x_{i}]=0$ for all $i$. As every element of $A$, $u$ can
be written as a linear combination of normal words: $u=\sum c_{j}w_{j}$, where
$w_{j}\in N$ are pairwise distinct and $c_{j}$ are non-zero constants. Then
$0=[u,x_{2}]=\sum c_{j}(w_{j}x_{2}-x_{2}w_{j})$. Since
$w_{j}x_{2},x_{2}w_{j}\in N$, the last equality holds if and only if it holds
in the free algebra. Hence $\sum c_{j}w_{j}$ commutes with $x_{2}$ in the free
algebra and therefore $u\in{\mathbb{K}}[x_{2}]$. The same holds for any other
$x_{j}$, $j\neq 1$ (they enter the game symmetrically) and therefore $u$ is in
the intersection of ${\mathbb{K}}[x_{j}]$ as subalgebras of $A$. Since this
intersection is ${\mathbb{K}}$, $u\in{\mathbb{K}}$. Note, that it is essential
here that we have at least two $x$th: $r\geqslant 2$. ∎

From (7.1), $[x_{j},u_{0}]=0$ in $B$ for all $i$, according to Lemma 7.10. By
the centralizer lemma $u_{0}$ is a constant in $B$. Since $m\geqslant 2$,
$u_{0}=0$ in $B$. Hence

$u_{0}=\sum s_{i}\Delta t_{i}$

in the free algebra. Thus

$u=\sum\xi s_{i}\Delta t_{i}+\sum a_{i}\xi b_{i}.$

Now we substitute $u$ with $u^{\prime}=u({\rm mod}\,{\rm Im(d)})$, where

$u^{\prime}=u-d(\sum(-1)^{{\rm deg}_{\delta}s_{i}}\xi s_{i}\xi t_{i}).$

After cancelations, we get

$u^{\prime}=\sum a_{i}\xi
b_{i}-\sum_{i,j}(-1)^{\sigma}\delta_{j}[x_{j},s_{i}\xi t_{i}]$

and therefore $u^{\prime}$ has no terms starting with $\xi$. Thus

$u^{\prime}=\sum\delta_{i}u_{i}$

and we fall into the situation of the differential $d_{\widehat{\zeta}}$ on
the complex $\hat{\zeta}$:

$d(u^{\prime})=\sum\delta_{i}d_{\widehat{\zeta}}(u_{i})\iff
d_{\widehat{\zeta}}(u_{i})=0.$

By Theorem 7.2, $u_{i}=d_{\widehat{\zeta}}(w_{i})$ and

$u^{\prime}=\sum\delta_{i}d_{\widehat{\zeta}}(w_{i})=d(-\sum\delta_{i}w_{i}),$

which yields that $u^{\prime}$ and therefore $u$ belongs to ${\rm Im}\,d$. ∎

Now let $\hbox{\tt\rm deg}\,_{\xi}u\geqslant 2.$ Suppose $du=0.$ We will show
that $u\in{\rm Im}\,d.$ As before present it as $u=\xi u_{0}+v,$ where $v$
does not start with $\xi$. Then $0=du=\xi
d_{\widehat{\zeta}}u_{0}+v^{\prime}$, where $v^{\prime}$ does not start with
$\xi$, hence $d_{\widehat{\zeta}}u_{0}=0$. By Theorem 7.2
$u_{0}=d_{\widehat{\zeta}}s$ for some $s$. Thus take $u^{\prime}=u-d(\xi
s)=u-\xi d_{\widehat{\zeta}}s-...=\xi d_{\widehat{\zeta}}s+v-\xi
d_{\widehat{\zeta}}s-...$, and we have a presentation of $u$ modulo the ideal
${\rm Im}\,d$ as an element with no $\xi$ at the first position:
$u=\sum\delta_{i}u_{i}$. Thus, $du=d_{\widehat{\zeta}}u$ and we can use
Theorem 7.2 to ensure that $u\in{\rm Im}d$. Indeed, since
$0=du=d_{\widehat{\zeta}}u$ and
$d_{\widehat{\zeta}}u=-\sum\delta_{i}d_{\widehat{\zeta}}u_{i},\quad
d_{\widehat{\zeta}}u_{i}=0$ for all $i$. Since $\hbox{\tt\rm
deg}\,_{\xi}u_{i}\geqslant 1$, by Theorem 7.2 we have
$u_{i}=d_{\widehat{\zeta}}(w_{i})$ for some $w_{i}$. Thus
$u=\sum\delta_{i}u_{i}=-\sum\delta_{i}d_{\widehat{\zeta}}w_{i}=d(\sum\delta_{i}w_{i})$.
The latter equality
$d(\sum\delta_{i}w_{i})=-\sum\delta_{i}d_{\widehat{\zeta}}w_{i}$ holds because
$\delta_{i}w_{i}$ not starting with $\xi$. So $u\in{\rm Im}d$, and this
completes the proof of Theorem 7.8.

∎

######  Lemma 7.12.

If the complex $(\tilde{\zeta},d)$ is homologically pure, and homology is
sitting in the last place w.r.t cohomological grading by $\xi$-degree, the
same is true for the complex $(\zeta,d)$.

###### Proof.

The statement about $\zeta$ follows from the fact the cyclization of the
complex commutes with the differential in our case. This in turn deduced from
the fact that the differential, given precisely at the beginning of this
section obviously commute with ${\mathbb{Z}}_{N}$ action. ∎

This lemma together with Theorem 7.8 completes the proof of Theorem 7.1.

∎

###  7.2 Homological purity of the higher Hochschild complex: quiver path
algebra case

We consider here the case when $A$ is the path algebra of the Kronecker
quiver, in stead of the case of free algebra. It allows to construct the
notions of noncommutative projective geometry, as explained in [27]. Then we
analogously proceed with the arbitrary quiver with at least two vertices.

The Gröbner bases theory, which in particular relates the generators of the
syzygy module with ambiguities on the relations, as in lemma 7.7 works not
only for ideals in free associative algebras, but in bigger generality of
algebras with multiplicative basis [15]. This includes path algebras of
quivers. Namely, the algebra $A$ should have a linear basis ${\mathcal{B}}$,
which is multiplicative in a sense that for any
$m_{1},m_{2}\in{\mathcal{B}},m_{1}\cdot_{A}m_{2}\in{\mathcal{B}}$ or
$m_{1}\cdot_{A}m_{2}=0$. Moreover, most essential (and not always present)
condition is that there should exist a well-ordering on ${\mathcal{B}}$
(ordering satisfying d.c.c.), which is compartible with multiplication, that
is for all $m_{1},m_{2},c\in{\mathcal{B}},m_{1}\leqslant m_{2}$ implies
$m_{1}\cdot_{A}c\leqslant m_{2}\cdot_{A}c$ and $c\cdot_{A}m_{1}\leqslant
c\cdot_{A}m_{2}.$ This ordering is extended to the whole algebra from the
basis ${\mathcal{B}}$ by saying that the element with the higher ’tip’ (the
highest element of ${\mathcal{B}}$, appearing with nonzero coefficient in the
linear decomposition of an element from the algebra) is higher.

Path algebra of a quiver ${\mathcal{Q}}$ is generated by ${\mathcal{Q}}_{0}$
\- the set of vertices of ${\mathcal{Q}}$ and ${\mathcal{Q}}_{1}$ \- the set
of arrows of ${\mathcal{Q}}$ subject to obvious relations. It has a
multiplicative linear basis ${\mathcal{B}}$ consisting of all pathes along the
quiver. The product of two pathes
$p,q\in{\mathcal{B}},\,p=v_{1}a_{1}...a_{n-1}v_{n},q=u_{1}b_{1}...b_{k-1}u_{k}$
is $p\cdot_{A}q=pq$, if $v_{n}=u_{1}$, and zero otherwise. The admissible
well-ordering on ${\mathcal{B}}$ can be as in the free associative algebra,
for example, deg-lex ordering, when some ordering on generators
${\mathcal{Q}}_{0}\cup{\mathcal{Q}}_{1}$ is fixed.

Lemma 7.7 in the setting of path algebras sounds as follows.

######  Lemma 7.13.

(Version of Diamond Lemma [15]) Let $A=P{\mathcal{Q}}/{\rm
Id}(r_{1},\dots,r_{m})$, where $P{\mathcal{Q}}$ is the path algebra of the
quiver ${\mathcal{Q}}$. Let $M$ be the syzygy module of the relations
$r_{1},\dots,r_{m}$, that is $M$ is the submodule of the free
${\mathcal{Q}}$-bimodule generated by the symbols
$\widehat{r_{1}},\dots\widehat{r_{m}}$ consisting of $\sum
f_{i}\widehat{r_{s_{i}}}g_{i}$ such that $\sum f_{i}r_{s_{i}}g_{i}=0$.

Now the proof of Theorem 7.1 will work for the case of path algebra of the
Kronecker quiver even with some simplifications. comparing to the case of free
algebras. For, example, the centraliser lemma 7.11 will get simplified, the
requirement that the number of arrows is bigger than two can be dropped in
this case.

We give now this version of the proof here. Let ${\mathcal{K}}_{r}$ be the
Kronecker quiver, namely the quiver with two vertices ($y=x_{0}$, $z=x_{r+1}$)
and $r$ arrows $x_{1},...,x_{r}$ form $y$ to $z$. Denote by
$P{\mathcal{K}}_{r}$ the path algebra of this quiver with generators
$x_{0}=y,x_{1},...,x_{r},x_{r+1}=z$. Consider the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex
$C^{(\bullet)}(P{\mathcal{K}}_{r})$, it will have a small quasi-isomorphic
(since $P{\mathcal{K}}_{r}$ is smooth) subcomplex $\zeta(P{\mathcal{K}}_{r})$,
defined in section 5. At some point in section 5, when we depicted the basis
of $\zeta(A)$ (pic.1), we started to work with free associative algebra $A$.
The new description of the $\xi\delta$ basis in the case of
$P{\mathcal{K}}_{r}$ one can obtain by saying that in pic.1 the monomials
$u_{i}$ are monomials on $\delta_{i},\delta^{\prime}_{i},i=0,r+1$ and the
linear basis of $P{\mathcal{K}}_{r}$
${\mathcal{B}}_{{\mathcal{K}}}=\\{yx_{i},yx_{i}z,x_{i}z\\}$ (or from the
linear basis of a path algebra $P{\mathcal{Q}}$, in case of an arbitrary
quiver ${\mathcal{Q}}$). In other words, $u_{i}$ are monomials from the free
product
$P{\mathcal{K}}_{r}*\mathbb{K}\langle\delta_{0},...,\delta_{r+1},\delta^{\prime}_{0},...,\delta^{\prime}_{r+1}\rangle.$
The differential in $\zeta(P{\mathcal{K}}_{r})$ in terms of
$\xi\delta$-monomials over $A=P{\mathcal{K}}_{r}$ will be:

$d(u_{1}\xi u_{2}\xi\dots u_{n})=\sum(-1)^{g(u_{1}\xi
u_{2}\xi...u_{i})}u_{1}\xi u_{2}\xi\dots u_{i}\Delta^{\prime}u_{i+1}\dots
u_{n}$

if $u_{1}\neq\varnothing$ ($u_{1}$ starting with $\delta_{i}$), here
$\Delta^{\prime}=\sum\limits_{i=0}^{r+1}\delta_{i}x_{i}-x_{i}\delta_{i}$, and

$d(\xi u_{1}\xi u_{2}\xi\dots u_{n})=\xi d(u_{1}\xi u_{2}\xi\dots
u_{n})+\sum_{i=0}^{r+1}[\delta_{i}x_{i}u_{1}\xi u_{2}\xi\dots
u_{n}-\delta_{i}u_{1}\xi u_{2}\xi\dots u_{n}x_{i}]$

if the monomial starts with $\xi.$ Here $u_{i}\in
P{\mathcal{K}}_{r}*\mathbb{K}\langle\delta_{0},...,\delta_{r+1},\delta^{\prime}_{0},...,\delta^{\prime}_{r+1}\rangle.$

######  Theorem 7.14.

The homology of the complex $\zeta(A)=(\zeta,d)$, $\zeta=\oplus\zeta^{k}_{m}$,
for $A=P{\mathcal{K}}_{r}$ is sitting in the diagonal $k=m$, consequently, the
complex $\zeta=\oplus\zeta(l)$,
$\zeta(l)=\mathop{\oplus}\limits_{m-k=l}\zeta^{k}_{m}$ is pure, that is the
homology is sitting only in the last place of this complex with respect to
cohomological grading by $\xi$-degree. Thus the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex over $A=P{\mathcal{K}}_{r}$
is pure as well.

###### Proof.

To prove this, we consider related complex $\widehat{\zeta}$ with the
following differential:

$d_{\widehat{\zeta}}(u_{1}\xi u_{2}\xi\dots u_{n})=\sum(-1)^{g(u_{1}\xi
u_{2}\xi...u_{i})}u_{1}\xi u_{2}\xi\dots u_{i}\Delta^{\prime}u_{i+1}\dots
u_{n},$

where
$\Delta^{\prime}=\sum\limits_{i=0}^{r+1}\delta_{i}x_{i}-x_{i}\delta_{i}$.

We first prove that homologies are sitting in one place in the complex
$(\widehat{\zeta},d_{\widehat{\zeta}})$ and then reduce the considerations for
$(\tilde{\zeta},d)$ to this. After that argument of Lemma 7.21 shows that for
the subcomplex $(\zeta,d)\subset(\tilde{\zeta},d)$ the homology is also
sitting in one place, if it is the case for $(\tilde{\zeta},d)$. Since $\zeta$
is quasi-isomorphis to the ${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex
$C^{(\bullet)}(A)$ for smooth algebras, we get that $C^{(\bullet)}(A)$ is also
pure.

######  Theorem 7.15.

The $m$-th slice of the complex $(\widehat{\zeta},d_{\widehat{\zeta}})$ over
$A=P{\mathcal{K}}_{r}$

$\widehat{\zeta}_{m}=\\{u\in{\mathbb{K}}\langle\xi,x_{i},\delta_{i}\rangle:w(u)={\rm
deg}_{\xi,\delta}u=m\\}$

has non-trivial homology only in the last place with respect to cohomological
grading by $\xi$-degree.

###### Proof.

Induction by $m$. For arbitrary $m$ we will first need to consider the case of
cohomological degree one, that is the case of one $\xi$.

######  Lemma 7.16.

Consider the place in $(\widehat{\zeta}_{m},d_{\widehat{\zeta}})$, where ${\rm
deg}_{\xi}u=1$, $u\in\widehat{\zeta}_{m}$ (one but last place in the complexes
$\widehat{\zeta}_{m}$). Then the homology in this place is trivial.

###### Proof.

Let $d_{\widehat{\zeta}}(u)=0$ for $u\in\widehat{\zeta}$ with ${\rm
deg}_{\xi}u=1$. We show that $u\in{\rm Im}\,d_{\widehat{\zeta}}$. Since ${\rm
deg}_{\xi}u=1$, $u$ has the shape

$u=\sum a_{i}\xi b_{i},\ \ a_{i},b_{i}\in
E=P{\mathcal{K}}_{r}*\mathbb{K}\langle\delta_{0},...,\delta_{r+1},\delta^{\prime}_{0},...,\delta^{\prime}_{r+1}\rangle.$

Then

$d_{\widehat{\zeta}}u=\sum(-1)^{g(a_{j})}a_{j}\Delta^{\prime}b_{j}=0.$

Consider the ideal $I$ in
$E=P{\mathcal{K}}_{r}*\mathbb{K}\langle\delta_{0},...,\delta_{r+1},\delta^{\prime}_{0},...,\delta^{\prime}_{r+1}\rangle$
generated by $\Delta^{\prime}$: $I={\rm Id}(\Delta^{\prime})$. To use the
Gröbner bases theory for $E$ we will present it as a quotient of the path
algebra of another quiver ${\mathcal{R}}$. Namely,
$E=P{\mathcal{K}}_{r}*\mathbb{K}\langle\delta_{0},...,\delta_{r+1},\delta^{\prime}_{0},...,\delta^{\prime}_{r+1}\rangle=P{\mathcal{R}}/{\rm
Id}(\delta_{i}^{\prime}=\delta_{i})$, where ${\mathcal{R}}$ is a quiver with
two vertices $x_{0}=y,x_{r+1}=z$, $r$ arrows between them and a rose in each
vertex, consisting of arrows $\delta_{0},...,\delta_{r+1}$ in vertex $x_{0}$,
and of arrows $\delta^{\prime}_{0},...,\delta^{\prime}_{r+1}$ in vertex
$x_{r+1}$. If we factor out relations identifying each $\delta_{i}$ with
$\delta_{i}^{\prime}$, we get exactly the free product
$P{\mathcal{K}}_{r}*\mathbb{K}\langle\delta_{0},...,\delta_{r+1},\delta^{\prime}_{0},...,\delta^{\prime}_{r+1}\rangle$.
Thus now we consider an ideal $I$ in the path algebra of a quiver
${\mathcal{R}}$, $F=P{\mathcal{R}}$ generated by two groups of relations:
$I={\rm Id}(\delta^{\prime}_{i}=\delta_{i},\Delta^{\prime})$, and use the
Gröbner bases theory in $F$ to show when the above equality may happen.

Let us fix the ordering
$\delta^{\prime}_{0}>\dots>\delta^{\prime}_{r+1}>\delta_{0}>\dots>\delta_{r+1}>\dots>x_{0}>x_{1}>\dots>x_{r+1}$.
Then the leading term of the polynomial $\Delta^{\prime}$ is
$\delta_{0}x_{0}$. It does not produce any ambiguities with itself, or with
relations $\delta^{\prime}_{i}=\delta_{i}$. The latter relations does not
produce any ambiguities between themselves either. Hence by Lemma 7.13
(version of Diamond Lemma), the corresponding syzygy module $M$ is generated
by trivial syzygies, and therefore

$(*)\quad\quad\sum a_{j}\widehat{\Delta}b_{j}=\sum
u_{k}(\widehat{\Delta}v_{k}\Delta^{\prime}-\Delta^{\prime}v_{k}\widehat{\Delta})w_{k}$

After we know this we can construct an element

$g=\sum{\gamma_{k}}u_{k}\xi v_{k}\xi w_{k}$

where $\gamma_{k}\in{\mathbb{C}}$ are chosen in such a way that in the
following sum all summands have positive signs

$d_{\widehat{\zeta}}(g)=\sum(u_{k}\xi
v_{k}\Delta^{\prime}w_{k}-u_{k}\Delta^{\prime}v_{k}\xi w_{k})$

We can see then that the latter expression is just the same as the above
formula $(*)$ with $\widehat{\Delta}$ substituted by $\xi$, hence

$\sum(u_{k}\xi v_{k}\Delta^{\prime}w_{k}-u_{k}\Delta^{\prime}v_{k}\xi
w_{k})=\sum a_{j}\xi b_{j}=u.$

And we finally have

$d_{\widehat{\zeta}}(g)=\sum a_{j}\xi b_{j}=u.$

∎

To continue the proof of Theorem 7.15, we need a basis of induction. So we
prove that in the complex $(\widehat{\zeta}_{2},d_{\widehat{\zeta}})$

$0\to\dots\xi\dots\xi\dots\to\dots\xi\dots\delta\dots\to\dots\delta\dots\delta\dots\to
0$

the homology is sitting in the last place.

Since we already have Lemma 7.16, which deals with the case of one $\xi$ it is
equivalent to proving exactness only in one term, where $\xi$-degree is equal
to two. That is, we need to show that if ${\rm deg}_{\xi}u=2$, ${\rm
deg}_{\delta_{i}}u=0$, and $d_{\widehat{\zeta}}(u)=0$, then $u=0$. Write
$u=\xi u_{0}+v$, where $v$ does not have $\xi$ on the first position. Then we
have

$0=d_{\widehat{\zeta}}(u)=\xi
d_{\widehat{\zeta}}(u_{0})+\Delta^{\prime}u_{0}+d_{\widehat{\zeta}}(v).$

The only term starting with $\xi$ is $\xi d_{\widehat{\zeta}}(u_{0})$, so
$d_{\widehat{\zeta}}(u_{0})=0$. Since ${\rm deg}_{\xi}u_{0}=1$, we are in
situation of Lemma 7.16 and $u_{0}=d_{\widehat{\zeta}}(v_{0})$. Since $u_{0}$
is free from $\delta_{i}$, we have $u_{0}=0$. Thus $u=v=\sum x_{i}u_{i}$ and
$0=d_{\widehat{\zeta}}(u)=\sum x_{i}d_{\widehat{\zeta}}(u_{i})$ implies
$d_{\widehat{\zeta}}(u_{i})=0$. Applying the same argument to $u_{i}$
repeatedly, we arrive at $u=0$, as required.

Step of induction in the proof of Theorem 7.15. Let ${\rm
deg}_{\xi,\delta_{i}}u=m$ and $u$ is homogeneous with respect to $\xi$ as well
as with respect to $x_{i},\xi,\delta_{i}$ and $u$ is not in the last term of
the complex: ${\rm deg}_{\xi}u\geqslant 1$. We need to show that $u\in{\rm
Im}(d_{\widehat{\zeta}})$ provided $d_{\widehat{\zeta}}(u)=0$. We present
$u=\xi u_{0}+v$, where $v$ is not starting from $\xi$. Then

$d_{\widehat{\zeta}}(u)=\Delta^{\prime}u_{0}+\xi
d_{\widehat{\zeta}}(u_{0})+d_{\widehat{\zeta}}(v)=0.$

The only term starting with $\xi$ can not cancel with anything, so
$d_{\widehat{\zeta}}(u_{0})=0$. Now ${\rm deg}_{\xi,\delta_{i}}(u_{0})=m-1$.
By induction hypothesis and Lemma 7.16 (if $m=2$),
$u_{0}=d_{\widehat{\zeta}}(w)$. Consider $d_{\widehat{\zeta}}(\xi
w)=\Delta^{\prime}w+\xi d_{\widehat{\zeta}}(w)$. Then

$u^{\prime}=u-d_{\widehat{\zeta}}(\xi w)=\xi u_{0}+v-\Delta^{\prime}w-\xi
d_{\widehat{\zeta}}(w)=v-\Delta^{\prime}w.$

Thus $u^{\prime}$ equals $u$ modulo ${\rm Im}\,d_{\widehat{\zeta}}$ and does
not have $\xi$ in the first position:

$u^{\prime}=\sum x_{j}\xi u_{j}+\sum\delta_{j}\xi v_{j}+v,$

where $\xi$ is absent from $v$ in the first two positions. Then

$0=d_{\widehat{\zeta}}(u^{\prime})=\sum
x_{j}\Delta^{\prime}u_{j}-\sum\delta_{j}\Delta^{\prime}v_{j}+\sum x_{j}\xi
d_{\widehat{\zeta}}(u_{j})+\sum\delta_{j}\xi
d_{\widehat{\zeta}}(v_{j})+d_{\widehat{\zeta}}(v).$

Considering terms with $\xi$ in the second position, we deduce
$d_{\widehat{\zeta}}(u_{j})=d_{\widehat{\zeta}}(v_{j})=0$ for all $j$. By the
induction hypothesis $u_{j}=d_{\widehat{\zeta}}(w_{j})$ and
$v_{j}=d_{\widehat{\zeta}}(s_{j})$. Now $u^{\prime\prime}$ equals $u^{\prime}$
modulo ${\rm Im}\,d_{\widehat{\zeta}}$ and $u^{\prime\prime}$ has no $\xi$ in
the first two positions, where

$u^{\prime\prime}=u-d_{\widehat{\zeta}}(\sum x_{j}\xi w_{j}+\sum\delta_{j}\xi
s_{j}).$

After repeating this procedure, at the end we get $u=t\xi^{m}$ modulo ${\rm
Im}\,d_{\widehat{\zeta}}$, where $\hbox{\tt\rm deg}\,_{\xi}t=0$. Now
$0=d_{\widehat{\zeta}}(t\xi^{m})=td_{\widehat{\zeta}}(\xi^{m})$ and therefore
$t=0$ since
$d_{\widehat{\zeta}}(\xi^{m})=\Delta^{\prime}\xi^{m-1}+\dots+\xi^{m-1}\Delta^{\prime}\neq
0$. Hence $u\in{\rm Im}\,d_{\widehat{\zeta}}$. ∎

Now we prove the theorem for $(\tilde{\zeta},d)$.

######  Theorem 7.17.

The $m$-th slice of the complex $(\bar{\zeta},d)$,

$\bar{\zeta}_{m}=\\{u\in{\mathbb{K}}\langle\xi,x_{i},\delta_{i}\rangle:w(u)={\rm
deg}_{\xi,\delta}u=m\\}$

for each $m\geqslant 2$ has non-trivial homology only in the last place with
respect to cohomological grading by $\xi$-degree.

###### Proof.

First we need preliminary exactness result for the case of one $\xi$.

######  Lemma 7.18.

Consider the place in $(\tilde{\zeta}_{m},d)$ for any $m\geqslant 2$, where
${\rm deg}_{\xi}u=1$, $u\in\tilde{\zeta}_{m}$ (one but last place in the
complex). Then the homology in this place is trivial.

###### Proof.

Let $u\in\tilde{\zeta}$ be such that ${\rm deg}_{\xi}u=1$, ${\rm
deg}_{\xi,\delta_{i}}u\geqslant 2$ and $d_{\tilde{\zeta}}(u)=0$. We have to
show that $u\in{\rm Im}(d)$. Write

$u=\xi u_{0}+\sum a_{i}\xi b_{i},\ \ \text{$a_{i}\neq$const}.$

Then

$0=d(u)=\sum\delta_{i}[x_{i},u_{0}]+\sum(-1)^{\sigma}a_{i}\Delta^{\prime}b_{i}.$

Thus the following equality holds in $G=P{\mathcal{R}}/{\rm
Id}(\Delta^{\prime},\delta^{\prime}_{i}=\delta_{i})$

$0=d(u)=\sum\delta_{i}[x_{i},u_{0}].$ (7.2)

######  Lemma 7.19.

The equality $\sum\delta_{i}[x_{i},u]=0$ in $G$ implies $[x_{i},u]=0$ in $G$
for any $i$.

###### Proof.

Let us consider ordering
$x_{0}>x_{1}>...>\delta^{\prime}_{0}>\delta^{\prime}_{1}>...>\delta_{0}>\delta_{1}>...$,
then $\Delta^{\prime}$ and relations $\delta^{\prime}_{i}=\delta_{i}$ form a
Gröbner basis. Take a normal form $N([x_{i},u])\in P{\mathcal{R}}$ with
respect to the Gröbner basis of the ideal ${\rm
Id}(\Delta^{\prime},\delta^{\prime}_{i}=\delta_{i})$. In other words, we
present the element $[x_{i},u]\in P{\mathcal{R}}$ as a sum of monomials which
does not contain $x_{0}\delta_{0}$ or $\delta^{\prime}_{i}$ as a submonomial.
Note that multiplication by $\delta_{i}$ from the left preserves normality of
the word. The element
$N(\sum\delta_{i}[x_{i},u])=\sum\delta_{i}N([x_{i},u])=0$ in
$F=P{\mathcal{R}}$, hence $N[x_{i},u]=0$ in $P{\mathcal{R}}$, which means
$[x_{i},u]=0$ in $G=P{\mathcal{R}}/{\rm
Id}(\Delta^{\prime},\delta^{\prime}_{i}=\delta_{i})$. ∎

######  Lemma 7.20.

(Centralizer) If in $G=P{\mathcal{R}}/{\rm
Id}(\Delta^{\prime},\delta^{\prime}_{i}=\delta_{i})$. $[u,x_{i}]=0$ for all
$i$, then $u\in{\mathbb{K}}$.

###### Proof.

Since in the Kronecker quiver we always have two vertices $x_{0},x_{r+1}$, and
they have a property that centraliser of each vertex in the path algebra is
only multipes of this vertex itself, it is enough to notice that $[u,x_{0}]=0$
implies $u=\alpha x_{0}$, $[u,x_{r+1}]=0$ implies $u=\beta x_{r+1}$, and since
$x_{0}\neq x_{r+1}$, we have that $u\in\mathbb{K}$. ∎

From (7.2), $[x_{j},u_{0}]=0$ in $G$ for all $i$, according to Lemma 7.19. By
the centralizer lemma $u_{0}$ is a constant in $G$. Since $m\geqslant 2$,
$u_{0}=0$ in $G$. Hence

$u_{0}=\sum s_{i}\Delta^{\prime}t_{i}$

in the free path algebra $P{\mathcal{R}}$. Thus

$u=\sum\xi s_{i}\Delta^{\prime}t_{i}+\sum a_{i}\xi b_{i}.$

Now we substitute $u$ with $u^{\prime}=u({\rm mod}\,{\rm Im(d)})$, where

$u^{\prime}=u-d(\sum(-1)^{{\rm deg}_{\delta}s_{i}}\xi s_{i}\xi t_{i}).$

After cancelations, we get

$u^{\prime}=\sum a_{i}\xi
b_{i}-\sum_{i,j}(-1)^{\sigma}\delta_{j}[x_{j},s_{i}\xi t_{i}]$

and therefore $u^{\prime}$ has no terms starting with $\xi$. Thus

$u^{\prime}=\sum\delta_{i}u_{i}$

and we fall into situation of the differential $d_{\widehat{\zeta}}$ on the
complex $\hat{\zeta}$:

$d(u^{\prime})=\sum\delta_{i}d_{\widehat{\zeta}}(u_{i})\iff
d_{\widehat{\zeta}}(u_{i})=0.$

By Theorem 7.2, $u_{i}=d_{\widehat{\zeta}}(w_{i})$ and

$u^{\prime}=\sum\delta_{i}d_{\widehat{\zeta}}(w_{i})=d(-\sum\delta_{i}w_{i}),$

which yields that $u^{\prime}$ and therefore $u$ belongs to ${\rm Im}\,d$. ∎

Now let $\hbox{\tt\rm deg}\,_{\xi}u\geqslant 2.$ Suppose $du=0.$ We will show
that $u\in{\rm Im}\,d.$ As before present it as $u=\xi u_{0}+v,$ where $v$
does not start with $\xi$. Then $0=du=\xi
d_{\widehat{\zeta}}u_{0}+v^{\prime}$, where $v^{\prime}$ does not start with
$\xi$, hence $d_{\widehat{\zeta}}u_{0}=0$. By Theorem 7.2
$u_{0}=d_{\widehat{\zeta}}s$ for some $s$. Thus take $u^{\prime}=u-d(\xi
s)=u-\xi d_{\widehat{\zeta}}s-...=\xi d_{\widehat{\zeta}}s+v-\xi
d_{\widehat{\zeta}}s-...$, and we have a presentation of $u$ modulo the ideal
${\rm Im}\,d$ as an element with no $\xi$ at the first position:
$u=\sum\delta_{i}u_{i}$. Thus, $du=d_{\widehat{\zeta}}u$ and we can use
Theorem 7.15 to ensure that $u\in{\rm Im}d$. Indeed, since
$0=du=d_{\widehat{\zeta}}u$ and
$d_{\widehat{\zeta}}u=-\sum\delta_{i}d_{\widehat{\zeta}}u_{i},\quad
d_{\widehat{\zeta}}u_{i}=0$ for all $i$. Since $\hbox{\tt\rm
deg}\,_{\xi}u_{i}\geqslant 1$, by Theorem 7.15 we have
$u_{i}=d_{\widehat{\zeta}}(w_{i})$ for some $w_{i}$. Thus
$u=\sum\delta_{i}u_{i}=-\sum\delta_{i}d_{\widehat{\zeta}}w_{i}=d(\sum\delta_{i}w_{i})$.
The latter equality
$d(\sum\delta_{i}w_{i})=-\sum\delta_{i}d_{\widehat{\zeta}}w_{i}$ holds because
$\delta_{i}w_{i}$ not starting with $\xi$. So $u\in{\rm Im}d$, and this
completes the proof of Theorem 7.17.

∎

The passing from the complex $\tilde{\zeta}$ to the complex $\zeta$ goes
exactly as before.

######  Lemma 7.21.

If the complex $(\tilde{\zeta},d)$ is homologically pure, and homology is
sitting in the last place w.r.t cohomological grading by $\xi$-degree, the
same is true for the complex $(\zeta,d)$.

This lemma together with Theorem 7.17 completes the proof of Theorem 7.14

∎

We considered here in details the case of the Kronecker quiver, since this
case produces the notions of noncommutative projective geometry, as it is
explained in [27]. But the same argument works in the case of the path algebra
of an arbitrary quiver as well. One should consider the initial quiver, where
additionally in each vertex there is a rose quiver with $n$ petals, where $n$
is a number of generators (vertices and arrows) of the initial path algebra.
Petals are labeled by $\delta_{i}^{j}$, $i=1,n$, $j$ stands for the number of
the vertex. Then we consider path algebra of this quiver, factorised by the
ideal generated by the relations $\delta_{i}^{j}=\delta_{i}^{k}$ for all
$j,k$. The resulting algebra is isomorphic to the free product of the
$P{\mathcal{Q}}*\mathbb{K}\langle\delta_{i}^{j}\rangle$, where ${\mathcal{Q}}$
is an initial quiver, and we apply the Gröbner bases theory for this algebra
exactly as before. The centralizer lemma for arbitrary quiver holds as soon as
quiver has at least two vertices, since then there exist a pair of vertices,
for which the intersection of their centralizers in the path algebra is
$\mathbb{K}$.

The resulting fact is that the theorem 7.14 holds for the path algebra of an
arbitrary quiver with at least two vertices.

######  Theorem 7.22.

Let $A$ be a path algebra of an arbitrary quiver ${\mathcal{Q}}$ with at least
two vertices, $A=P{\mathcal{Q}}$. The homology of the complex
$\zeta(A)=(\zeta,d)$, $\zeta=\oplus\zeta^{k}_{m}$, is sitting in the diagonal
$k=m$, consequently, the complex $\zeta=\oplus\zeta(l)$,
$\zeta(l)=\mathop{\oplus}\limits_{m-k=l}\zeta^{k}_{m}$ is pure, that is the
homology is sitting only in the last place of this complex with respect to
cohomological grading by $\xi$-degree. Thus the
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ complex over $A=P{\mathcal{Q}}$ is
pure as well.

Combinations of theorems 7.14 and 7.22 gives the result for path algebra
$A=P{\mathcal{Q}}$ of an arbitrary quiver ${\mathcal{Q}}$, except for the
quiver with one vertex and one loop.

###  7.3 Formality

The important consequence of the result on the homological purity of the
higher cyclic Hochschild complex is that we derive formality for these
complexes in $L_{\infty}$ sense [23]. Various aspects of formality have been
studied extensively (for example [11, 6, 7, 37, 33]), some of them are
famously difficult. The main point of [23] is that for the purposes of
deformation theory the weaker property of $L_{\infty}$ formality is what is
essential. It was shown there that not only two quasi-isomorphic DGLAs give
rise to the isomorphic deformation functors, but there is much weaker
equivalence relation on DGLAs, that of $L_{\infty}$-equivalence, which
actually defines the deformation functor.

######  Definition 7.23.

The DGLA $(C,d)$ is called formal if it is quasi-isomorphic to its
cohomologies $H^{\bullet}C.$

######  Definition 7.24.

The DGLA $(C,d)$ is $L_{\infty}-formal$ if it is $L_{\infty}$ quasi-isomorphic
to its cohomologies $(H^{\bullet}C,0)$, considered with zero differential,
that is there exists an $L_{\infty}$-morphism, which is a quasi-isomorphism of
complexes.

######  Theorem 7.25.

The higher cyclic Hochschild complex
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}(A)=C^{(\bullet)}(A)=\prod\limits_{N}C_{cycl}^{(N)}(A)$
is $L_{\infty}$-formal, for free algebra $A=\mathbb{K}\langle
x_{1},...x_{n}\rangle$ with at least two generators or path algebra
$P{\mathcal{Q}}$ of an arbitrary quiver ${\mathcal{Q}}$ with at least two
vertices.

In other words, the theorem holds for path algebra $P{\mathcal{Q}}$ of an
arbitrary quiver${\mathcal{Q}}$, except for the quiver with one vertex and one
loop.

###### Proof.

Remind that in the higher cyclic Hochschild complex
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}=\mathop{\prod}\limits_{N}C_{cycl}^{(N)}(A)$
we have the following grading and the subcomplex $\zeta$ quasi-isomorphic to
this complex is situated with respect to this grading in the following way:
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}=\mathop{\oplus}\limits_{i\in{\mathbb{Z}}}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}}{(i)}$,
where $i$ is a number of outputs minus number of inputs of corresponding
operation. Thus the embedding of $\zeta$ into
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$, described in section 5 place zero
component of $\zeta$ into zero component of
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$:
$\zeta(0)\subseteq{\mathcal{H}}{\mathcal{C}}{\mathcal{H}}(0)$, where $\zeta$
considered again with the cohomological grading by $\xi$-degree. Hence our
main Theorem 7.1 ensures that the homology of
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ is sitting in the zero place of the
grading. Let us consider the group action on
${\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ induced by scaling, namely,
${\mathbb{C}}^{*}$ acts by $\lambda(u)=\lambda^{m}u$ for
$u\in{\mathcal{H}}{\mathcal{C}}{\mathcal{H}}(m)$. This means that the action
uniquely defines the grading.

Now consider $(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},\infty)$,
the $L_{\infty}$-structure on the homologies obtained by the homotopy transfer
of Kadeishvili [20], constructive description of which is given in [22], one
can find explanations also in [38]. Since we had a reductive group action on
$({\mathcal{H}}{\mathcal{C}}{\mathcal{H}},0)$ this action can be pulled
through to $(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},\infty)$ and
so will be compatible with the new $L_{\infty}$-structure on
$H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}}$ again. Thus the grading
on $(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},\infty)$, being
defined by this action, is also natural, i.e. only zero component of it will
be nontrivial.

Obviously, if there is only one component in the grading of
$L_{\infty}$-algebra, then only one multiplication from $L_{\infty}$-structure
can be non-zero. Since we shown that homology
$(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},\infty)$ is sitting in
zero component only, and we are using convention where binary multiplication
in the infinity structure has degree zero, only multiplication $m_{2}$ can be
present. Thus in the $L_{\infty}$-structure of
$(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},\infty)$ $m_{n}=0$ for
$n\geqslant 3$, and this implies formality. Indeed, for formality we need to
show that $({\mathcal{H}}{\mathcal{C}}{\mathcal{H}},d)$ is quasi-isomorphic to
$(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},0)$. Since for the
$L_{\infty}$-structure obtained by the homotopy transfer it is always true
that $(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},\infty)$ is qiso to
$({\mathcal{H}}{\mathcal{C}}{\mathcal{H}},d)$, it is enough to show that
$(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},0)$ is qiso to
$(H^{\bullet}{\mathcal{H}}{\mathcal{C}}{\mathcal{H}},\infty)$, and this is
obviously the case when $m_{n}=0,n\geqslant 3.$

∎

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Natalia Iyudu, School of Mathematics, The University of Edinburgh, JCMB, The
King’s Buildings, Edinburgh, Scotland EH9 3FD

E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>
Maxim Kontsevich, Institut des Hautes Études Scientifiques, 35 route de
Chartres, F - 91440 Bures-sur-Yvette

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