Calabi–Yau Algebras

Introduction

A Calabi–Yau algebra of dimension $d$ is an algebra whose diagonal bimodule is self-dual with a shift: writing $A^{\mathrm{e}} = A\otimes_k A^{\mathrm{op}}$ for the enveloping algebra, the condition is

$$ \operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}}) \;\cong\; A[-d] $$

in the derived category of $A^{\mathrm{e}}$-modules, together with a finiteness condition on $A$ as a module over $A^{\mathrm{e}}$. The condition is a duality condition on the bimodule structure of $A$ alone; it implies a Poincaré duality between the Hochschild homology and the Hochschild cohomology of $A$ with a shift by $d$, and it is equivalent to the existence of a nondegenerate trace-like pairing on $A$.

The article is the twelfth and last of the theory of the category. It follows Koszul Duality, Differential Graded Categories and A-Infinity and L-Infinity Algebras, whose language it uses throughout, and it is followed, which closes the category with its application. Its content is the definition and the two equivalent formulations of the Calabi–Yau condition, the transfer of the condition along Koszul duality, the standard examples — the polynomial algebra, the Weyl algebra, the path algebras of quivers with potentials — and the precise relation to the geometric notion of the same name.

The name is shared with a geometric notion, and the two are different articles. A Calabi–Yau manifold is a complex manifold whose canonical bundle is trivial, an object of Part II where manifolds, forms and cohomology are available; of the geometry agent treats it. A Calabi–Yau algebra is the algebraic condition above, a condition on an algebra over a commutative ring, and it is treated here. The two are related — the derived category of coherent sheaves on a Calabi–Yau manifold is a Calabi–Yau category in the sense below — but the relation needs the manifold, and it is stated in Part II and not used here.

The article is algebraic. The words smooth and proper are used below in their homological senses: an algebra is homologically smooth if it has a finite resolution by finitely generated projective bimodules, and proper if its homology is finite-dimensional in each degree. These are algebraic conditions on resolutions and dimensions, not topological or geometric ones, and no manifold, no form, no distance and no topology is used anywhere in the article.

Throughout, $k$ is a field (the definition works over a commutative ring, with flatness hypotheses); $A$ is a $k$-algebra, graded or differential graded; $A^{\mathrm{e}} = A\otimes_kA^{\mathrm{op}}$ is the enveloping algebra of Separable Algebras; $D(A^{\mathrm{e}})$ is the derived category of DG $A^{\mathrm{e}}$-modules of Differential Graded Categories; and $HH_\bullet(A)$, $HH^\bullet(A)$ are the Hochschild homology and cohomology of Deformation Quantization.

The Definition of a Calabi–Yau Algebra

Definition. A $k$-algebra $A$ is homologically smooth if $A$, regarded as a DG module over $A^{\mathrm{e}}$, is a compact object of $D(A^{\mathrm{e}})$; equivalently, if the diagonal bimodule $A$ admits a finite resolution by finitely generated projective $A^{\mathrm{e}}$-modules. $A$ is proper if the graded $k$-modules $A$ have finite total dimension, and in the differential graded case if $H^\bullet(A)$ is finite-dimensional in each degree.

Definition (Ginzburg). Let $A$ be a homologically smooth $k$-algebra and let $d$ be an integer. Then $A$ is a Calabi–Yau algebra of dimension $d$, or $d$-CY, if there is an isomorphism

$$ \operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}}) \;\cong\; A[-d] $$

in the derived category $D(A^{\mathrm{e}})$ of DG $A^{\mathrm{e}}$-modules. The shift convention is fixed once and for all as follows: $\operatorname{RHom}^{\bullet}_{A^{\mathrm{e}}}(A,-)$ is computed by a projective resolution $P_\bullet$ of $A$ whose term $P_i$ sits in homological degree $i$, with $\operatorname{Hom}_{A^{\mathrm{e}}}(P_i,-)$ placed in cohomological degree $i$; with this convention the polynomial algebra in $n$ variables has dimension $n$, as verified below, and the sign of $d$ in the display is the sign of the dimension. Other sources differ by the sign of the shift, and the invariant content is the pair (existence of the isomorphism, absolute value of the shift) together with the convention.

Proposition. If $A$ is $d$-CY then the integer $d$ is determined by $A$ once the convention on the shift is fixed, because $\operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})$ has cohomology concentrated in a single degree.

Proof. The condition gives $\operatorname{Ext}^i_{A^{\mathrm{e}}}(A,A^{\mathrm{e}}) = 0$ for $i\neq d$ and $\operatorname{Ext}^d_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})\cong A$; a change of $d$ would change the vanishing statement, which is intrinsic.

Proposition. The class of $d$-CY algebras is closed under the following operations:

the tensor product: if $A$ is $d$-CY and $B$ is $e$-CY then $A\otimes_kB$ is $(d+e)$-CY, since $(A\otimes_kB)^{\mathrm{e}} = A^{\mathrm{e}}\otimes_kB^{\mathrm{e}}$ and the duality is the tensor product of the two dualities together with the Künneth isomorphism;
the matrix algebra: $M_n(A)$ is $d$-CY whenever $A$ is $d$-CY, because $M_n(A)$ is Morita equivalent to $A$ and the Calabi–Yau condition is Morita invariant in the sense of the next proposition.

Proof. Statement 1 is the Künneth formula for the enveloping algebra $A^{\mathrm{e}}\otimes_kB^{\mathrm{e}} = (A\otimes_kB)^{\mathrm{e}}$ and the compatibility of the two dualities; statement 2 follows from the fact that $M_n(A)$ and $A$ have equivalent categories of modules over their enveloping algebras in the appropriate sense.

Proposition (Morita invariance). Let $A$ and $B$ be homologically smooth algebras that are derived Morita equivalent in the sense of Differential Graded Categories. Then $A$ is $d$-CY if and only if $B$ is $d$-CY.

Proof. A derived Morita equivalence induces an equivalence of the derived categories of DG modules over the enveloping algebras, and the object $A\in D(A^{\mathrm{e}})$ corresponds to $B\in D(B^{\mathrm{e}})$ under the equivalence up to the shift; an equivalence of triangulated categories preserves the vanishing of cohomology in all but one degree and the isomorphism class of the remaining cohomology, since it is compatible with the shift functor. Hence the isomorphism $\operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})\cong A[-d]$ translates to the corresponding statement for $B$.

The Hochschild Formulation

The Calabi–Yau condition is a duality for the Hochschild theories, and this is its most useful form for calculation.

Theorem (standard). Let $A$ be a homologically smooth $k$-algebra. Then $A$ is $d$-CY if and only if there is an isomorphism of graded $k$-modules

$$ HH^i(A,A) \;\cong\; \bigl(HH_{d-i}(A,A)\bigr)^* , $$

the Hochschild Poincaré duality: the Hochschild cohomology in degree $i$ is dual to the Hochschild homology in degree $d-i$, functorially for the $A$-bimodule structures.

Proof (outline). For a homologically smooth $A$ there is an isomorphism $HH^\bullet(A,A)\cong \operatorname{RHom}_{A^{\mathrm{e}}}(A,A)$ with a shift, and $HH_\bullet(A,A)\cong A\otimes^{\mathbb{L}}_{A^{\mathrm{e}}}A$; the duality between $\operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})$ and the diagonal bimodule, together with the standard duality between $\operatorname{Hom}$ and $\otimes^{\mathbb{L}}$ for the compact objects of $D(A^{\mathrm{e}})$, converts the isomorphism of the definition into the displayed duality between the Hochschild groups.

Corollary. A $d$-CY algebra satisfies $\dim_kHH^i(A,A) = \dim_kHH_{d-i}(A,A)$ in each degree, when these dimensions are finite; in particular the Hochschild homology and cohomology of a CY algebra have the same total dimension and their Betti numbers are palindromic about $d/2$.

Example (the polynomial algebra). Let $A = \operatorname{Sym}(V) = k[x_1,\dots,x_n]$ with $\dim_kV = n$. The Hochschild–Kostant–Rosenberg theorem gives

$$ HH^i(A,A) \cong \wedge^i\operatorname{Der}_k(A), \qquad HH_i(A,A)\cong\Omega^i_{A/k}, $$

that is, polyvector fields in degree $i$ and degree-$i$ differential forms in the algebraic sense, each of rank $\binom{n}{i}$ as a module over $A$. The duality $HH^i\cong(HH_{n-i})^*$ is then the identity $\binom{n}{i} = \binom{n}{n-i}$, and $A$ is $n$-CY. This is the basic commutative example, and it is the reason the dimension of a Calabi–Yau algebra is required to match the number of variables: the two sides of the Hochschild duality are spanned by the polyvector fields and the algebraic differential forms, and their degrees complement each other only for $d = n$. The verification uses only the algebraic differentials of the symmetric algebra; the geometric differential forms on a manifold, and de Rham's theorem, belong to Part II.

Example (the Weyl algebra). Let $A_d$ be the $d$-th Weyl algebra, the algebra generated by $x_1,\dots,x_d$, $y_1,\dots,y_d$ with the relations $y_ix_j - x_jy_i = \delta_{ij}$. Then $A_d$ is homologically smooth, and it is a Calabi–Yau algebra of dimension $2d$: the duality of the definition is satisfied with the shift $[2d]$, so the algebra is $2d$-CY. It is not proper — the algebra is infinite-dimensional over $k$ — so it is the standard example showing that the Calabi–Yau condition is not confined to finite-dimensional algebras. Two of the Hochschild groups of $A_d$ are computed by elementary means. The zeroth cohomology is the centre, $HH^0(A_d,A_d) = Z(A_d) = k$, because $A_d$ is simple; and the first cohomology vanishes, $HH^1(A_d,A_d) = \operatorname{Der}(A_d)/\operatorname{Inn}(A_d) = 0$, because every derivation is inner. For $d = 1$ the last statement is visible in the Poincaré–Birkhoff–Witt basis $x^iy^j$: a derivation is a pair $(a,b) = (D(x),D(y))$, the inner derivations are the pairs $(-\partial_yz,\partial_xz)$ obtained from an element $z$, and the derivation condition $[b,x] = [a,y]$ reads $\partial_xa+\partial_yb = 0$ in the formal derivatives of the basis, an equation solved by such a pair. The zeroth homology vanishes as well, $HH_0(A_d,A_d) = A_d/[A_d,A_d] = 0$, because $1 = [x_1,y_1]$ is a commutator and the commutator subspace is a two-sided ideal; so the Calabi–Yau condition is a duality between the Hochschild groups, not a statement about their size.

The Trace and the Bimodule Formulation

Definition. Let $A$ be a graded algebra with finite-dimensional graded pieces and let $d$ be an integer. A Calabi–Yau trace of degree $d$ on $A$ is a $k$-linear map $\operatorname{tr} : A\to k$ of degree $-d$ that vanishes on the graded commutators and makes the $k$-bilinear pairing

$$ \langle a,b\rangle = \operatorname{tr}(ab), \qquad a,b\in A , $$

nondegenerate in the sense that $\langle a,a'\rangle = 0$ for all $a$ implies $a' = 0$.

Theorem (Van den Bergh, standard). Let $A$ be a homologically smooth, proper, graded $k$-algebra with finite-dimensional graded pieces. Then $A$ is $d$-CY if and only if $A$ carries a Calabi–Yau trace of degree $d$; the trace is unique up to multiplication by a unit of $k$ when it exists.

Remark. The trace formulation makes the analogy with the classical situation explicit and shows why the notion is a duality: the pairing $\langle a,b\rangle = \operatorname{tr}(ab)$ is the algebraic substitute for an integration pairing on a manifold, and the vanishing on commutators is the algebraic substitute for the statement that an integral of an exact form vanishes. The geometric integration theory, the Hodge theory and the de Rham complex of a manifold belong to Part II; the algebraic pairing above is defined entirely in terms of the multiplication of $A$ and the linear functional $\operatorname{tr}$.

Example (the preprojective algebra). Let $Q$ be a finite quiver and let $\overline Q$ be its double, obtained by adjoining a reverse arrow $\alpha^*$ for each arrow $\alpha$, with the preprojective relation $\sum_{\alpha}\bigl(\alpha\alpha^* - \alpha^*\alpha\bigr) = 0$ summed over the arrows of $Q$. The preprojective algebra $\Pi(Q) = k\overline Q/(\text{the preprojective relation})$ is homologically smooth; it is $2$-CY when $Q$ is a non-Dynkin quiver, and the CY trace is the functional that reads off the coefficient of the longest path. For a Dynkin quiver the same algebra is not CY, and the criterion detecting the failure is the Hochschild duality of the previous section.

Example (quivers with potentials). Let $Q$ be a finite quiver with a potential $W$, that is, a formal linear combination of cyclic paths of length $\geq3$ taken up to cyclic rotation, and let

$$ A = kQ/(\partial_\alpha W : \alpha\in Q_1) $$

be the Jacobian algebra of the pair $(Q,W)$, where $\partial_\alpha$ is the cyclic derivative with respect to the arrow $\alpha$. Then $A$ is homologically smooth when the potential is "nondegenerate" in the appropriate sense, and it is $3$-CY; this is the standard source of noncommutative CY algebras of dimension $3$, and the construction is the algebraic counterpart of the description of a three-dimensional CY geometry by a quiver with a potential. The geometric statement requires the manifold and belongs to Part II; the algebraic statement is the one recorded here.

Koszul Duality and the Transfer of the Calabi–Yau Condition

Theorem (standard). Let $A$ be a Koszul algebra in the sense of Koszul Duality with Koszul dual $A^!$, both homologically smooth and proper. Then $A$ is $d$-CY if and only if $A^!$ is $d$-CY.

Proof (outline). The Koszul complex identifies the derived categories of graded $A$-modules and graded $A^!$-modules, and the Koszul dual is $\operatorname{Ext}^\bullet_A(k,k)$; the diagonal bimodule of $A$ corresponds under the equivalence to the diagonal bimodule of $A^!$, and the self-duality of the one with shift $d$ corresponds to the self-duality of the other with the same shift, because the equivalence is compatible with the shift and with the duality functors. The precise comparison is the content of the Koszul duality for Hochschild (co)homology recorded in Koszul Duality; the Hochschild duality of the previous section is then transferred verbatim.

Corollary. Let $V$ be a finite-dimensional vector space of dimension $n$. Then $\operatorname{Sym}(V)$ is $n$-CY and $\Lambda(V^*)$ is $n$-CY, consistently with $\operatorname{Sym}(V)^! = \Lambda(V^*)$; the transfer is the Koszul-dual form of the fact that the polynomial algebra and the exterior algebra have the same dimension in the Calabi–Yau sense. More generally, the $q$-deformations of a Koszul CY algebra along a Poisson bracket are again CY for the same dimension when the deformation is defined, since the Koszul dual deforms compatibly.

Example (the case of dimension two and the quantum plane). The quantum plane $k_q[x,y]$ of Quantum Groups, with $yx = qxy$, is homologically smooth and graded-proper, and it is $2$-CY; its Koszul dual, the quantum exterior algebra with $x^{*2} = y^{*2} = 0$ and $x^*y^* = -q\,y^*x^*$, is $2$-CY by the theorem. The example is the simplest noncommutative deformation of a Koszul Calabi–Yau algebra, and it shows that the Calabi–Yau condition survives a deformation along a Poisson bracket of the kind considered in Deformation Quantization.

The Calabi–Yau Completion

The Calabi–Yau condition is not merely a property an algebra may have; every algebra of finite global dimension has a universal CY algebra built from it, and this construction is the standard source of the examples of the previous sections.

Definition. Let $A$ be a $k$-algebra of finite global dimension and let $d\geq1$. A $d$-Calabi–Yau completion of $A$ is a $d$-CY algebra $\Pi_d(A)$ together with a homomorphism $A\to\Pi_d(A)$ that is universal among the homomorphisms from $A$ to $d$-CY algebras whose image has the appropriate finiteness properties; the universality is expressed by the requirement that the functor $\operatorname{Hom}_{\text{alg}}(\Pi_d(A),-)$ from $d$-CY algebras to $A$-algebras be an equivalence onto its image, in the appropriate homotopy-theoretic sense.

Theorem (Keller, standard). Let $A$ be a $k$-algebra of finite global dimension and let $d\geq1$. Then the $d$-Calabi–Yau completion $\Pi_d(A)$ exists; it is generated over $A$ by the extension bimodule $\operatorname{Ext}^2_A(DA,A)$, with the dimension $d$ entering through the shift that defines the completion, and its relations are determined by the requirement that the CY condition hold. For $A$ of finite global dimension the construction is functorial and the resulting algebra is $d$-CY; in the graded case the algebra is the tensor algebra over $A$ on the generating bimodule modulo the relations imposed by the duality, and in the non-graded case an algebraic completion is taken in the sense.

Example (the preprojective algebra as a completion). Let $A = kQ$ be the path algebra of a finite quiver. Taking $d = 2$ the completion $\Pi_2(kQ)$ is the preprojective algebra $\Pi(Q)$ of the earlier section: $\operatorname{Ext}^2_A(DA,A)$ is the span of the reverse arrows, the relations of the completion are the preprojective relations, and the resulting algebra is $2$-CY exactly when $Q$ is not Dynkin. The preprojective algebra is therefore not an isolated example but the value at $d = 2$ of a functorial construction.

Example (the Ginzburg algebra as a completion). With $A = kQ$ and $d = 3$ the completion $\Pi_3(kQ)$ is the Ginzburg DG algebra, whose zeroth cohomology is the Jacobian algebra $kQ/(\partial_\alpha W)$ of a potential $W$ when the potential is chosen to match the generating bimodule; the potential formalism of the earlier section is thus the presentation of the $3$-CY completion, and the theorem explains why the relations of a quiver with potential have the shape of cyclic derivatives: they are the relations imposed by the Calabi–Yau condition on the tensor algebra generated by the extension bimodule.

Corollary. Every algebra of finite global dimension is the quotient of a CY algebra in the appropriate sense, and every CY algebra of dimension $\geq2$ that is generated in low degrees arises as a completion of an algebra of finite global dimension. In particular the classification of CY algebras of small dimension is, in the appropriate sense, the classification of pairs (an algebra of finite global dimension, a generating bimodule), and the classification is carried out in the literature on the graded CY algebras of dimension three.

Calabi–Yau Algebras and the Categories of this Part

Definition. A $k$-linear triangulated category $\mathcal{T}$ is a Calabi–Yau category of dimension $d$ if it is proper, if it is homologically smooth in the sense that the Hom functors are of finite dimension in each degree, and if there is a Serre functor $\mathbb{S}$ on $\mathcal{T}$ with $\mathbb{S}\cong[d]$, the shift by $d$. Equivalently, for every pair of objects $x,y$ there is a natural isomorphism $\operatorname{Hom}(y,x)\cong\operatorname{Hom}(x,\mathbb{S}y)^*$.

Theorem (standard). Let $A$ be a homologically smooth proper algebra. Then $A$ is $d$-CY if and only if the triangulated category $H^0(\mathrm{perf}(A))$ of compact DG $A$-modules is a Calabi–Yau category of dimension $d$.

Proof (outline). The Serre functor on $H^0(\mathrm{perf}(A))$ is the functor $-\otimes^{\mathbb{L}}_A\operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})$ up to the appropriate identification; when $\operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})\cong A[-d]$ the functor is the shift $[d]$, and conversely the identification of the Serre functor determines the dualising bimodule.

This is the form in which the Calabi–Yau condition is a condition on the derived category of a DG category, and it is the reason the notion belongs to this Part rather than to a later one: everything in the statement — the DG category, the compact objects, the derived category, the shift — has been introduced in Differential Graded Categories and A-Infinity and L-Infinity Algebras. The Calabi–Yau manifold, by contrast, is the geometric object whose derived category is expected to be Calabi–Yau; that statement, and the construction of the manifold, belong to Part II and to the geometry agent's articleand they are not used here.

Example (the category of a quiver with potential). For the Jacobian algebra $A$ of a quiver with potential, the DG category $H^0(\mathrm{perf}(A))$ is Calabi–Yau of dimension $3$; the relevant Serre functor is the shift by $3$, and the statement is the algebraic form of the three-dimensionality of the geometry. The compact DG modules of Differential Graded Categories are the objects among which the duality is stated, and their shifts and cones are the operations used in the formulation; the higher operations of A-Infinity and L-Infinity Algebras are generally nonzero on $H^0$ of the derived category, and it is the $A_\infty$-structure that makes the Serre functor computation possible.

Summary

A $k$-algebra $A$ is homologically smooth if the diagonal bimodule $A$ is compact over the enveloping algebra $A^{\mathrm{e}} = A\otimes_kA^{\mathrm{op}}$, and proper if its homology is finite-dimensional in each degree. It is Calabi–Yau of dimension $d$, or $d$-CY, if there is an isomorphism $\operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})\cong A[-d]$ in $D(A^{\mathrm{e}})$, with the convention on the shift fixed so that the polynomial algebra in $n$ variables has dimension $n$. The class is closed under grading shifts (with $d$ increased by the shift), tensor products (with $d$ the sum of the dimensions) and matrix algebras, and it is invariant under derived Morita equivalence. Equivalently, a homologically smooth $A$ is $d$-CY if and only if its Hochschild cohomology is dual to its Hochschild homology with complementary degrees, $HH^i(A,A)\cong(HH_{d-i}(A,A))^*$; this is the Hochschild Poincaré duality, and for the polynomial algebra it reduces to the identity $\binom{n}{i} = \binom{n}{n-i}$ between polyvector fields and algebraic differential forms. Equivalently again, a homologically smooth proper graded algebra with finite-dimensional pieces is $d$-CY if and only if it carries a Calabi–Yau trace $\operatorname{tr}$ of degree $-d$, vanishing on graded commutators, whose associated pairing $\langle a,b\rangle = \operatorname{tr}(ab)$ is nondegenerate.

The standard examples are: $k$ in dimension $0$; the polynomial algebra and the exterior algebra in $n$ variables in dimension $n$, the two being exchanged by Koszul duality; the $d$-th Weyl algebra in dimension $2d$; the preprojective algebra of a non-Dynkin quiver in dimension $2$ , for a Dynkin quiver, not at all; and the Jacobian algebra of a quiver with potential in dimension $3$. The Calabi–Yau condition transfers along Koszul duality: a Koszul algebra is $d$-CY if and only if its Koszul dual is $d$-CY. Finally, the condition on an algebra is equivalent to the condition on its compact DG modules that the shift $[d]$ be a Serre functor, so that $H^0(\mathrm{perf}(A))$ is a Calabi–Yau category of dimension $d$; this is the form in which the notion belongs to the language of Differential Graded Categories and A-Infinity and L-Infinity Algebras. The Calabi–Yau manifold of Part II is a different object — a complex manifold with trivial canonical bundle — whose derived category is expected to be Calabi–Yau; that geometric statement belongs to Part II, and treats it.

Summary of Notation

Symbol	Meaning
$k$, $A$	field, $k$-algebra (graded or differential graded)
$A^{\mathrm{e}} = A\otimes_kA^{\mathrm{op}}$	enveloping algebra
$\operatorname{RHom}_{A^{\mathrm{e}}}(A,A^{\mathrm{e}})\cong A[-d]$	Calabi–Yau condition of dimension $d$
$D(A^{\mathrm{e}})$	derived category of DG $A^{\mathrm{e}}$-modules
$d$-CY	Calabi–Yau of dimension $d$
homologically smooth	$\operatorname{RHom}$ of compact objects, finite projective resolution
proper	finite-dimensional homology in each degree
$HH^\bullet(A,A)$, $HH_\bullet(A,A)$	Hochschild cohomology, Hochschild homology
$HH^i\cong(HH_{d-i})^*$	Hochschild Poincaré duality
$\operatorname{Der}_k(A)$, $\Omega^i_{A/k}$	derivations (polyvector fields), algebraic differential forms
$\operatorname{tr}: A\to k$, $\langle a,b\rangle = \operatorname{tr}(ab)$	Calabi–Yau trace and its pairing
$\mathrm{perf}(A)$, $H^0$	compact DG modules, their triangulated homotopy category
$\mathbb{S}$, $[d]$	Serre functor, shift by $d$
$\Pi(Q)$, $kQ$, $W$, $\partial_\alpha W$	preprojective algebra, path algebra, potential, cyclic derivative
$\Pi_d(A)$	$d$-Calabi–Yau completion of $A$, generated by $\operatorname{Ext}^2_A(DA,A)$