2.1 Hochschild and negative cyclic homology

We denote by the category of chain complexes over k. We warn the reader that we use the homological grading instead of the cohomological grading used in our previous papers [Reference Bozec, Calaque and Scherotzke3, Reference Bozec, Calaque and Scherotzke4]. In particular, differentials have degree $-1$ , whereas mixed differentials have degree $+1$ . Apart from this change, throughout this paper we borrow the convention and notation from op. cit., to which we refer for more details. For instance, whenever is a model category, we write $\mathbf {M}$ for the corresponding $\infty $ -category obtained by localizing along weak equivalences.

A dg-category is a

-enriched category, and the category of dg-categories with dg-functors is denoted by

. We refer to [Reference Keller18, Reference Toën24] for a detailed introduction to dg-categories and their homotopy theory. The Hochschild chains $\infty $ -functor is then defined as

where $\mathcal {C}^e:=\mathcal {C}\otimes \mathcal {C}^{\mathrm {op}}$ . We write

for the i-th homology of

.

There is an explicit description of the derived tensor product $\mathcal {C}\underset {\mathcal {C}^e}{\overset {\mathbb {L}}{\otimes }}\mathcal {C}^{\mathrm {op}}$ , which uses the normalized bar resolution of $\mathcal {C}$ as a $\mathcal {C}$ -bimodule, and that leads to standard normalized Hochschild chains that we denote $\big (C_{*}(\mathcal {C}),b\big )$ :

$$\begin{align*}C_{*}(\mathcal{C})=\bigoplus_{\substack{n\geq0 \\ a_0,\dots, a_n\in\mathrm{Ob}(\mathcal{C})}} \mathcal{C}(a_n,a_0){\otimes}\bar{\mathcal{C}}(a_{n-1},a_n){\otimes}\cdots{\otimes} \bar{\mathcal{C}}(a_1,a_2){\otimes} \bar{\mathcal{C}}(a_0,a_1)[-n], \end{align*}$$

with $\bar {\mathcal {C}}(a,a')={\mathcal {C}}(a,a')$ if $a\neq a'$ and $\bar {\mathcal {C}}(a,a)={\mathcal {C}}(a,a)/k\cdot \mathrm {id}_a$ .

Hochschild chains carry a mixed structure (i.e., given on the standard normalized model by Connes’s B-operator). We refer to [Reference Bozec, Calaque and Scherotzke3, Reference Bozec, Calaque and Scherotzke4] and references therein for the homotopy theory of mixed complexes and explicit formulas.Footnote ¹ The negative cyclic complex of , denoted by , is defined as the homotopy fixed points of with respect to the mixed structure; it comes with a natural transformation . In concrete terms, is given by $\big (C_{*}(\mathcal {C})[\![u]\!],b-uB\big )$ , where u is a degree $-2$ variable.

We can view every dg-algebra with a finite set $(e_i)_{i\in I}$ of orthogonal nonzero idempotents such that $1=\sum _{i\in I}e_i$ is a dg-category with object set I. Conversely, we can associate to every dg-category $\mathcal {C}$ with finitely many objects its path algebra given by the complex

$$\begin{align*}A_{\mathcal{C}}:=\bigoplus_{(a, b) \in Ob(\mathcal{C})\times Ob(\mathcal{C})} \mathcal{C}(a, b) \end{align*}$$

with product given by composition of morphisms. The dg-algebra $A_{\mathcal {C}}$ is an R-algebra, where $R=\oplus _{c\in Obj(c)}ke_c$ . Note that the construction is in general not functorial, meaning that a functor does not necessarily give a morphism between the corresponding dg-algebras (unless the functor is injective on objects). This can be seen very easily in the following example, which will play an important role in the next section.

Example 2.1. The dg-category coproduct $k\coprod k$ is the dg-category given by two objects $1$ and $2$ and endomorphism ring $k=\mathrm {End}(1)$ respectively $k=\mathrm {End}(2)$ at each object, but zero Hom-spaces between the two objects. Hence, its path algebra $A_{ k \coprod k}$ is isomorphic to $k \oplus k$ . There is a dg-functor

$$\begin{align*}k \coprod k \to k \end{align*}$$

sending $1$ and $2$ to $pt$ , which denotes the only object of k, but there is no map of k-linear dg-algebras $ k \oplus k \to k$ .

Nevertheless, $\mathcal {C}$ and $A_{\mathcal {C}}$ are Morita equivalent, so that their Hochschild (resp. negative cyclic) homology is isomorphic. More precisely, we have an inclusion of mixed complexes $\big (C_{*}(\mathcal {C}),b,B\big )\hookrightarrow \big (C_{*}(A_{\mathcal {C}}),b,B\big )$ , which is a weak equivalence (here, we view $A_{\mathcal {C}}$ as a dg-category with one object).

2.2 Noncommutative forms

Consider a unital associative k-algebra A, along with a subalgebra R. We fix a complementary subspace $\bar A\simeq A/R$ of R. Denote by $d:A\to \bar A$ the associated quotient map. We will systematically use the $\bar {~}$ notation for the quotient by R. The graded algebra $\Omega _R^* A$ of noncommutative differential forms is defined as the quotient of $T_R(A\oplus \bar A[-1])$ by the relations

$$\begin{align*}a\otimes b=ab\qquad\text{and}\qquad d(ab)=a\otimes d(b)+d(a)\otimes b \end{align*}$$

for every $a,b\in A$ . It comes equipped with a mixed differential, that is the derivation induced by d and that we denote by the same symbol. The mixed differential d, descends to the Karoubi–de Rham graded vector space $\mathrm {DR}^*_R A:=\Omega ^*_R A/[\Omega ^*_R A,\Omega ^*_R A]$ , first introduced in [Reference Karoubi17].

In order to define a differential on $\Omega ^*_R A$ , turning it into a mixed complex, we consider the distinguished double derivation $E:a\mapsto a\otimes 1-1\otimes a$ , denoted by $\Delta $ in [Reference Crawley-Boevey, Etingof and Ginzburg9]. Recall that the A-bimodule of (R-linear) double derivations is defined as

$$\begin{align*}D_{A/R}:=\mathrm{Der}_R(A,A\otimes A)\simeq \Omega_R^1 A^{\vee}, \end{align*}$$

where the derivations are taken with respect to the outer A-bimodule structure on $A\otimes A$ , and the remaining A-bimodule structure on $D_{A/R}$ comes from the inner one on $A\otimes A$ . Here, $\Omega _R^1A$ is the kernel of the multiplication $A\otimes _R A\to A$ and inherits its A-bimodule structure from the outer one on $A\otimes A$ ; it is isomorphic to $A\otimes _R\bar A$ as a left A-module ( $1\otimes da\in A\otimes _R\bar A$ being identified with $E(a)\in \Omega _R^1A$ ). As a matter of notation, we will often write $\Omega _{A/R}:=\Omega _R^1 A$ .

There is an obvious graded algebra isomorphism $\Omega ^*_R A\simeq T_A(\Omega _R^1A[-1])$ , as well as a left A-module isomorphism $\Omega _R^n A\simeq A\otimes _R\bar A^{\otimes _R n}$ (see [Reference Cuntz and Quillen11]). For later purposes, we also introduce the graded algebra of polyvector fields $D^*_RA=T_A(D_{A/R}[-1])$ from [Reference Van den Bergh30].

Following [Reference Crawley-Boevey, Etingof and Ginzburg9], we define, for any R-linear double derivation $\delta \in D_{A/R}$ of A, a graded double derivation

$$\begin{align*}i_{\delta}:\Omega^*_R A\rightarrow \Omega^*_R A \otimes \Omega^*_R A \end{align*}$$

of $\Omega ^*_R A$ by setting

$$\begin{align*}i_{\delta}(a):=0\qquad \mathrm{and}\qquad i_{\delta}(da):=\delta(a) \end{align*}$$

for any $a\in A$ . On $\Omega _R^2A$ , we thus have, for instance,

$$\begin{align*}i_{\delta}(pdqdr)=p\delta(q)'\otimes\delta(q)"dr-pdq\delta(r)'\otimes\delta(r)"\in A\otimes\Omega_R^1A+\Omega_R^1A\otimes A, \end{align*}$$

where we use Sweedler’s sumless notation $\delta (a)=\delta (a)'\otimes \delta (a)"$ . The graded double derivation $i_{\delta }$ induces a linear contraction operator

$$\begin{align*}\iota_{\delta}:={}^{\circ} i_{\delta}:\Omega_R^* A\rightarrow\Omega_R ^{*-1}A, \end{align*}$$

where ${}^{\circ }(\alpha \otimes \beta )=(-1)^{kl}\beta \otimes \alpha $ for $\alpha \otimes \beta \in \Omega _R^kA\otimes \Omega _R^lA$ . Our differential will be given by the contraction operator $\iota _E:\Omega _R^* A\to \Omega _R ^{*-1}A$ , which has the following properties thanks to [Reference Crawley-Boevey, Etingof and Ginzburg9, Lemma 3.1.1]: it is explicitly given by the formula

$$\begin{align*}\iota_E(a_0da_1\dots da_n)=\sum_{l=1}^n(-1)^{(l-1)(n-1)+1}[a_l,da_{l+1}\dots da_na_0da_1\dots da_{l-1}]. \end{align*}$$

It vanishes on $[\Omega ^*_RA,\Omega ^*_RA]$ (and thus factors though $\mathrm {DR}_R^* A$ ), and it takes vales in $[\Omega ^*_RA,\Omega ^*_RA]^R$ (in particular, $\iota _E^2=0$ ), and $[\iota _E,d]=0$ . As a consequence, we obtain that $\big (\Omega ^*_RA,\iota _E,d)$ is a mixed complex.

2.3 Hochschild chains versus noncommutative forms

Below, we rephrase some constructions and results of [Reference Ginzburg and Schedler15] in terms of mixed complexes. Beware that the notation used here is not exactly the same as in op. cit.. For the moment, we only assume that A is a k-algebra.

Through the identification $C_*(A)\simeq \Omega ^*_kA$ , the Hochschild differential b reads as

$$\begin{align*}b(\alpha da)=(-1)^{|\alpha|}[\alpha,a]. \end{align*}$$

The Karoubi operator on $\Omega _k^* A$ , given by

$$\begin{align*}\kappa(\alpha da)=(-1)^{|\alpha|}da\alpha, \end{align*}$$

allows one to define a harmonic decomposition $\bar \Omega _k^* A=P\bar \Omega _k^* A\oplus P^{\perp }\bar \Omega _k^*A$ , where

$$\begin{align*}P\bar\Omega_k^* A=\ker(1-\kappa)^2\qquad\text{and}\qquad P^{\perp}\bar\Omega_k^*A=\mathrm{ima}(1-\kappa)^2.\end{align*}$$

The following identites hold:

$$\begin{align*}\iota_E=bN|_P\qquad\text{and}\qquad B=Nd|_P, \end{align*}$$

where N is the grading operator and B is the Connes mixed differential.

Hence, we have the following chain of morphisms of mixed complexes

(2.1)

such that, according to [Reference Ginzburg and Schedler15], $\overline {[d\Omega _k^*A,d\Omega _k^*A]}\hookrightarrow (\ker (P)[\![u]\!],\iota _E-ud)$ is a quasi-isomorphism, where u is a degree $-2$ formal variable, $N!$ is an isomorphism and the rightmost inclusion is a quasi-isomorphism. We thus get a quasi-isomorphism

$$\begin{align*}\left(\dfrac{\bar\Omega_k^* A[\![u]\!]}{\overline{[d\Omega_k^*A,d\Omega_k^*A]}},\iota_E-ud\right) \longrightarrow (\bar\Omega_RA[\![u]\!],b-uB), \end{align*}$$

and the homology of both complexes yields the reduced negative cyclic homology $\overline {\mathrm {HC}}^-(A)$ .

Hence, when $A=A_{\mathcal {C}}$ , for $\mathcal {C}$ a genuine k-linear category with a finite set I of objects, and $R=\oplus _{i\in I}ke_i$ , we have a zig-zag

where only the last bottom arrow may not be a quasi-isomorphism.

2.4 Computations for $A=k[x^{\pm 1}]$

As a matter of convention, we always mean $(dx)y$ if no brackets appear in $dxy$ . We want to find a harmonic cyclic lift for $\alpha _1:=x^{-1}dx\in \bar \Omega ^1A$ which is closed for the mixed structure $(P\bar \Omega ,\iota _E,d)$ . That means that A is $1$ -pre-Calabi–Yau according to the terminology of [Reference Bozec, Calaque and Scherotzke3]. This was already proved in [Reference Bozec, Calaque and Scherotzke4] using the standard normalized Hochschild complex, but we reprove it here on the ‘de Rham side’ and check consistency afterwards to illustrate (2.1).

Set $\alpha _n=(x^{-1}dx)^{2n-1},\beta _n=\kappa (\alpha _n)=(dxx^{-1})^{2n-1}\in \bar \Omega ^{2n-1}A$ . Then

$$\begin{align*}\kappa(\beta_n)=\kappa(-\beta_{n-1}dxdx^{-1})=-dx^{-1}\beta_{n-1}dx=\alpha_n.\end{align*}$$

Hence, $\alpha _n+\beta _n\in P\bar \Omega A$ and $\alpha _n-\beta _n=\frac {1}{2}(1-\kappa )^2(\alpha _{n})\in P^{\perp }\bar \Omega A$ . Then

$$ \begin{align*} \iota_E\alpha_n&=\frac{1}{2}(2n-1)b(\alpha_n+\beta_n)\\ &=\frac{1}{2}(2n-1)([\alpha_{n-1}x^{-1}dxx^{-1},x]-[\beta_{n-1}dx,x^{-1}])\\ &=\frac{1}{2}(2n-1)(x^{-1}\beta_{n-1}dx+\alpha_{n-1}x^{-1}dx-\beta_{n-1}dxx^{-1}-x\alpha_{n-1}x^{-1}dxx^{-1})\\ &=(2n-1)((x^{-1}dx)^{2n-2}-(dxx^{-1})^{2n-2}). \end{align*} $$

However, $d\alpha _1=-(x^{-1}dx)^2$ , and if we assume $d\alpha _{n-1}=-(x^{-1}dx)^{2n-2}$ , we get

$$ \begin{align*} d\alpha_n&=d(x^{-1}dx(x^{-1}dx)^{2n-2})\\ &=d(x^{-1}dx)(x^{-1}dx)^{2n-2}-x^{-1}dxd((x^{-1}dx)^{2n-2})\\ &=-x^{-1}dxx^{-1}dx(x^{-1}dx)^{2n-2}-x^{-1}dxd^2\alpha_{n-1}\\ &=-(x^{-1}dx)^{2n}. \end{align*} $$

Similarly, $d\beta _n=(dxx^{-1})^{2n}$ for all n. Thus, as $\iota _E\alpha _n=\iota _E\beta _n$ ,

$$\begin{align*}\iota_E(\alpha_n+\beta_n)=2\iota_E\alpha_n=-2(2n-1)d(\beta_{n-1}+\alpha_{n-1}).\end{align*}$$

As a consequence, $(\iota _E-ud)(\gamma )=0$ , where $\gamma _k=\frac {1}{2}(\alpha _k+\beta _k)\in P\bar \Omega ^{2k-1}k[x^{\pm 1}]$ and

$$\begin{align*}\gamma=\sum_{k\ge0}\dfrac{k!}{(2k+1)!}(-u)^k\gamma_{k+1},\end{align*}$$

where u is a formal degree $-2$ variable.

Let us check now that this is coherent with [Reference Bozec, Calaque and Scherotzke4]. Through (2.1) and the isomorphism $\Omega ^nA\simeq A\otimes \bar A^{\otimes n}$ , $\gamma $ is mapped to

$$\begin{align*}\sum_{k\ge0}{k!}u^k\dfrac{(x^{-1}\otimes x)^{\otimes(k+1)}-(x\otimes x^{-1})^{\otimes(k+1)}}{2}\end{align*}$$

as

$$ \begin{align*} \alpha_{k+1}&=(x^{-1}dx)^{2k+1}=(-1)^kx^{-1}(dxdx^{-1})^kdx,\\ \beta_{k+1}&=(dxx^{-1})^{2k+1}=(-1)^{k+1}x(dx^{-1}dx)^kdx^{-1},\\ \text{and }\gamma_{k+1}&\in P\bar\Omega^{2k+1}, \end{align*} $$

all of which is consistent with [Reference Bozec, Calaque and Scherotzke4, 3.1.1].

3.1 Fusion as a pushout

Recall that Van den Bergh defines in [Reference Van den Bergh30] the fusion algebra which identifies two pairwise orthogonal idempotents. We use the notation $(-)^+$ instead of $\overline {(-)}$ as in [Reference Van den Bergh30] since it is already used.

One can see that ${A}^+$ is isomorphic to $A \coprod _{R} {R^+}$ and that $R^+=M_2(k)\oplus R_{\geq 3}$ and $R^f=ke_1\oplus R_{\geq 3}$ , where $R_{\geq 3}:=ke_3\oplus \cdots \oplus ke_n$ .

Now, let $\mathcal {C}$ be a dg-category with a finite set of objects $I=\{1,\dots ,n\}$ , $n\ge 2$ . We define

$$\begin{align*}\mathcal{C}^f:=\mathcal{C} \coprod_{k\coprod k} k, \end{align*}$$

where the functor $k\coprod k\to \mathcal {C}$ is given by the units of the first two objects $1$ and $2$ . Note that the strict pushout is (categorically equivalent to) a homotopy pushout.

Examples 3.2. (1) The category $(k[x]\amalg k[y])^f$ (when defined using the strict pushout) is isomorphic to $k\langle x,y\rangle $ . Similarly, $(k[x^{\pm 1}]\amalg k[y^{\pm 1}])^f$ is isomorphic $k\langle x^{\pm },y^{\pm 1} \rangle $ . As a consequence, we get that

$$\begin{align*}\mathcal{C}^f\simeq \mathcal{C} \coprod_{k[x_1^{\square}] \coprod k[x_2^{\square}]} k\langle x_1^{\square},x_2^{\square} \rangle, \end{align*}$$

where $\square \in \{\emptyset ,\pm 1\}$ and $k[x_i^{\square }] \to \mathrm {End}_{\mathcal {C}}(i)$ .

(2) If $\mathcal R= \coprod _{i\in I} k$ , then $\mathcal R^f= k \amalg \mathcal R_{\geq 3}$ , where $\mathcal R_{\geq 3}:=\coprod _{i\geq 3} k$ . As a consequence, we get that

$$\begin{align*}\mathcal{C}^f:=\mathcal{C} \coprod_{\mathcal R} \big( k\amalg \mathcal R_{\ge3}\big), \end{align*}$$

where the functor $\mathcal R \rightarrow \mathcal {C}$ is uniquely determined by mapping the object of the i-th copy of k to i, and the functor $\mathcal R \to k\amalg \mathcal R_{\ge 3}$ maps the first two objects of $\mathcal R$ to the object of the first copy of k.

Proof. We can assume without loss of generality that $\mathcal {C}$ has only two objects $1$ and $2$ . We denote $e_1$ and $e_2$ their respective identity map. The dg-category $\mathcal {C} \coprod _{k\coprod k} k$ has exactly one object which we denote $pt$ . Let us show that the endomorphism ring $B:=\mathrm {End}(pt)$ is isomorphic to the fusion algebra $A^f$ of $A:=A_{\mathcal {C}}$ . By the pushout property, there are algebra homomorphisms

$$ \begin{align*} f:\mathrm{End}_{\mathcal{C}}(1) &\simeq e_1 A e_1 \to B\\ g:\mathrm{End}_{\mathcal{C}}(2) &\simeq e_2 A e_2 \to B, \end{align*} $$

and bimodule morphisms $ e_1A e_2 \simeq \mathcal {C}(2, 1) \to B, e_1 a e_2 \mapsto e_1 a e_{21}$ and $e_2A e_1 \simeq \mathcal {C}(1, 2) \to B, e_2 a e_1 \mapsto e_{12} a e_1$ such that

commutes. The algebra homomorphism $k \to B$ is then uniquely determined.

We have injective algebra morphisms $ \mathrm {End}_{\mathcal {C}}(1) \simeq e_1 A e_1 \to A^f, a \mapsto a$ , $\mathrm {End}_{\mathcal {C}}(2) \simeq e_2 A e_2 \to A^f, a \mapsto e_{12} a e_{21}$ . Similarly, we have injective morphisms of bimodules $\mathcal {C}(2,1) \simeq e_1 A e_2 \to A^f, a \mapsto a e_{21} $ and $\mathcal {C}(1,2) \simeq e_2 A e_1 \to A^f, a \mapsto e_{12} a$ compatible with the composition of morphisms. Hence, we obtain a unique injective algebra homomorphism $B \to A^f$ . As the image of the above maps generates $A^f$ , this morphism is also surjective, and hence, $B=A_{\mathcal {C}^f} \simeq A^f$ .

3.2 Trace maps

Acccording to Van den Bergh [Reference Van den Bergh30], we consider the following situation: an R-algebra A and an idempotent e in R such that $ReR=R$ . One writes $1= \sum _i p_i e q_i$ with $p_i, q_i\in R$ and defines a trace map

$$\begin{align*}\mathrm{Tr}:A\rightarrow eAe~;~a\mapsto\sum_ieq_iap_ie.\end{align*}$$

We recall a series of standard results, for which we provide full proofs for the sake of completeness; the main point is to be able to describe the trace map on $\Omega _RA$ and $\mathrm {DR}_RA$ .

Proof. First of all, the trace map $\mathrm {Tr}$ sends commutators to commutators. Indeed,

$$ \begin{align*} \mathrm{Tr}(ab-ba) & = \sum_{i}(eq_iabp_ie-eq_ibap_ie) \\ & = \sum_{i,j}eq_iap_jeq_jbp_ie-eq_ibp_jeq_jap_ie \\ & = \sum_{i,j}eq_iap_jeq_jbp_ie-eq_jbp_ieq_iap_je\in[eAe,eAe]. \end{align*} $$

Then, one can check that it is a k-linear inverse modulo commutator, to the algebra morphism $eAe\to A$ . Indeed, on the one hand, $a=\sum _ip_ieq_i a=\mathrm {Tr}(a)~\mathrm {mod}~[A,A]$ , and on the other hand, $eae=\sum _iep_ieq_i eae=\mathrm {Tr}(eae)~\mathrm {mod}~[eAe,eAe]$ . Since the morphism $eAe\to A$ does not depend on the decomposition of $1$ , its inverse (modulo commutator) does not either.

Lemma 3.5. For any two A-bimodules M and N, the canonical morphism $Me \otimes _{eRe} eN \to M \otimes _R N$ of A-bimodules is inversible with the inverse given by

$$\begin{align*}\Psi_{M,N}:M \otimes_R N \to Me \otimes_{eRe} eN~;~m \otimes n \mapsto \sum_i mp_i e \otimes eq_i n. \end{align*}$$

Proof. Let us check that it is well-defined. Consider $r\in R$ and write $r= \sum _j h_j e l_j$ for some $h_j, l_j \in R$ . Then

$$ \begin{align*} \Psi_{M,N} (mr \otimes n ) & = \sum_imrp_i e \otimes eq_i n = \sum_{i,j} m h_j e l_j p_i e \otimes eq_i n\\ &= \sum_{i,j} m h_je \otimes e l_j p_i eq_i n = \sum_{j} m h_je \otimes e l_j n\\ &= \sum_{i,j} m p_i e q_i h_je \otimes e l_j n = \sum_{i,j} m p_i e \otimes e q_i h_je l_j n \\ &=\sum_i m p_i e \otimes e q_i r n = \Psi_{M,N}( m \otimes r n ). \end{align*} $$

We finally observe that $\Psi _{M,N}$ is an inverse to the canonical morphism $Me \otimes _{eRe} eN \to M \otimes _R N$ . Indeed, in $M \otimes _R N$ , $\sum _i mp_i e \otimes eq_i n=\sum _i m\otimes p_i e q_i n=m\otimes n$ , and in $Me \otimes _{eRe} eN$ , $\sum _i mep_i e \otimes eq_i en=\sum _i me\otimes p_i e q_i e n=me\otimes en$ .

As a matter of notation, we introduce $\Psi _M:=\Psi _{M,M}$ .

Lemma 3.6. The isomorphism $\Psi _{\Omega _{A/ R}}$ induces an isomorphism $e(\Omega _RA)e \simeq \Omega _{eRe}(eAe)$ , through which the trace map of $\Omega _RA$ reads as follows:

$$ \begin{align*} \mathrm{Tr}: \Omega_RA& \rightarrow e(\Omega_RA)e \simeq \Omega_{eRe}(eAe) \\ a_0da_1\dots da_m&\longmapsto \sum_{i_0,\dots,i_m}eq_{i_0}a_0p_{i_1}ed(eq_{i_1}a_1p_{i_2}e)\dots d(eq_{i_m}a_mp_{i_0}e). \end{align*} $$

Moreover, it induces a k-linear isomorphism

$$\begin{align*}\mathrm{Tr}: \mathrm{DR}_R ( A ) \to \mathrm{DR}_{eRe}( eAe ) \end{align*}$$

that does not depend on the decomposition $1=\sum _i p_ieq_i$ .

Proof. Thanks to the previous lemma, the isomorphism $\Psi _{\Omega _{A/ R}}$ induces an isomorphism of tensor algebras $e( T_{A} \Omega _{A/R} )e\simeq T_{eAe} ( e\Omega _{A/ R} e) $ . Using $\Psi _A$ , we also have

$$ \begin{align*} \Omega_{eAe/eRe}&=\ker(eAe\otimes_{eRe}eAe\rightarrow eAe)\\ &\simeq\ker(eA\otimes_{R}Ae\rightarrow eAe)\\ &=e\ker(A\otimes_{R}A\rightarrow A)e\\ &=e\Omega_{A/R}e. \end{align*} $$

Combining these, we get

$$\begin{align*}e (\Omega_RA) e := e( T_{A} \Omega_{A/R} )e\simeq T_{eAe} ( e\Omega_{A/ R} e)\simeq T_{eAe} \Omega_{eAe/ eRe} = :\Omega_{eRe}(eAe). \end{align*}$$

Through this identification, an element $edae=ea\otimes e-e\otimes ae\in e\Omega _{A/R}e$ becomes, in $\Omega _{eAe/eRe}$ ,

$$\begin{align*}\sum_ieap_ie\otimes eq_ie-ep_ie\otimes eq_iae=eae\otimes e-e\otimes eae=:d(eae)\in \Omega_{eAe/eRe}. \end{align*}$$

Thus, the trace map reads

$$ \begin{align*} \Omega_RA\ni a_0da_1\dots da_m&\mapsto\sum_{i_0}eq_{i_0}a_0da_1\dots da_mp_{i_0}e\in e(\Omega_RA)e\\ &\mapsto\sum_{i_0,i_1,\dots,i_m}eq_{i_0}a_0p_{i_1}ed(eq_{i_1}a_1p_{i_2}e)\dots d(eq_{i_m}a_mp_{i_0}e)\in \Omega_{eRe}(eAe). \end{align*} $$

The last part of the claim follows from lemma 3.4.

3.3 Functoriality

We now apply the constructions from the previous section 3.2 to the idempotent $\epsilon =1-e_2$ of ${R^+}$ (see definition 3.1), where $1=\epsilon \epsilon \epsilon +e_{21}\epsilon e_{12}$ . Precomposing with the algebra morphism $A\to {A^+}$ , we get maps $\Omega _RA\to \Omega _{R^f}A^f$ and $ \mathrm {DR}_R(A) \to \mathrm {DR}_{R^f}A^f$ that we denote by $(-)^f$ . Since $\epsilon e_{12}=e_{12}$ and $e_{21}\epsilon =e_{21}$ , we have $\mathrm {Tr}(a)= \epsilon a \epsilon + e_{12} a e_{21}$ for all $a\in {A^+}$ . Actually, the trace map in this situation also has a simpler expression on forms.

Lemma 3.7. On $\Omega _{{A^+}/{R^+}}$ , we have

$$\begin{align*}\mathrm{Tr}(adb )= \epsilon a d b \epsilon + e_{12} ad b e_{21},\end{align*}$$

and dually, we have a trace map on double derivations

$$\begin{align*}\mathrm{Tr}:D_{{R^+}}{A^+}\to D_{R^f}A^f~,~\delta\mapsto \epsilon \delta \epsilon + e_{12} \delta e_{21}.\end{align*}$$

More generally, if $\omega \in \Omega _{{R^+}}{A^+}$ , we have $\mathrm {Tr}(\omega )=\epsilon \omega \epsilon + e_{12} \omega e_{21}$ .

Proof. Thanks to lemma 3.6, we have on $1$ -forms

$$ \begin{align*} \mathrm{Tr}(adb )&= \epsilon a \epsilon d (\epsilon b \epsilon)+e_{12} a \epsilon d(\epsilon b e_{21}) + e_{12} a e_{21} d (e_{12} b e_{21})+ \epsilon a e_{21}d ( e_{12} b \epsilon) \\ &= \epsilon a \epsilon d b \epsilon+e_{12} a \epsilon d b e_{21}+ e_{12} a e_{2} d b e_{21}+ \epsilon a e_{2}d b \epsilon. \end{align*} $$

If $a\in Ae_2$ and $b\in e_2A$ , as $\epsilon e_2=e_2\epsilon =0$ , we get

$$\begin{align*}\mathrm{Tr}(adb )= e_{12} ad b e_{21}+ \epsilon a d b \epsilon.\end{align*}$$

If $a\in Ae_i$ and $b\in e_iA$ for some $i\neq 2$ , as $\epsilon e_i=e_i\epsilon =e_i$ , we again have

$$\begin{align*}\mathrm{Tr}(adb )= \epsilon a d b \epsilon + e_{12} ad b e_{21}.\end{align*}$$

It generalizes to all forms.

We go back to the context of a dg-category $\mathcal {C}$ with a finite set of objects I and set $A:=A_{\mathcal {C}}$ . We define idempotents $e_i=\mathrm {id_i}$ and set $R=\oplus _{i\in I}ke_i$ , a subalgebra of A. Recall that $R^f\simeq \oplus _{i\neq 2}ke_i$ and consider the k-linear map $C_{*}(\mathcal {C})\to \Omega ^*_R A$ given by

$$\begin{align*}a_0\otimes a_1\otimes\dots\otimes a_m\mapsto a_0da_1\dots da_m. \end{align*}$$

Since there is a functor $\mathcal {C}\to \mathcal {C}^f$ , we have a natural map $\nu :C_{*}(\mathcal {C})\to C_{*}(\mathcal {C}^f)$ .

Lemma 3.8. The following diagram commutes:

Proof. Thanks to lemma 3.6, the map $ \Omega _R^*(A) \to \Omega _{R^f}^*(A^f)$ is given by

$$\begin{align*}(a_0d a_1\cdots a_m)^f=\sum_{i_0,\dots,i_m}q_{i_0}a_0p_{i_1}d(q_{i_1}a_1p_{i_2})\dots d(q_{i_m}a_mp_{i_0}). \end{align*}$$

Since $p_{i_j}\epsilon =p_{i_j}$ and $\epsilon q_{i_j}=q_{i_j}$ in our situation, that is either $p_{i_j}=\epsilon =q_{i_j}$ or $p_{i_j}=e_{21},q_{i_j}=e_{12}$ . Now, if $a_0\otimes \cdots a_m$ belongs to the Hochschild complex of $\mathcal {C}$ , then these elements are completely determined by the $a_j$ ’s. Indeed, if $a_j\in \mathcal {C}(x_{j+1},x_j)$ , then $q_{i_j}=\epsilon $ whenever $x_j\neq 2$ and $p_{i_{j+1}}=\epsilon $ whenever $x_{j+1}\neq 2$ .

From the proof of proposition 3.3, we have that $\mathcal {C}(x,y)\to A^f$ is given by $a\mapsto q a p$ , with

• • $q=\epsilon $ if $y\neq 2$ , and $e_{12}$ otherwise.

• • $p=\epsilon $ if $x\neq 2$ , and $e_{21}$ otherwise.

Hence, the composed map $ C_{*}(\mathcal {C})\to C_{*}(\mathcal {C}^f)\to \Omega _{R^f}^*(A^f)$ is given by

$$\begin{align*}a_0 \otimes \cdots \otimes a_m \mapsto q_{i_0}a_0p_{i_1} \otimes q_{i_1} a_2 p_{i_2} \otimes \cdots \otimes q_{i_m} a_m p_{i_0}, \end{align*}$$

with the same $p_{i_j}$ ’s and $q_{i_j}$ ’s as above, proving the commutativity.

Lemma 3.9. Let $\omega \in \Omega ^2_R(A)$ . Then $\omega $ induces a map $\iota (\omega ): D_{A/R}\to \Omega _{A/R}$ . Under the fusion process, the following diagram commutes:

Proof. The commutativity of the left-hand side square follows immediately from definitions, and the commutativity of the right-hand side square means that

$$\begin{align*}\iota_{\mathrm{Tr}(\delta)} (\mathrm{Tr}(\omega) ) = \mathrm{Tr} (\iota_{\delta} (\omega) ) \end{align*}$$

for all $\omega \in \Omega ^2_{{R^+}}({A^+})$ and $\delta \in D_{{A^+}/{R^+}}$ . We prove this now. Recall that the bimodule structure on $D_{A/R}$ is induced by the inner one on $A\otimes _RA$ . We know from the proof of [Reference Crawley-Boevey, Etingof and Ginzburg9, Lemma 2.8.6] that $\iota _{a\delta b}=a\iota _{\delta } b$ . Thanks to lemma 3.7, we thus have

$$ \begin{align*} \iota_{\mathrm{Tr}(\delta)} (\mathrm{Tr}(\omega) )&=\iota_{\epsilon\delta\epsilon+e_{12}\delta e_{21}} (\mathrm{Tr}(\omega) )\\ &=\epsilon\iota_{\delta} (\mathrm{Tr}(\omega) )\epsilon+e_{12}\iota_{\delta } (\mathrm{Tr}(\omega) )e_{21}\\ &=\epsilon\iota_{\delta} (\epsilon\omega\epsilon+e_{12}\omega e_{21})\epsilon+e_{12}\iota_{\delta } (\epsilon\omega\epsilon+e_{12}\omega e_{21} )e_{21}\\ &=\epsilon\iota_{\delta} (\omega)\epsilon+e_{12}\iota_{\delta}(\omega) e_{21}\\ &=\mathrm{Tr} (\iota_{\delta} (\omega) ), \end{align*} $$

as wished.

3.4 Fusion and $1$ -smoothness

We start with the following notion simply called ‘smoothness’ in [Reference Crawley-Boevey, Etingof and Ginzburg9] or [Reference Van den Bergh30].

It implies that A has a projective dimension at most 1 and that we may (and will) use short resolutions. Note that it implies smoothness of associated representation schemes, but we call it $1$ -smooth in order to emphasize that it is way more demanding than the notion of (homological) smoothness we use in previous works [Reference Bozec, Calaque and Scherotzke3, Reference Bozec, Calaque and Scherotzke4] for dg-categories (see also section 4.1), following, for example, [Reference Keller18].

In the sequel, assume that $A=A_{\mathcal {C}}$ , where $\mathcal {C}$ has a finite number of objects, and $R=\oplus _{e\in Ob(\mathcal {C})}ke$ .

Proof. Recall that ${A^+}=A \otimes _R {R^+}$ . By definition, $\Omega _{{A^+}/{R^+}} $ is the kernel of the multiplication map $m^+:{A^+} \otimes _{{R^+}} {A^+} \rightarrow {A^+} $ which can be identified with

Since R-modules are $Ob(\mathcal {C})\times Ob(\mathcal {C})$ -graded k-vector space, $R^+$ is flat over R and

$$\begin{align*}\Omega_{{A^+}/{R^+}}\simeq (R^+)^e\otimes_{R^e} \Omega_{A/R} \simeq (R^+)^e\otimes_{R^e}A^e\otimes_{A^e} \Omega_{A/R} \simeq (A^+)^e\otimes_{A^e} \Omega_{A/R}. \end{align*}$$

Since $\Omega _{A/R}$ is a projective A-bimodule, $\Omega _{{A^+}/{R^+}} $ is a projective ${A^+}$ -bimodule.

Then, we know that $\Omega _{A^f/R^f}= e \Omega _{{A^+}/{R^+} }e$ from lemma 3.6. Since $ \Omega _{{A^+}/{R^+} } $ is a projective ${A^+} $ -bimodule, there exists $r\in \mathbb N$ such that $\Omega _{A^f/R^f}$ is a direct summand of $e( {A^+} \otimes _{R^+} {A^+})^re=(e {A^+} \otimes _{R^+} {A^+}e)^r\simeq (A^f\otimes _{R^f} A^f)^r$ by lemma 3.5. Hence, $\Omega _{A^f/R^f}$ is a projective $A^f$ -bimodule.

4.1 Calabi–Yau structures, absolute and relative

Our notation follows [Reference Bozec, Calaque and Scherotzke3, Reference Bozec, Calaque and Scherotzke4]. A dg-category $\mathcal {A}$ is called (homologically) smooth if $\mathcal {A}$ is a perfect $\mathcal {A}^e$ -module. In this case, we have the following equivalence:

where $\mathcal {A}^{\vee }$ is the dualizing bimodule.

Relative Calabi–Yau structures on morphisms and cospans of dg-categories where introduced by Brav–Dyckerhoff [Reference Brav and Dyckerhoff5] following Toën [Reference Toën25, §5.3].

Definition 4.2. An n-Calabi–Yau structure on a cospan

of smooth dg-categories is a homotopy commuting diagram

whose image under $(-)^{\natural }$ is non-degenerate in the following sense: $c_{\mathcal {A}}^{\natural }$ and

are non-degenerate, and the homotopy commuting square

is cartesian. We say that a morphism $g: \mathcal {A} \longrightarrow \mathcal {C}$ is relative n-Calabi–Yau if the copsan

is n-Calabi–Yau.

We will also use the fact that by [Reference Brav and Dyckerhoff5, Theorem 6.2], n-Calabi–Yau cospans compose. It is immediate with the above definitions that an n-Calabi–Yau structure on is the same as an $(n+1)$ -Calabi–Yau structure on . Finally, recall (see, for example, [Reference Bozec, Calaque and Scherotzke4, Proposition 2.3]) that a non-degenerate Hochschild class on a smooth dg-category $\mathcal {A}$ concentrated in degree zero admits a unique cyclic lift, making $\mathcal {A}$ a Calabi–Yau category.

The next example of a Calabi–Yau cospan was investigated thoroughly in Section 3.3 of [Reference Bozec, Calaque and Scherotzke4] and related to the pair-of-pants.

Example 4.4 (Pair-of-pants).

The cospan

(4.1) $$ \begin{align} k[x^{\pm1}] \amalg k[y^{\pm1}] \longrightarrow k\langle x^{\pm1} ,y^{{\pm1}} \rangle\longleftarrow k[z^{\pm1}] \end{align} $$

where the rightmost map is $z\mapsto xy$ , is a relative $1$ -Calabi–Yau cospan with the Calabi–Yau structures $\alpha _1(x)+\alpha _1(y)-\alpha _1(z) =b(\beta _1)\sim 0$ and homotopy $\beta _1:=y^{-1}\otimes x^{-1}\otimes xy-y\otimes y^{-1}x^{-1}\otimes x$ .

We prove here the additive version of the previous example which we refer to as the additive pair-of-pants, as opposed to the multiplicative pair-of-pants of the previous example.

Lemma 4.5. There exists a relative $1$ -Calabi–Yau structure on

(4.2) $$ \begin{align} k[x]\coprod k[y] \longrightarrow k\langle x,y\rangle \longleftarrow k[z]\,, \end{align} $$

where the rightmost map is $z\mapsto x+y$ , such that the underlying absolute $1$ -Calabi–Yau structures on $k[x]$ , $k[y]$ and $k[z]$ are the natural ones.

Proof. The algebra ${\mathcal B}:=k\langle x,y \rangle $ has a small resolution as a ${\mathcal B}$ -bimodule:

$$\begin{align*}({\mathcal B}^e)^{\oplus2}[1] \oplus {\mathcal B}^e \end{align*}$$

with differential sending $(1\otimes 1,0)$ to $x\otimes 1-1\otimes x$ , and $(0,1\otimes 1)$ to $y\otimes 1-1\otimes y$ . Therefore,

$$\begin{align*}{\mathcal B}^{\vee}\simeq {\mathcal B}^e\oplus ({\mathcal B}^e)^{\oplus2}[-1] \end{align*}$$

with differential sending $1\otimes 1$ to $(x\otimes 1-1\otimes x,y\otimes 1-1\otimes y)$ .

The canonical Calabi–Yau structures on $\mathcal A:=k[x]$ are given by $\alpha _1(x)= 1\otimes x \in \mathrm {HH}_1(\mathcal {A})$ . Note that $\alpha _1$ has a unique cyclic lift by Proposition 2.3 of [Reference Bozec, Calaque and Scherotzke4] which we denote $\alpha $ . The following diagram induced by the natural Calabi–Yau structures on $\mathcal A$ is strictly commutative:

Using the small resolution of $\mathcal {A}$ , we find ${\mathcal A}\underset {{\mathcal A}^e}{\otimes }{\mathcal B}^e \simeq {\mathcal B}^e[1] \oplus {\mathcal B}^e$ , with differential sending $1\otimes 1$ to $x \otimes 1 -1\otimes x$ . Hence, we get that the diagram is cartesian. The zero homotopy is the unique lift in cyclic homology between $\alpha (z)$ and $\alpha (x)+\alpha (y)$ . Therefore, the cospan (4.2) carries a relative $1$ -Calabi–Yau structure.

4.2 Bisymplectic structures and fusion

Let A be an R-algebra, where $R= ke_1 \oplus \cdots \oplus ke_n$ is based on pairwise orthogonal idempotents as usual. We define gauge elements $E_i=(a\mapsto ae_i\otimes e_i-e_i\otimes e_ia)\in D_{A/R}$ and recall notions introduced in [Reference Crawley-Boevey, Etingof and Ginzburg9].

Definition 4.6. We call $\omega \in \Omega ^2_R(A)$ a bisymplectic structure on A if

• • $\omega $ is closed; that is, $d \omega =0 \in \mathrm {DR}_R(A)$ ,

• • $\omega $ is non-degenerate that is, $\iota (\omega ): D_{A/R} \to \Omega _{A/R}, \delta \mapsto \iota _{\delta }(\omega )$ is an isomorphism.

An element $\mu =(\mu _i)\in \oplus _ie_iAe_i$ is a moment map for a bisymplectic algebra $(A,\omega )$ if

$$\begin{align*}d\mu_i=\iota_{E_i}(\omega)\end{align*}$$

for all $i\in I$ .

A moment map always exists; see [Reference Van den Bergh30, A.7]. Now we discuss fusion of bisymplectic structures and aim to prove [Reference Van den Bergh30, Proposition 2.6.6]. We use the notation of section 3. Recall that we have trace maps $A \to A^f, a \mapsto a^f=\epsilon a \epsilon + e_{12} a e_{21}$ , $\Omega _R^* (A)\to \Omega _{R^f}^* (A^f)$ and $D_R^*(A) \to D_{R^f}^*(A^f)$ . Let A be an algebra equipped with a bisymplectic structure $\omega $ , with moment map $\mu $ . We define $\mu _i^{f\!\!f}=\mu _i^f=\mu _i $ for $i\ge 3$ and

$$\begin{align*}\mu_1^{f\!\!f}= \mu_1 +e_{12} \mu_2 e_{21} = \mu_1^f+ \mu_2^f.\end{align*}$$

Proof. By definition, $\omega ^f \in \Omega ^2_{R^f}A^f$ is a closed form. We need to show that $\iota (\omega ^f): D_{A^f/R^f} \to \Omega _{A^f/R^f}$ is an isomorphism. Recall from lemma 3.9 that we have the following commutative diagram:

Now, $\iota ({\omega ^+})$ is an isomorphism as it is obtained from $\iota (\omega )$ by an extension of rings $-\otimes _R {R^+}$ , where R is semi-simple.

We observe that the map $\mathrm {Tr}: \Omega _{{A^+}/{R^+}} \to \Omega _{A^f/R^f}$ is surjective. As $\iota ({\omega ^+})$ is surjective, $\iota (\omega ^f)$ is also surjective by lemma 3.9. Furthermore, the kernel of $\mathrm {Tr}: \Omega _{{A^+}/{R^+}} \to \Omega _{A^f/R^f}$ is given by $ \epsilon \Omega _{A/R} e_2 + e_2 \Omega _{A/R} \epsilon $ and the kernel of $\mathrm {Tr}: D_{{A^+}/{R^+}} \to D_{A^f/R^f}$ is $\epsilon D_{{A^+}/ {R^+}} e_2+ e_2D_{{A^+}/ {R^+}} \epsilon $ . The morphism $\iota ({\omega ^+})$ maps the two kernels bijectively to each other as it is an ${A^+} \otimes _{{R^+} } {A^+}$ -linear isomorphism. Furthermore, $\mathrm {Tr}: D_{{A^+}/ {R^+}} \to D_{A^f/R^f}$ is surjective. As a consequence, $\iota (\omega ^f)$ is also an isomorphism proving that $\omega ^f$ is non-degenerate. This shows that $\omega ^f$ is a bisymplectic structure o, $A^f$ . The moment map $\mu :=( \mu _i )_i$ associated to $\omega $ is determined by the condition $d \mu _i =\iota _{E_i} (\omega )$ . Denote by $F_i$ for $i\neq 2$ the gauge elements in $A^f$ . By lemma 3.9,

$$\begin{align*}d (\mu^f_i)=(d\mu_i)^f =( \iota_{E_i} (\omega))^f= \iota_{E_i^f} (\omega^f)= \iota_{F_i} (\omega^f)\end{align*}$$

for $ i \not = 1,2$ . We know from [Reference Van den Bergh30, Lemma 5.3.3] that $F_1=E_1^f+E_2^f$ , so

$$\begin{align*}d(\mu_1^{f\!\!f})=d(\mu_1^f+\mu_2^f)= \iota_{E_1^f} (\omega^f) + \iota_{E_2^f} (\omega^f) = \iota_{F_1} (\omega^f),\end{align*}$$

as expected.

4.3 From Calabi–Yau structures to bisymplectic structures

Let ${\mathcal C}$ be a k-linear category with set of objects $I=\{1, \dots , n\}$ (in particular, we assume that $\mathcal {C}$ is concentrated in degree $0$ ). Set $e_i=\mathrm {id}_i$ , $R= \oplus _{i\in I} ke_i$ , $\hat {\mathcal R}= \coprod _{i\in I} k[x_i]$ and $A=A_{\mathcal {C}}$ . Note that $\hat {R}:=A_{\hat {\mathcal R}}\simeq \bigoplus _{i\in I} k[x_i]$ . We assume that we are given an endomorphism of each object i. This amounts to having a k-linear functor $\mu : \hat {\mathcal R} \to {\mathcal C}$ or, equivalently, an R-algebra morphism $\hat {\mathcal R}\to A$ . Let us set $\mu _i:=\mu (x_i) \in e_iA e_i$ .

Proof. The $1$ -Calabi–Yau structure gives a homotopy $0 \sim \mu ( \sum _{i=1}^n 1\otimes x_i) = \sum _{i=1}^n 1\otimes \mu _i$ which yields, thanks to section 2.3, an element $\omega _1 \in \Omega _R^2(A)$ satisfying $ \iota _{E} (\omega _1)= \sum _{i=1}^n d\mu _i$ . Hence, $ \mu $ is a moment map for $\omega _1$ .

It remains to show that $\omega _1$ is closed and non-degenerate. First, note that $\gamma :=\sum _{i=1}^n 1\otimes x_i\in \Omega _R^1\hat {R}$ trivially lifts in negative cyclic homology as $B(\gamma )=0$ . Then the Calabi–Yau structure is given by a family $\omega _k \in \bar {\Omega }_R^{2k} A$ , satisfying

$$\begin{align*}(\iota_E-ud )\left(\sum_{k\ge0} u^k\omega_{k+1}\right)= \mu( \gamma),\end{align*}$$

which implies $ d\omega _1=\iota _E (\omega _2)=0\in \overline {\mathrm {DR}}_RA$ . This proves the closedness of $\omega _1$ .

The (Calabi–Yau) non-degeneration property yields the homotopy fiber sequence

$$\begin{align*}A^{\vee}[1]\to R^{\vee}[1]\otimes_{R^e}A^e\overset{\gamma_1}{\simeq} R\otimes_{R^e}A^e\to A. \end{align*}$$

Using short resolutions (thanks to the $1$ -smoothness of A), we get the homotopy commuting diagram

The homotopy is given by $\iota (\omega _1): D_{A/R}\to \Omega _{A/R}$

Now, as the Calabi–Yau structure is non-degenerate, we have

$$\begin{align*}A^{\vee}[1]\simeq\mathrm{hofib}\left( R^{\vee}[1]\otimes_{R^e}A^e\overset{\gamma_1}{\simeq} R\otimes_{R^e}A^e\to A\right).\end{align*}$$

In short resolutions, this yields a quasi-isomorphism between the vertical complexes

which, in particular, gives an isomorphism $\iota (\omega _1): D_{A/R} \to \Omega _{A/R}.$

We next investigate the relationship between fusion of bisymplectic structures and relate them to the compositions of Calabi–Yau cospans. Consider a dg-category $\mathcal {C}$ with object set I, along with a relative $1$ -Calabi–Yau structure $\mu : \hat {\mathcal R}\to \mathcal {C}$ that induces natural absolute Calabi–Yau structures on each $k[x_i]$ . Set $\hat {\mathcal R}_{\ge 3}= \coprod _{i\ge 3} k[x_i]$ . We can consider the composition of cospans

defining ${\mathcal C}^f$ , where z is mapped to $x_1+x_2$ . This yields a relative Calabi–Yau structure on

(4.3) $$ \begin{align} k[z] \amalg\hat{\mathcal R}_{\ge3} \to {\mathcal C}^f.\end{align} $$

Proof. Set $A=A_{\mathcal {C}}$ . We know, thanks to proposition 3.3, that $A^f \simeq A_{ {\mathcal C}^f}$ . As the bisymplectic structure is compatible with the relative 1-Calabi–Yau structure, we have that the image of z under this isomorphism is $\mu (x_1)^f+\mu (x_2)^f$ . Hence, the moment map of the fusion bisymplectic structure is induced from the Calabi–Yau cospan. Let $\omega \in \Omega _R^2( A)$ denote the homotopy $ \mu (1\otimes (\sum _{i\in I} x_i)) \sim 0$ of the Calabi–Yau structure which induces by assumption the bisymplectic structure on A. Since the homotopy between the 1-forms in the cospan

$$\begin{align*}k[z] \amalg\hat{\mathcal R}_{\ge3} \longrightarrow k\langle x_1, x_2 \rangle \amalg\hat{\mathcal R}_{\ge3} \longleftarrow \hat{\mathcal R} \end{align*}$$

is trivial, the zero-homotopy of the composition of Calabi–Yau cospans is given by the image of $\omega $ under the map $\nu $ from lemma 3.8. But it is proven there that this image is $\omega ^f$ , which is precisely what we want.

To summarize, we have proven that the following diagram commutes, with $R^f\simeq \oplus _{i\in I\setminus \{2\}}ke_i$ and $\hat {\mathcal R}^f\simeq \amalg _{i\in I\setminus \{2\}}k[x_i]$ .

5.1 Quasi-bisymplectic structures

Consider an R-algebra A.

Recall from [Reference Van den Bergh31, Theorem 7.1] the $A\otimes _RA$ -linear map $T:\Omega _{A/R} \stackrel {e} \rightarrow A E^* A \stackrel {T^0} \rightarrow A d\Phi A \stackrel {c} \rightarrow \Omega _{A/R}$ , where c denotes the canonical embedding, e denotes the adjoint of c and $T^0$ is uniquely determined by $T^0(E^*)=\Phi ^{-1} d\Phi -d\Phi \Phi ^{-1}$ .

What is proved by [Reference Van den Bergh31, Theorem 7.1] is that each quasi-bisymplectic structure of corresponds to a unique non-degenerate double quasi-Poisson bracket in $(D_RA/[D_RA,D_RA])_2$ . We will not recall the definition of the latter here.

Assume that $R=\oplus _{i\in I}ke_i$ is based on pairwise orthogonal idempotents. Let $(\omega , P,\Phi )$ be a compatible triple on A such that $(\omega ,\Phi )$ is quasi-bisymplectic and assume that $\Phi =(\Phi _i)_{i\in I}\in \oplus _{i\in I}e_i A^{\ast } e_i$ . Set $\Phi _1^{f\!\!f}=\Phi _1^f\Phi _2^f$ and $\Phi _i^{f\!\!f}=\Phi _i^f=\Phi _i$ if $i>2$ . The following rather computational result is the noncommutative analog of [Reference Alekseev, Kosmann-Schwarzbach and Meinrenken1, Proposition 10.7].

Proof. We need to prove that $\iota (\omega ^{f\!\!f})\iota (P^{f\!\!f})=1-\frac {1}{4}T^{f\!\!f}$ , which is equivalent to

(5.1)

Note that $ A^+ \to A^f$ , $a \mapsto \mathrm {Tr}(a)$ is surjective. Hence, it is sufficient to show compatibility on all images of $da\in \Omega _{{A^+}/{R^+}}$ . We will systematically use the notation $(-)^f=\mathrm {Tr}(-)$ in the rest of this proof.

We have $\Phi _1^{f\!\!f}=\Phi _1^f\Phi _2^f=\Phi _1^+e_{12}\Phi _2^+e_{21}$ and $\Phi _i^{f\!\!f}=\Phi _i^f=\Phi _i$ if $i>2$ . We abusively note $\Phi _i=\Phi _i^+$ , as they do not involve $e_{ij}$ ’s, so that $\Phi _i^f=\Phi _i$ when $i\neq 2$ , $\Phi _2^f=e_{12}\Phi _2e_{21}$ and we set $\Psi =\Phi _1^{f\!\!f}$ . Then for any $a\in A^+$ ,

$$ \begin{align*} (\mathrm{V})(da^f)&=T^{f\!\!f}(da^f)\\ &=[a^f,(\Phi^{f\!\!f})^{-1}d\Phi^{f\!\!f}-d\Phi^{f\!\!f}(\Phi^{f\!\!f})^{-1}]\\ &=[a^f,\Psi^{-1}d\Phi_1\Phi_2^f+(\Phi_2^f)^{-1}d\Phi_2^f-d\Phi_1\Phi_1^{-1}-\Phi_1d\Phi_2^f\Psi^{-1}]\\ &\qquad\qquad +\sum_{i>2}[a^f,\Phi_i^{-1}d\Phi_i-d\Phi_i\Phi_i^{-1}]\\ &=[a^f,\Psi^{-1}d\Phi_1\Phi_2^f+(\Phi_2^f)^{-1}d\Phi_2^f-d\Phi_1\Phi_1^{-1}-\Phi_1d\Phi_2^f\Psi^{-1}]\\ &\qquad\qquad +\sum_{i>2}\epsilon[a,\Phi_i^{-1}d\Phi_i-d\Phi_i\Phi_i^{-1}]\epsilon, \end{align*} $$

whereas, thanks to lemma 3.9,

$$ \begin{align*} (\mathrm{I})(da^f) &= \iota(\omega^f)\iota(P^f)(da^f)\\ &=\iota(\omega^f) \big(\iota(P)(da)\big)^{\!f} \\ &= \big( \iota(\omega) \iota( P) (da)\big)^{\!f}\\ &= \big(a-\frac{1}{4}T(da)\big)^{\!f} \\ &=a^f-\dfrac{1}{4}\epsilon[a,\Phi_1^{-1}d\Phi_1-d\Phi_1\Phi_1^{-1}]\epsilon-\dfrac{1}{4}e_{12}[a,\Phi_2^{-1}d\Phi_2-d\Phi_2\Phi_2^{-1}]e_{21}\\ &\qquad\qquad-\frac{1}{4}\sum_{i>2}\epsilon[a,\Phi_i^{-1}d\Phi_i-d\Phi_i\Phi_i^{-1}]\epsilon\\ &=a^f-\dfrac{1}{4}[\epsilon a\epsilon,\Phi_1^{-1}d\Phi_1-d\Phi_1\Phi_1^{-1}]\epsilon-\dfrac{1}{4}[e_{12}ae_{21},(\Phi_2^f)^{-1}d\Phi_2^f-d\Phi_2^f(\Phi_2^f)^{-1}]\\ &\qquad\qquad-\frac{1}{4}\sum_{i>2}\epsilon[a,\Phi_i^{-1}d\Phi_i-d\Phi_i\Phi_i^{-1}]\epsilon.\end{align*} $$

Recall that for every $\delta \in D_{A^f}$ ,

$$ \begin{align*} 2\iota(\omega_{\mathrm{cor}})(\delta)&={}^{\circ} i_{\delta}(\Phi_1^{-1}d\Phi_1d\Phi^f_2(\Phi_2^f)^{-1})\\ &={}^{\circ}(\Phi_1^{-1}\delta\Phi_1d\Phi^f_2(\Phi_2^f)^{-1}-\Phi_1^{-1}d\Phi_1\delta\Phi^f_2(\Phi_2^f)^{-1})\\ &=\delta(\Phi_1)"d\Phi^f_2\Psi^{-1}\delta(\Phi_1)'-\delta(\Phi_2^f)"\Psi^{-1}d\Phi_1\delta(\Phi^f_2)', \end{align*} $$

and that for every $a\in A$ , we have $\iota (P)(da)=H_a$ , the Hamiltonian vector field which satisfies

$$\begin{align*}H_a(\Phi)=-\dfrac{1}{2}(\Phi E+E\Phi)(a)^{\circ}.\end{align*}$$

It implies (recall that the bimodule structure on double derivations is induced by the inner one on $A\otimes _RA$ )

$$ \begin{align*} 2H^f_a(\Phi_1^f)&=2(\epsilon H_a\epsilon+e_{12}H_ae_{21})(\Phi_1)\\ &=-(\epsilon\Phi_1E_1\epsilon+\epsilon E_1\Phi_1\epsilon+e_{12}\Phi_1E_1e_{21}+e_{12} E_1\Phi_1e_{21})(a)^{\circ}\\ &=-(\Phi_1E_1\epsilon+\epsilon E_1\Phi_1)(a)^{\circ}\\ &=-(a\epsilon\otimes \Phi_1-\epsilon\otimes \Phi_1a+a\Phi_1\otimes \epsilon-\Phi_1\otimes \epsilon a)^{\circ}\\ &=-\Phi_1\otimes a\epsilon+\Phi_1a\otimes \epsilon-\epsilon\otimes a\Phi_1+\epsilon a\otimes \Phi_1\\ &=-\Phi_1\otimes \epsilon a \epsilon+\Phi_1\epsilon a \epsilon\otimes e_1-e_1\otimes \epsilon a \epsilon\Phi_1+\epsilon a \epsilon\otimes \Phi_1 \end{align*} $$

and

$$ \begin{align*} 2H^f_a(\Phi_2^f)&=2e_{12}(\epsilon H_a\epsilon+e_{12}H_ae_{21})(\Phi_2)e_{21}\\ &=-\big(e_{12}(\epsilon\Phi_2E_2\epsilon+\epsilon E_2\Phi_2\epsilon+e_{12}\Phi_2E_2e_{21}+e_{12} E_2\Phi_2e_{21})(a)e_{21}\big)^{\!\circ}\\ &=-\big(e_{12}(e_{12}\Phi_2E_2e_{21}+e_{12} E_2\Phi_2e_{21})(a)e_{21}\big)^{\!\circ}\\ &=-\big(e_{12}ae_{21}\otimes e_{12}\Phi_2e_{21}-e_{12}e_{21}\otimes e_{12} \Phi_2ae_{21}\\ &\qquad\qquad+e_{12}a\Phi_2e_{21}\otimes e_{12}e_2e_{21}-e_{12}\Phi_2e_{21}\otimes e_{12}e_2ae_{21}\big)^{\!\circ}\\ &=-\big(e_{12}ae_{21}\otimes \Phi_2^f-e_{1}\otimes\Phi_2^fe_{12}ae_{21}+e_{12}ae_{21}\Phi_2^f\otimes e_{1}-\Phi_2^f\otimes e_{12}ae_{21}\big)^{\!\circ}\\ &=-\Phi_2^f\otimes e_{12}ae_{21}+\Phi_2^fe_{12}ae_{21}\otimes e_{1}-e_1\otimes e_{12}ae_{21}\Phi_2^f+e_{12}ae_{21}\otimes\Phi_2^f. \end{align*} $$

We thus obtain

$$ \begin{align*} 4(\mathrm{III})(da^f)&=4\iota(\omega_{\mathrm{cor}})\iota(P^f)(da^f)\\ &=4\iota(\omega_{\mathrm{cor}})(H_a^f)\\ &=2H^f_a(\Phi_1)"d\Phi_2^f\Psi^{-1}H^f_a(\Phi_1)'-2H^f_a(\Phi_2^f)"\Psi^{-1}d\Phi_1H^f_a(\Phi_2^f)'\\ &=-e_1ad\Phi_2^f\Psi^{-1}\Phi_1+d\Phi_2^f\Psi^{-1}\Phi_1ae_1-e_1a\Phi_1d\Phi_2^f\Psi^{-1}+\Phi_1d\Phi_2^f\Psi^{-1}ae_1\\ &+e_{12}ae_{21}\Psi^{-1}d\Phi_1\Phi_2^f-\Psi^{-1}d\Phi_1\Phi_2^fe_{12}ae_{21}+e_{12}ae_{21}\Phi_2^f\Psi^{-1}d\Phi_1-\Phi_2^f\Psi^{-1}d\Phi_1e_{12}ae_{21}\\ &=-\epsilon a \epsilon d\Phi_2^f(\Phi_2^f)^{-1}+d\Phi_2^f(\Phi_2^f)^{-1}\epsilon a \epsilon-\epsilon a \epsilon\Phi_1d\Phi_2^f\Psi^{-1}+\Phi_1d\Phi_2^f\Psi^{-1}\epsilon a \epsilon\\ &+e_{12}ae_{21}\Psi^{-1}d\Phi_1\Phi_2^f-\Psi^{-1}d\Phi_1\Phi_2^fe_{12}ae_{21}+e_{12}ae_{21}\Phi_1^{-1}d\Phi_1-\Phi_1^{-1}d\Phi_1e_{12}ae_{21}\\ &=-[\epsilon a \epsilon,d\Phi_2^f(\Phi_2^f)^{-1}+\Phi_1d\Phi_2^f\Psi^{-1}]+[e_{12}ae_{21},\Psi^{-1}d\Phi_1\Phi_2^f+\Phi_1^{-1}d\Phi_1]. \end{align*} $$

Also,

$$ \begin{align*} 2\iota(\omega_{\mathrm{cor}})\iota(E_1^fE_2^f)(da^f)&=2\iota(\omega_{\mathrm{cor}}){}^{\circ}(i_{da^f}(E_1^f)E_2^f-E_1^fi_{da^f}(E_2^f))\\ &=2\iota(\omega_{\mathrm{cor}}){}^{\circ}(E_1^f(a^f)E_2^f-E_1^fE_2^f(a^f))\\ &=2\iota(\omega_{\mathrm{cor}})(e_1E_2^f\epsilon ae_1-e_1a\epsilon E_2^fe_1-e_{1}E_1^fe_{12}ae_{21}+e_{12}ae_{21}E_1^fe_{1})\\ &=e_1\iota(2\omega_{\mathrm{cor}})(E_2^f)\epsilon ae_1-e_1a\epsilon\iota(2\omega_{\mathrm{cor}})(E_2^f)e_1\\ &\qquad-e_1{\iota(2\omega_{\mathrm{cor}})({E_1^f})}e_{12}ae_{21}+e_{12}ae_{21}\iota(2\omega_{\mathrm{cor}})(E_1^f)e_{1}. \end{align*} $$

But

$$ \begin{align*} E_1^f(a^f)&=\epsilon E_1^+(a)\epsilon+e_{12} E_1^+(a)e_{21}\\ &=\epsilon ae_1\otimes e_1\epsilon-\epsilon e_1\otimes e_1a\epsilon+e_{12}ae_1\otimes e_1e_{21}-e_{12}e_1\otimes e_1ae_{21}\\ &=\epsilon a\epsilon \otimes e_1- e_1\otimes \epsilon a\epsilon \end{align*} $$

and

$$ \begin{align*} E_2^f(a^f)&=\epsilon(e_{12} E_2^+e_{21})(a)\epsilon+e_{12}(e_{12} E_2^+e_{21})(a)e_{21}\\ &=\epsilon ae_{21}\otimes e_{12}\epsilon-\epsilon e_{21}\otimes e_{12}a\epsilon+e_{12}ae_{21}\otimes e_{12}e_{21}-e_{12}e_{21}\otimes e_{12}ae_{21}\\ &=e_{12}ae_{21}\otimes e_{1}-e_{1}\otimes e_{12}ae_{21} \end{align*} $$

imply $E_1^f(\Phi _1)=E_1(\Phi _1)$ , $E_1^f(\Phi _2^f)=0$ , $E_2^f(\Phi _1)=0$ , $E_2^f(\Phi _2^f)=E_1(\Phi _2^f)$ and

$$ \begin{align*} \iota(2\omega_{\mathrm{cor}})(E_1^f)&=d\Phi^f_2(\Phi_2^f)^{-1}-\Phi_1d\Phi^f_2\Psi^{-1}\\ \iota(2\omega_{\mathrm{cor}})(E_2^f)&=-\Psi^{-1}d\Phi_1\Phi^f_2+\Phi_1^{-1}d\Phi_1. \end{align*} $$

Hence,

$$ \begin{align*} 2(\mathrm{II})(da^f)&=2\iota(\omega_{\mathrm{cor}})\iota(E_1^fE_2^f)(da^f)\\ &=e_1(-\Psi^{-1}d\Phi_1\Phi^f_2+\Phi_1^{-1}d\Phi_1)\epsilon a\epsilon-\epsilon a\epsilon(-\Psi^{-1}d\Phi_1\Phi^f_2+\Phi_1^{-1}d\Phi_1)e_1\\ &-e_1(d\Phi^f_2(\Phi_2^f)^{-1}-\Phi_1d\Phi^f_2\Psi^{-1})e_{12}ae_{21}+e_{12}ae_{21}(d\Phi^f_2(\Phi_2^f)^{-1}-\Phi_1d\Phi^f_2\Psi^{-1})e_{1}\\ & =[e_{12} a e_{21}, d\Phi^f_2 (\Phi^{f}_2)^{-1}- \Phi_1d\Phi_2^f \Psi^{-1}] +[\epsilon a \epsilon, \Psi^{-1}d\Phi_1 \Phi^{f}_2-\Phi_1^{-1}d \Phi_1 ]. \end{align*} $$

Similarly, using $\iota (2\omega ^f)(E_i^f)=(\Phi _i^{-1}d\Phi _i+d\Phi _i\Phi _i^{-1})^f$ , one gets

$$ \begin{align*} 2(\mathrm{IV})(da^f)&=2\iota(\omega^f)\iota(E_1^fE_2^f)(da^f)\\ &=e_1\iota(2\omega^f)(E_2^f)\epsilon a\epsilon- \epsilon a\epsilon\iota(2\omega^f)(E_2^f)e_1\\ &\qquad-e_1{\iota(2\omega^f)({E_1^f})}e_{12}ae_{21}+e_{12}ae_{21}\iota(2\omega^f)(E_1^f)e_{1}\\ &=e_1(\Phi_2^{-1}d\Phi_2+d\Phi_2\Phi_2^{-1})^f\epsilon a \epsilon-\epsilon a\epsilon(\Phi_2^{-1}d\Phi_2+d\Phi_2\Phi_2^{-1})^fe_1\\ &\qquad-e_1(\Phi_1^{-1}d\Phi_1+d\Phi_1\Phi_1^{-1})^fe_{12}ae_{21}+e_{12}ae_{21}(\Phi_1^{-1}d\Phi_1+d\Phi_1\Phi_1^{-1})^fe_{1}\\ &=e_{12}(\Phi_2^{-1}d\Phi_2+d\Phi_2\Phi_2^{-1})e_{21} \epsilon a \epsilon-\epsilon a \epsilon e_{12}(\Phi_2^{-1}d\Phi_2+d\Phi_2\Phi_2^{-1})e_{21}\\ &\qquad-(\Phi_1^{-1}d\Phi_1+d\Phi_1\Phi_1^{-1})e_{12}ae_{21}+e_{12}ae_{21}(\Phi_1^{-1}d\Phi_1+d\Phi_1\Phi_1^{-1}) \\ &= [e_{12} a e_{21}, \Phi^{-1}_1 d \Phi_1+d\Phi_1 \Phi_1^{-1}] -[\epsilon a \epsilon , (\Phi_2^{f})^{-1} d\Phi_2^f+d\Phi_2^f (\Phi_2^{f})^{-1}]. \end{align*} $$

Putting everything together yields (5.1) as expected.

5.2 From Calabi–Yau structures to quasi-bisymplectic structures

Again, let ${\mathcal C}$ be a k-linear category with objects set $I=\{1,\dots ,n\}$ . Set $e_i=\mathrm {id_i}$ , $R=\oplus _{i\in I}ke_i$ and ${\mathcal T}:= \coprod _{i\in I} k[x^{\pm 1}_i]$ .

Proof. Define $\Phi :k[x^{\pm 1}]\to A$ by $\Phi (x)=\sum _{i=1}^n \mu (x_i)\in \oplus _{i\in I}e_i A^{\ast } e_i$ . Since $\mu $ is $1$ -Calabi–Yau, using the notation of section 2.4, we know that there exists $\omega _k\in \bar \Omega _R^{2k}A$ for all k such that

$$\begin{align*}(\iota_E-ud)\bigg(\sum_{k\ge0}u^k\omega_{k+1}\bigg)=\Phi(\gamma),\end{align*}$$

or equivalently,

$$ \begin{align*} \iota_E\omega_1&=\Phi(\gamma_1)=\dfrac{1}{2}(\Phi^{-1}d\Phi+d\Phi\Phi^{-1})&&(\mathbb{B}2)\\ \iota_E\omega_2-d\omega_1&=-\dfrac{1}{6}\Phi(\gamma_2)\Rightarrow d\omega_1=\dfrac{1}{6}(\Phi^{-1}d\Phi)^3\mod [-,-]&&(\mathbb B1)\\ \iota_E\omega_3-d\omega_2&=\dfrac{2!}{5!}\Phi(\gamma_3)\\ \vdots\\ \iota_E\omega_{k+1}-d\omega_{k}&=(-1)^k\dfrac{k!}{(2k+1)!}\Phi(\gamma_{k+1})&&k\ge1. \end{align*} $$

For $(\mathbb {B}3)$ , set $T=k[x^{\pm 1}]$ and write the relative $1$ -pre-Calabi–Yau structure

$$\begin{align*}A^{\vee}[1]\to T^{\vee}[1]\otimes_{T^e}A^e\overset{\gamma}{\simeq} T\otimes_{T^e}A^e\to A\end{align*}$$

with short resolutions (thanks to our $1$ -smoothness assumption) to get the homotopy commuting diagram

where the homotopy $D_{A/R}\to \Omega _{A/R}$ gives $\iota _E\omega _1=(\Phi ^{-1}d\Phi +d\Phi \Phi ^{-1})/2$ .

Now assume that our Calabi–Yau structure is non-degenerate; that is,

$$\begin{align*}A^{\vee}[1]\simeq\mathrm{hofib}\left( T^{\vee}[1]\otimes_{T^e}A^e\overset{\gamma}{\simeq} T\otimes_{T^e}A^e\to A\right).\end{align*}$$

In short resolutions, this yields a quasi-isomorphism (between vertical complexes)

which, in particular, gives a surjection $D_{A/R} \to \Omega _{A/R}/\langle d\Phi \rangle $ , that is $(\mathbb B_3)$ .

5.3 Fusion

Set ${\mathcal T}_{\ge 3}=\amalg _{i\ge 3}k[x_i^{\pm 1}]$ and consider the following composition of $1$ -Calabi–Yau cospans:

(5.2)

where the leftmost one is induced by the pair-of-pants. We want to prove the following multiplicative analog of theorem 4.10.

Theorem 5.6. Consider a $1$ -Calabi–Yau functor ${\mathcal T}\rightarrow {\mathcal C}$ inducing the natural $1$ -Calabi–Yau structure on each $k[x_i^{\pm 1}]$ , and assume that $A_{{\mathcal C}}$ is $1$ -smooth. Then the quasi-bisymplectic structure on $\mathcal {C}^f$ induced, thanks to theorem 5.5, by the $1$ -Calabi–Yau functor

$$\begin{align*}k[z^{\pm1}]\amalg{\mathcal T}_{\ge3}\rightarrow\mathcal{C}^f\end{align*}$$

is the one obtained by fusion of $1$ and $2$ from the quasi-bisymplectic structure of $A_{\mathcal {C}}$ induced by theorem 5.5.

Proof. Denote by $\Phi _1^f,\Phi _2^f$ the images of $x=x_1,y=x_2$ in the pushout $\mathcal {C}^f$ . The extra difficulty here with respect to the proof of theorem 4.10 is that the homotopy $\beta _1$ involved in the pair-of-pants cospan is nontrivial; see example 4.4. This non-degenerate homotopy

$$\begin{align*}\beta_1=\dfrac{1}{2}\Big(y^{-1}\otimes x^{-1}\otimes xy-y\otimes y^{-1}x^{-1}\otimes x\Big)\in\overline{\mathrm{HH}}_2k\langle x^{\pm1} ,y^{\pm1}\rangle\end{align*}$$

is mapped in $\overline {\mathrm {DR}}^2 k\langle x^{\pm 1} ,y^{\pm 1}\rangle $ to

$$ \begin{align*} \omega&=\dfrac{1}{4}\Big(y^{-1}d x^{-1}d (xy)-yd( y^{-1}x^{-1})d x\Big)\\ &=\dfrac{1}{4}\Big(-y^{-1}x^{-1}d xx^{-1}(xd y+dxy)+d yy^{-1}x^{-1}d x+x^{-1}dxx^{-1}dx\Big)\\ &=\dfrac{1}{4}\Big(-y^{-1}x^{-1}d xd y-y^{-1}x^{-1}d xx^{-1}dxy+d yy^{-1}x^{-1}d x+x^{-1}dxx^{-1}dx\Big)\\ &\equiv-\dfrac{1}{2}x^{-1}dxdyy^{-1}\quad\mod[-,-], \end{align*} $$

which is mapped to

$$\begin{align*}-\dfrac{1}{2}(\Phi_1^f)^{-1}d\Phi_1^fd\Phi^f_2(\Phi_2^f)^{-1}\in \overline{\mathrm{DR}}^2_{R^f} \mathcal{C}^f.\end{align*}$$

The proposition 5.4 allows us to conclude, thanks to the uniqueness [Reference Van den Bergh31, Theorem 7.1] of compatibility and [Reference Van den Bergh31, Theorem 8.2.1].

To summarize, we have proven that the following diagram commutes, where $R^f=\oplus _{i\in I\setminus \{2\}}ke_i$ and ${{\mathcal T}}^f=\amalg _{i\in I\setminus \{2\}}k[x_i^{\pm 1}]$ .

5.4 Examples

5.4.1 An elementary quiver

Consider the quiver $A_2=(V=\{1,2\},E=\{e:1\to 2\})$ , with orthogonal idempotents $e_1$ and $e_2$ satisfying $1=e_1+e_2$ , $R=ke_1\oplus ke_2$ , and set

$$\begin{align*}a_1=e_1+e^*e\text{ and }a_2=e_2+ee^*.\end{align*}$$

Let us denote by A the localization $(k\overline {A_2})_{a_{1},a_2}$ . Recall that we have given in [Reference Bozec, Calaque and Scherotzke4] a relative $1$ -Calabi–Yau structure on $ \Phi : k[x^{\pm 1}] \to A$ defined by

$$\begin{align*}\Phi_1(x_1)=a_1^{-1}\quad \text{and}\quad \Phi_2(x_2)=a_2. \end{align*}$$

Define $\partial /\partial e$ and $\partial /\partial e^*$ in $D_RA$ by $\partial e/\partial e=e_2\otimes e_1$ , $\partial e^*/\partial e=0$ , $\partial e^*/\partial e^*=e_1\otimes e_2$ and $\partial e/\partial e^*=0$ .

In the previous section, we proved that this Calabi–Yau structure induces a quasi-bisymplectic one $\omega _1\in \overline {\mathrm {DR}}_R^2A$ on A. We want to prove the following.

Proposition 5.7. The double quasi-Poisson bracket compatible with $\omega _1$ through [Reference Van den Bergh31, Theorem 7.1] is the one described in [Reference Van den Bergh31, §8.3]:

$$\begin{align*}P=\dfrac{1}{2} \left(\left(1+ee^* \right) \dfrac{\partial}{\partial e^*} \frac{ \partial}{\partial e} -\left(1+e^*e\right) \dfrac{\partial}{\partial e}\frac{ \partial}{\partial e^*}\right) \in \left( D_RA/[D_RA, D_RA] \right)_2.\end{align*}$$

Note that we use the convention regarding concatenation of paths opposite to the one in [Reference Van den Bergh30]; that is, $e=e_2ee_1$ .

Proof. In [Reference Bozec, Calaque and Scherotzke3], one homotopy $\phi (\gamma _1) \sim 0$ is given by

(5.3) $$ \begin{align}\begin{split} \beta_1&=\dfrac{1}{2}\big(e^*\otimes e\otimes \Phi+\Phi\otimes e^*\otimes e- e^*\otimes \Phi^{-1}\otimes e-\Phi^{-1}\otimes e\otimes e^*\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+1\otimes e^*\otimes e \Phi-1\otimes e\Phi\otimes e^*\big),\end{split}\end{align} $$

where $\Phi =\Phi _1(x_1)+\Phi _2(x_2)$ . It yields an element ( $1/4$ appears because of the degree operator)

$$\begin{align*}\omega_1=\dfrac{1}{4}\big(e^*ded\Phi+\Phi de^*de-e^*d\Phi^{-1}de-\Phi^{-1}dede^*+de^*d(e\Phi)-d(e\Phi)de^*\big)\end{align*}$$

in $\overline {\mathrm {DR}}^2A= \left ( \overline \Omega A/[\overline \Omega A,\overline \Omega A] \right )_2$ . We can heavily simplify this expression working modulo $[\overline \Omega A,\overline \Omega A]$ . First, note that (again, $dab$ stands for $(da)b$ )

$$ \begin{align*} d\Phi&=-a_1^{-1}(de^*e+e^*de)a_1^{-1}+dee^*+ede^*=-\Phi(de^*e+e^*de)\Phi+dee^*+ede^*\\ d\Phi^{-1}&=de^*e+e^*de-a_2^{-1}(dee^*+ede^*)a_2^{-1}=de^*e+e^*de-\Phi^{-1}(dee^*+ede^*)\Phi^{-1}. \end{align*} $$

Thus, using $\Phi e\Phi =e$ and $\Phi e^*\Phi =e^*$ (cf [Reference Bozec, Calaque and Scherotzke4, (4.3)]),

$$ \begin{align*} 4\omega_1&=\Phi de^*de-\Phi^{-1}dede^*+e^*ded\Phi-e^*d\Phi^{-1}de+2de^*d(e\Phi)\\ &=\Phi de^*de-\Phi^{-1}dede^*-e^*de\Phi(de^*e+e^*de)\Phi\\ &\qquad+e^*\Phi^{-1}(dee^*+ede^*)\Phi^{-1}de+2de^*de\Phi-2de^*e\Phi(de^*e+e^*de)\Phi\\ &=\Phi de^*de-\Phi^{-1}dede^*-e^*de\Phi de^*e\Phi\\ &\qquad\underbrace{-e^*de\Phi e^*de\Phi+e^*\Phi^{-1} dee^*\Phi^{-1}de}_{\equiv0}+e^*\Phi^{-1}ede^*\Phi^{-1}de\\ &\qquad\qquad+2de^*de\Phi-2\underbrace{de^*e\Phi de^*e\Phi}_{\equiv0}-2de^*e\Phi e^*de\Phi\\ &\equiv3\Phi de^*de-\Phi^{-1}dede^*-e\Phi e^*de\Phi de^*+e^*\Phi^{-1}ede^*\Phi^{-1}de+2de^*e\Phi e^*de\Phi\\ &=3\Phi de^*de-\Phi^{-1}dede^*-ee^*\Phi_2^{-1}de\Phi de^*+e^*e\Phi_1 de^*\Phi^{-1}de+2de^*ee^*\Phi_2^{-1} de\Phi\\ &=3\Phi de^*de-\Phi^{-1}dede^*-de\Phi de^*+\Phi^{-1}de\Phi de^*\\ &\qquad+ de^*\Phi^{-1}de-\Phi de^*\Phi^{-1}de-2de^*de\Phi+2de^*\Phi^{-1}de\Phi\\ &\equiv2\Phi de^*de-2\Phi^{-1}dede^*. \end{align*} $$

We now need to prove that P and $\omega _1$ are compatible, meaning that

(5.4) $$ \begin{align} \iota(\omega_1)\iota(P)=1-\dfrac{1}{4}T\end{align} $$

with $T(dp)=[p,\Phi ^{-1}d\Phi -d\Phi \Phi ^{-1}]$ . For $p=e$ , the LHS is

$$ \begin{align*} \iota(\omega_1)\iota(P)(de)&=\dfrac{1}{2}\iota(\omega_1)\left(\dfrac{\partial}{\partial e^*}(1+e^*e)+(1+ee^*)\dfrac{\partial}{\partial e^*}\right)\\ &=\dfrac{1}{2}({}^{\circ} i_{{\partial}/{\partial e^*}}(\omega_1)(1+e^*e)+(1+ee^*){}^{\circ} i_{{\partial}/{\partial e^*}}(\omega_1)), \end{align*} $$

where

$$\begin{align*}i_{\delta}(pdqdr)=p\delta(q)'\otimes\delta(q)"dr-pdq\delta(r)'\otimes\delta(r)"\in A\otimes\Omega^1+\Omega^1\otimes A,\end{align*}$$

as stated earlier. Note that above we have used, for $\pi ,\nu \in A$ and $\delta \in D_{A/R}$ ,

$$ \begin{align*} {}^{\circ} i_{\pi\delta\nu}(pdqdr)&={}^{\circ}(p\delta(q)'\nu\otimes\pi\delta(q)"dr-pdq\delta(r)'\nu\otimes\pi\delta(r)")\\ &=\pi{}^{\circ} i_{\delta}(pdqdr)\nu \end{align*} $$

since the bimodule structure on $D_{A/R}$ is induced by the inner one on $A^e$ , as explained in the proof of [Reference Crawley-Boevey, Etingof and Ginzburg9, 2.8.6]. We have

$$ \begin{align*} {}^{\circ} i_{{\partial}/{\partial e^*}}(2\omega_1)={}^{\circ}(\Phi\otimes de+\Phi^{-1}de\otimes e_2)=de\Phi+\Phi^{-1}de.\end{align*} $$

Thus,

$$ \begin{align*} 4\iota(\omega_1)\iota(P)(da)&=(de\Phi+\Phi^{-1}de)(1+e^*e)+(1+ee^*)(de\Phi+\Phi^{-1}de)\\ &=2de+\Phi^{-1}de\Phi^{-1}+\Phi de\Phi, \end{align*} $$

whereas $4$ times the RHS of (5.4) evaluated at $de$ is

$$ \begin{align*} 4de-[e,\Phi^{-1}d\Phi-d\Phi\Phi^{-1}]&=4de-e\Phi^{-1}(-\Phi(de^*e+e^*de)\Phi+dee^*+ede^*)\\ &\qquad+e(-\Phi(de^*e+e^*de)\Phi+dee^*+ede^*)\Phi^{-1}\\ &\qquad\qquad+\Phi^{-1}(-\Phi(de^*e+e^*de)\Phi+dee^*+ede^*)e\\ &\qquad\qquad\qquad-(-\Phi(de^*e+e^*de)\Phi+dee^*+ede^*)\Phi^{-1}e\\ &=4de+ede^*e\Phi+ee^*de\Phi-e\Phi de^*e-e\Phi e^*de\\ &\qquad+\Phi^{-1}dee^*e+\Phi^{-1}ede^*e-dee^*\Phi^{-1}e-ede^*\Phi^{-1}e\\ &=4de+ee^*de\Phi-\Phi^{-1}e e^*de+\Phi^{-1}dee^*e-dee^*e\Phi\\ &=4de+\Phi de\Phi-de\Phi-de+\Phi^{-1}de\\ &\qquad+\Phi^{-1}de\Phi^{-1}-\Phi^{-1}de-de+de\Phi\\ &=2de+\Phi^{-1}de\Phi^{-1}+\Phi de\Phi, \end{align*} $$

as wished. Computations are similar to prove eq. (5.4) evaluated at $de^*$ .

5.4.2 Arbitrary quivers

Let us go back to the proof [Reference Bozec, Calaque and Scherotzke4, Theorem 4.8] of the $1$ -Calabi–Yau structure on the multiplicative moment map $\mu _Q:\coprod _{v\in V}k[z_v^{\pm 1}] \rightarrow k\overline {Q}_{loc} :=k\overline {Q}[(1+ee^*)^{-1}]_{e\in \overline E}$ defined by

$$\begin{align*}z_v \longmapsto \prod_{e\in E\cap t^{-1}(v)}(1+ee^*)\times \prod_{e\in E\cap s^{-1}(v)}(1+e^*e)^{-1}.\end{align*}$$

It is done by realizing this functor as successive compositions of Calabi–Yau cospans. Let us specify an order that better suits our purpose. As usual, we denote by $Q^{\mathrm {sep}}$ the quiver with same edge set E but vertex set $\overline E=\{v_e=s(e),v_{e^*}=t(e)\}$ . It is the disjoint union of $|E|$ copies of $A_2$ . We have a 1-Calabi–Yau morphism

(5.5) $$ \begin{align} \mu_{Q^{\mathrm{sep}}}:\coprod_{e\in E}(k[x_e^{\pm1}]\amalg k[y_e^{\pm1}])\longrightarrow k\overline {Q^{\mathrm{sep}}}_{loc} \end{align} $$

given by $x_e\mapsto (e_{s(e)}+e^*e)^{-1}$ and $y_e\mapsto e_{t(e)}+ee^*$ . We know, thanks to the previous section, that the quasi-bisymplectic structure on $k\overline {Q^{\mathrm {sep}}}_{loc}$ induced by this $1$ -Calabi–Yau multiplicative moment map matches the one described by Van den Bergh in [Reference Van den Bergh31].

We want to prove the same for Q by fusing pairs of vertices $(v_e,v_f)$ any time $s(e)=s(f)$ in $\overline Q$ . Precisely, pick a finite sequence of fusion of pairs of vertices that takes us from $Q^{\mathrm {sep}}$ to Q, and consider an intermediary step $Q^{\diamond }$ . Assume that the quasi-bisymplectic structure induced by the $1$ -Calabi–Yau one on $\mu _{Q^{\diamond }}$ matches Van den Bergh’s, and proceed to the next fusion in our sequence. Assume that we fuse $1$ and $2$ in the vertex set I of $Q^{\diamond }$ . By that, we mean that we precisely proceed to the composition (5.2), where $\mathcal {C}=k\overline {Q^{\diamond }}_{loc}$ . By induction, and using theorem 5.6, we get the following.

Theorem 5.8. The quasi-bisymplectic structure on $k\overline {Q}_{loc}$ induced by the $1$ -Calabi–Yau one on $\mu _Q$ matches the one given by Van den Bergh.

6.1 Lagrangian morphisms and quasi-hamiltonian spaces

Let X be a smooth algebraic variety. Since we will apply the following results to $X=X_V$ , we assume X to be affine for simplicity, but these results can be extended to the non-affine case. Assume that a reductive group G acts on X and consider a G-equivariant morphism $\mu :X\to {G}$ , which induces $[\mu ]:[X/G]\to [G/G]$ . Consider the standard 1-shifted symplectic structure on $[G/G]$ given by $\underline \omega =\underline \omega _0+\underline \omega _1$ , where $\underline \omega _0\in (\Omega ^1(G)\otimes \mathfrak g^*)^G$ and $\underline \omega _1\in \Omega ^3(G)^G$ .

We refer to [Reference Bozec, Calaque and Scherotzke3, §3] for a precise definition of the space $\mathcal A^{p,(\mathrm {cl})}(X,n)$ of (closed) p-forms of degree n on X. When $\alpha \in \Omega ^2(X)^G$ , we say that $(\alpha ,\mu )$ satisfies the multiplicative moment condition if

(𝕄) $$ \begin{align} \forall u\in\mathfrak g,~~i_{\vec u}\alpha=\langle\mu^*\underline\omega_0,u\rangle. \end{align} $$

This is condition (B2) in [Reference Van den Bergh31].

Proof. The cochain complex of $2$ -forms on $[X/G]$ is given by

The result follows from the fact that, by definition, $\partial $ is given by $\langle \partial \alpha ,u\rangle =i_{\vec u}\alpha $ for every $u\in \mathfrak g$ .

This can be extended to the following, where we recognize the extra condition (B1) of [Reference Van den Bergh31].

Lemma 6.3. The space of homotopies between $[\mu ]^*\underline \omega $ and $0$ in $\mathcal A^{2,\mathrm {cl}}([X/G],1)$ is discrete. It is the space of $2$ -forms $\alpha \in \Omega ^2(X)^G$ satisfying (𝕄) and

$$ \begin{align*} d_{\mathrm{dR}}\alpha&=\mu^*\underline\omega_1. \end{align*} $$

Proof. The de Rham (cochain) complex of $[X/G]$ in weight $\ge 2$ is the total (cochain) complex of the bicomplex

The space of 2-forms $\alpha \in \Omega ^2(X)^G$ mapped on $\mu ^*\omega \in \Omega ^3(X)^G\oplus (\Omega ^1(X)\otimes \mathfrak g^*)^G$ by $d_{\mathrm {dR}}\oplus \partial $ has the expected description.

Now thanks to [Reference Pantev, Toën, Vaquié and Vezzosi21], the non-degeneracy condition (that is, (B3) in [Reference Van den Bergh31]) defines a union of connected components in the space of (closed) $2$ -forms. Therefore, we have the following result (which is already implicit in [Reference Calaque7, Reference Safronov23]).

In particular, the space of lagrangian structures on $[\mu ]$ (or, equivalently, the set of quasi-hamiltonian structures on X with group valued moment map $\mu $ ) is a subset of $\Omega ^2(X)$ .

Remark 6.6. Here is how we understand geometrically the $2$ -form on X we get from an $\alpha $ satisfying (𝕄). The pull-back of $\underline \omega _0$ along the quotient $G\to [G/G]$ is zero. As $[\mu ]^*\underline \omega _0\sim 0$ via $\alpha $ , we get a self-homotopy of $0$ in the space $2$ -forms of degree $1$ on the fiber product

$$\begin{align*}[X/G]\underset{[G/G]}{\times}G\simeq X. \end{align*}$$

Such a self-homotopy is a $2$ -form of degree $0$ on X, which is nothing but $\alpha $ .

6.2 Identifying two lagrangian structures: proof of theorem 6.1

Consider the composition

$$\begin{align*}\mathrm{Spec}(A_V)=X_V\twoheadrightarrow[X_V/\mathrm{GL}_V]\hookrightarrow\mathrm{Perf}_A. \end{align*}$$

It is given by an $A-A_V$ -bimodule M which induces a chain

given by

$$\begin{align*}a_0\otimes a_1\otimes\dots\otimes a_n\mapsto\mathrm{tr}(\hat a_0)d\mathrm{tr}(\hat a_1)\dots d\mathrm{tr}(\hat a_n) \end{align*}$$

(that is, $\underline {\mathrm {tr}}$ again, cf (6.1)). Thus, the $2$ -forms match on $X_V$ , and therefore, the associated lagrangian structures as well thanks to the previous subsection.