Cohomologies, extensions and deformations of differential algebras with any weights
aa r X i v : . [ m a t h . R A ] M a r COHOMOLOGIES, EXTENSIONS AND DEFORMATIONS OF DIFFERENTIALALGEBRAS WITH ANY WEIGHTS
LI GUO, YUNNAN LI, YUNHE SHENG, AND GUODONG ZHOUA bstract . As an algebraic study of di ff erential equations, di ff erential algebras have been studiedfor a century and and become an important area of mathematics. In recent years the area hasbeen expended to the noncommutative associative and Lie algebra contexts and to the case whenthe operator identity has a weight in order to include di ff erence operators and di ff erence algebras.This paper provides a cohomology theory for di ff erential algebras of any weights. This gives auniform approach to both the zero weight case which is similar to the earlier study of di ff erentialLie algebras, and the non-zero weight case which poses new challenges. As applications, abelianextensions of a di ff erential algebra are classified by the second cohomology group. Furthermore,formal deformations of di ff erential algebras are obtained and the rigidity of a di ff erential algebrais characterized by the vanishing of the second cohomology group. C ontents
1. Introduction 11.1. Di ff erential algebras, old and new 21.2. Homology and deformations of di ff erential algebras 21.3. Layout of the paper 32. Di ff erential algebras and their bimodules 32.1. The category of bimodules over di ff erential algebras 32.2. Di ff erential bimodules in terms of monoid objects in slice categories 43. Cohomology of di ff erential algebras 73.1. Cohomology of di ff erential operators 73.2. Cohomology of di ff erential algebras 83.3. Relationship among the cohomologies 94. Abelian extensions of di ff erential algebras 105. Deformations of di ff erential algebras 13Appendix: Proof of Proposition 3.3 15References 201. I ntroduction This paper studies the cohomology theory, abelian extensions and formal deformations fordi ff erential algebras of any weights. Date : March 10, 2020.2010
Mathematics Subject Classification.
Key words and phrases. cohomology, extension, deformation, di ff erential algebra, di ff erence algebra, derivation. Di ff erential algebras, old and new. Classically, a di ff erential algebra is a commutative al-gebra equipped with a linear operator satisfying the Leibniz rule, modeled after the di ff erentialoperator in analysis. In fact, the origin of di ff erential algebras is the algebraic study of di ff er-ential equations pioneered by Ritt in the 1930s [26, 27]. Through the work of many mathemati-cians [12, 18, 24] in the following decades, the subject has been fully developed into a vast area inmathematics, comprising of di ff erential Galois theory, di ff erential algebraic geometry and di ff er-ential algebraic groups, with broad connections to other areas in mathematics such as arithmeticgeometry and logic, as well as computer science (mechanical proof of geometric theorems) andmathematical physics (renormalization in quantum field theory) [4, 6, 20, 31, 32].Another broadly used notion of di ff erential operators is the derivations on (co)chain complexes.There the operator d is assumed to satisfy the nilpotent condition d =
0, in which case a di ff er-ential algebra induces an associative diassociative algebra structure [15], in analog to the case ofRota-Baxter algebras that induce dendriform algebras and tridendriform algebras.In recent years, di ff erential algebras without the commutative or nilpotent conditions have beenconsidered, to include naturally arisen algebras such as path algebras and to have a more mean-ingful di ff erential Lie algebra theory generalizing the classical relationship between associativealgebras and Lie algebras [10, 22, 23]. In [16] di ff erential algebra is studied from an operadicpoint of view. Di ff erential algebras have also been applied to control theory and gauge theorythrough the BV-formalism [2, 30]. In [1], a notion of di ff erential algebras were generalized tonon(anti)commutative superspace by deformation in the study of instantons in string theory.In another direction, the Leibniz rule is generalized to include the di ff erence quotient f ( x + λ ) − f ( x ) λ before taking the limit λ
0, leading to the notion of a di ff erential algebra of weight λ [9].This generalized notion of di ff erential algebra provides a framework for a uniform approach ofthe di ff erential algebra (corresponding to the case when λ = ff erence algebra (corresponding to the case when λ =
1) [3, 13, 25], asan algebraic study of di ff erence equations. This notion also furnishes an algebraic context for thestudy of quantum calculus [11]. Di ff erential operators with weights on Virasoro algebras werealso investigated [14].1.2. Homology and deformations of di ff erential algebras. As noted above, nilpotent di ff eren-tial operators are fundamental notions for complexes to define cohomology which in turn plays akey role in deformation theory, either for specific algebraic structures, starting with the seminalworks of Gerstenhaber for associative algebras and of Nijenhuis and Richardson for Lie alge-bras [7, 8, 21], or for the general context of operads, culminated in the monographs [19, 17]. Asa further step in this direction, studies of deformations and the related cohomology have recentlyemerged for algebras with linear operators, including Rota-Baxter operators and di ff erential op-erators on Lie algebras [29, 28].The importance of di ff erential (associative) algebras makes it compelling to develop their co-homology theory, in both the zero weight case and nonzero weight case. The natural role playedby nilpotent di ff erential operators in cohomology makes it even more fascinating in this study.The purpose of this paper is to develop such a theory for di ff erential (associative) algebras of anyweight, and give its applications in the study of abelian extensions and formal deformations ofdi ff erential algebras.In comparison with the recent work [28] on cohomology and deformation of di ff erential Liealgebras, we note that the derivations there are of weight zero in which case its approach canbe adapted for di ff erential (associative) algebras. Our main emphasis on di ff erential algebras inthis paper is for the nonzero weight case, which is needed in order to study di ff erence operators IFFERENTIAL ALGEBRAS 3 and di ff erence algebras, but for which a di ff erent approach has to be taken. See comments in theoutline below and Remark 3.4.1.3. Layout of the paper.
The paper is organized as follows. In Section 2, we introduce thenotion of a bimodule over a di ff erential algebra of nonzero weight, and provide its characterizationin terms of a monoid object in slice categories.A di ff erential algebra is the combination of the underlying algebra and the di ff erential operator.In this light we build the cohomology theory of a di ff erential algebra by combining its compo-nents from the algebra and from the di ff erential operator. Thus in Section 3.1, we establish thecohomology theory for di ff erential operators of any weights, which is quite di ff erent from the onefor the underlying algebra unless the weight is zero. In Section 3.2, we combine the Hochschildcohomology for associative algebras and the just established cohomology for di ff erential oper-ators of any weights to define the cohomology of di ff erential algebras of any weights, with thecochain maps again posing extra challenges when the weight is not zero. Finally in Section 3.3,we establish a close relationship among these cohomologies. More precisely, we show that thereis a short exact sequence of cochain complexes for the algebra, the di ff erential operator and thedi ff erential algebra. The resulting long exact sequence gives linear maps from the cohomologygroups of the di ff erential algebra to those of the algebra, with the error terms (kernels and coker-nels) controlled by the cohomology groups of the di ff erential operator.As applications and further justification of our cohomology theory for di ff erential algebras, inSection 4, we apply the theory to study abelian extensions of di ff erential algebras of any weights,and show that abelian extensions are classified by the second cohomology group of the di ff erentialalgebras.Further, in Section 5, we apply the above cohomology theory to study formal deformationsof di ff erential algebras of any weights. In particular, we show that if the second cohomologygroup of a di ff erential algebra with coe ffi cients in the regular representation is trivial, then thisdi ff erential algebra is rigid. Notation.
Throughout this paper, k denotes a field of characteristic zero. All the vector spaces,algebras, linear maps and tensor products are taken over k unless otherwise specified.2. D ifferential algebras and their bimodules This section gives background on di ff erential algebras and first results on their bimodules, withan interpretation in the general context of monoid objects in slice categories [5].2.1. The category of bimodules over di ff erential algebras.Definition 2.1. ([9]) Let λ ∈ k be a fixed element. A di ff erential algebra of weight λ (also calleda λ -di ff erential algebra ) is an associative algebra A together with a linear operator d A : A → A such that(1) d A ( xy ) = d A ( x ) y + xd A ( y ) + λ d A ( x ) d A ( y ) , ∀ x , y ∈ A . If A is unital, it further requires that(2) d A (1 A ) = . Such an operator is called a di ff erential operator of weight λ or a derivation of weight λ . It isalso called a λ -di ff erential operator or a λ -derivation . LI GUO, YUNNAN LI, YUNHE SHENG, AND GUODONG ZHOU
Given two di ff erential algebras ( A , d A ) , ( B , d B ) of the same weight λ , a homomorphism ofdi ff erential algebras from ( A , d A ) to ( B , d B ) is an algebra homomorphism ϕ : A → B such that ϕ ◦ d A = d B ◦ ϕ . We denote by DA λ the category of λ -di ff erential algebras.To simply notations, for all the above notions, we will often suppress the mentioning of theweight λ unless it needs to be specified.Recall that a bimodule of an associative algebra A is a triple ( V , ρ l , ρ r ), where V is a vectorspace, ρ l : A → End k ( V ) , x ( v xv ) and ρ r : A → End k ( V ) , x ( v vx ) are homomor-phism and anti-homomorphism of associative algebras respectively such that ( xv ) y = x ( vy ) for all x , y ∈ A and v ∈ V . Definition 2.2.
Let ( A , d A ) be a di ff erential algebra.(i) A bimodule over the di ff erential algebra ( A , d A ) is a quadruple ( V , ρ l , ρ r , d V ), where d V ∈ End k ( V ), and ( V , ρ l , ρ r ) is a bimodule over the associative algebra A , such that for all x , y ∈ A , v ∈ V , the following equalities hold: d V ( xv ) = d A ( x ) v + xd V ( v ) + λ d A ( x ) d V ( v ) , d V ( vx ) = vd A ( x ) + d V ( v ) x + λ d V ( v ) d A ( x ) . (ii) Given two bimodules ( U , ρ Ul , ρ Ur , d U ) , ( V , ρ Vl , ρ Vr , d V ) over ( A , d A ), a linear map f : U → V is called a homomorphism of bimodules, if f ◦ d U = d V ◦ f and f ◦ ρ Ul ( x ) = ρ Vl ( x ) ◦ f , f ◦ ρ Ur ( x ) = ρ Vr ( x ) ◦ f , ∀ x ∈ A . We denote by ( A , d A )- Bimod the category of bimodules over the di ff erential algebra ( A , d A ). Example 2.3.
Any di ff erential algebra ( A , d A ) is a bimodule over itself with ρ l : A → End k ( A ) , x ( y xy ) , ρ r : A → End k ( A ) , x ( y yx ) . It is called the regular bimodule over the di ff erential algebra ( A , d A ).It is straightforward to obtain the following result. Proposition 2.4.
Let ( V , ρ l , ρ r , d V ) be a bimodule of the di ff erential algebra ( A , d A ) . Then ( A ⊕ V , d A ⊕ d V ) is a di ff erential algebra, where the associative algebra structure on A ⊕ V is given by ( x + u )( y + v ) = xy + xv + uy , ∀ x , y ∈ A , u , v ∈ V . Di ff erential bimodules in terms of monoid objects in slice categories. We now show thatthe above definition of bimodules for di ff erential algebras coincides with the notion obtained byapplying monoid objects in certain slice categories.2.2.1. Monoid objects in slice categories.
We first recall some general concepts [5].
Definition 2.5.
For a category C and an object A in C . The slice category C / A is the categorywhose • objects ( B , π ) are C -morphisms π : B → A , B ∈ C , and • morphisms ( B ′ , π ′ ) f → ( B ′′ , π ′′ ) are commutative diagrams of C -morphisms: B ′ f −−−−−→ B ′′ π ′ y y π ′′ A A . IFFERENTIAL ALGEBRAS 5
Definition 2.6.
Let C be a category with finite products and a terminal object T . A monoid objectin C is an object X ∈ Ob ( C ) together with two morphisms µ : X × X → X and η : T → X suchthat following diagrams commute: • the associativity of µ : X × X × X µ × Id X −−−−−→ X × X Id X × µ y y µ X × X −−−−−→ µ X , • the neutrality of η : X × X µ −−−−−→ X µ ←−−−−− X × X Id X × η x (cid:13)(cid:13)(cid:13)(cid:13) x η × Id X X × T ←−−−−− (Id X , t X ) X −−−−−→ ( t X , Id X ) T × X , where t X : X → T is the unique morphism.Let C m be the category whose objects are monoid objects ( X , µ, η ) in C as above and the hom-set Hom C m (( X , µ, η ) , ( X ′ , µ ′ , η ′ )) is the set of all f ∈ Hom C ( X , X ′ ) for which µ ′ ◦ ( f × f ) = f ◦ µ and η ′ = f ◦ η .2.2.2. The di ff erential algebra case. Fix a di ff erential algebra ( A , d ), and consider the slice cat-egory DA λ / A . The terminal object in DA λ / A is T = A Id → A . Given ( A , d ) , ( A , d ) ∈ DA λ and X = ( A , ϕ ) , X = ( A , ϕ ) ∈ DA λ / A , the product X × X is given by ( A × A A , ¯ ϕ ), where A × A A : = { ( a , a ′ ) ∈ A × A | ϕ ( a ) = ϕ ( a ′ ) } , and ¯ ϕ : A × A A → A , ( a , a ′ ) ϕ ( a ) . For any ( A ′ , d ′ ) ∈ DA λ , X = A ′ ϕ → A ∈ DA λ / A is a monoid object if and only if there existdi ff erential algebra homomorphisms M : A ′ × A A ′ → A ′ and ι : A → A ′ such that¯ ϕ = ϕ M , ϕι = Id A , M ( M ( a , b ) , c ) = M ( a , M ( b , c )) , M ( ι ( ϕ ( a )) , b ) = b , M ( a , ι ( ϕ ( b ))) = a . for any ( a , b , c ) ∈ A ′ × A A ′ × A A ′ . Consequently, M ( a , a ′ ) = M ( a − ι ( ϕ ( a )) , + M (0 , a ′ − ι ( ϕ ( a ′ ))) + M ( ι ( ϕ ( a )) , ι ( ϕ ( a ′ ))) = a − ι ( ϕ ( a )) + a ′ − ι ( ϕ ( a ′ )) + ι ( ϕ ( a ′ )) = a + a ′ − ι ( ϕ ( a )) , for any ( a , a ′ ) ∈ A ′ × A A ′ . Let V : = ker ϕ as a di ff erential ideal of A ′ . Then A ′ = ι ( A ) ⊕ V . Since M is also a di ff erential algebra homomorphism, we have uv = M ( u , M (0 , v ) = M (( u , , v )) = M (0 , = , for any u , v ∈ V . On the other hand, as ι d = d ′ ι and ϕ d ′ = d ϕ , it is clear that V is an A -bimodulewith di ff erential d V : = d ′ | V by letting ρ l ( x ) v = ι ( x ) v , ρ r ( x ) v = v ι ( x ) , for any x ∈ A , v ∈ V . LI GUO, YUNNAN LI, YUNHE SHENG, AND GUODONG ZHOU
Conversely, given any bimodule ( V , d V ) of di ff erential algebra ( A , d A ), one can define a di ff er-ential algebra structure on A ⊕ V naturally with di ff erential operator ( d A , d V ) by letting( x , u )( y , v ) = ( xy , xv + uy ) , ∀ x , y ∈ A , u , v ∈ V . We say this di ff erential algebra the semi-direct product of A and V , and denote it as ( A ⋉ V , d ⋉ ).Let p : A ⋉ V → A be the canonical projection, and associate it with( A ⋉ V ) × A ( A ⋉ V ) = { (( x , u ) , ( x , v )) ∈ ( A ⋉ V ) × ( A ⋉ V ) | x ∈ A , u , v ∈ V } . Define di ff erential algebra homomorphisms M ⋉ : ( A ⋉ V ) × A ( A ⋉ V ) → A ⋉ V , (( x , u ) , ( x , v )) ( x , u + v )and i ⋉ : A → A ⋉ V , x ( x , . Then the following lemma is easy to check.
Lemma 2.7.
For any A-bimodule V, X = A ⋉ V p → A with morphisms µ ⋉ , η ⋉ determined byM ⋉ , i ⋉ respectively is a monoid object in DA λ / A. Theorem 2.8.
The functors ( DA λ / A ) m ker . . ( A , d A ) - Bimod ⋉ m m induce an equivalence of categories.Proof. First note that ker ◦ ⋉ = Id ( A , d A ) - Bimod . Define Ψ : Id ( DA λ / A ) m → ⋉ ◦ ker by letting Ψ X : ( A ′ ϕ → A , M , ι ) → ( A ⋉ ker ϕ p → A , M ⋉ , i ⋉ ) , a ( ϕ ( a ) , a − ι ( ϕ ( a ))) . for any X = A ′ ϕ → A ∈ ( DA λ / A ) m .By definition ϕ = p ◦ Ψ X , thus Ψ X is a morphism in ( DA λ / A ) m . The inverse of Ψ X is Ψ − X : ( A ⋉ ker ϕ p → A , M ⋉ , i ⋉ ) → ( A ′ ϕ → A , M , ι ) , ( x , u ) u + ι ( x ) . Also, for any ( a , a ′ ) ∈ A ′ × A A ′ , x ∈ A , we have M ⋉ ( Ψ X ( a ) , Ψ X ( a ′ )) = M ⋉ (( ϕ ( a ) , a − ι ( ϕ ( a ))) , ( ϕ ( a ′ ) , a ′ − ι ( ϕ ( a ′ )))) = ( ϕ ( a ) , a + a ′ − ι ( ϕ ( a ))) = Ψ X ( a + a ′ − ι ( ϕ ( a ))) = Ψ X ( M ( a , a ′ )) , Ψ X ( ι ( x )) = ( ι ( x ) , ι ( x ) − ι ( ϕ ( ι ( x )))) = ( ι ( x ) , = i ⋉ ( x ) , Ψ X ( d ′ ( a )) = ( ϕ ( d ′ ( a )) , d ′ ( a ) − ι ( ϕ ( d ′ ( a )))) = ( d ( ϕ ( a )) , d ′ ( a − ι ( ϕ ( a )))) = ( d , d ′ ) Ψ X ( a ) . For any X = A ϕ → A , X = A ϕ → A ∈ ( DA λ / A ) m and morphism f : X → X , we have Ψ X ( f ( a )) = ( ϕ ( f ( a )) , f ( a ) − ι ( ϕ ( f ( a )))) = ( ϕ ( a ) , f ( a ) − ι ( ϕ ( a ))) = ( ϕ ( a ) , f ( a − ι ( ϕ ( a )))) = (Id A , f )( Ψ X ( a )) . Hence, Ψ is a natural isomorphism. The categories ( DA λ / A ) m and ( A , d A )- Bimod are equivalentto each other. (cid:3)
IFFERENTIAL ALGEBRAS 7
3. C ohomology of differential algebras
Let V be a bimodule of an associative algebra A . Denote by C n Alg ( A , V ) = Hom( ⊗ n A , V ). Inparticular, C Alg ( A , V ) = V . Recall that the Hochschild cochain complex is the cochain complex( C ∗ Alg ( A , V ) = ⊕ ∞ n = C n Alg ( A , V ) , ∂ ), where the coboundary operator ∂ : C n Alg ( A , V ) −→ C n + Alg ( A , V )is given by ∂ f ( x , . . . , x n + ) = x f ( x , . . . , x n + ) + n X i = ( − i f ( x , . . . , x i x i + , . . . , x n + ) + ( − n + f ( x , . . . , x n ) x n + for all f ∈ C n Alg ( A , V ) , x , . . . , x n + ∈ A . The corresponding Hochschild cohomology is denotedby HH ∗ Alg ( A , V )3.1. Cohomology of di ff erential operators. Let ( A , d A ) be a di ff erential algebra of weight λ andlet ( V , d V ) be a bimodule over ( A , d A ). In this subsection, we define the cohomology of di ff erentialoperators.First we make the following observation, noting that the bimodule structure coincides with theregular bimodule when the weight λ is zero. Lemma 3.1.
A di ff erential algebra ( A , d A ) admits a new bimodule structure on ( V , d V ) given by:x ⊢ λ v = ( x + λ d A ( x )) v , v ⊣ λ x = v ( x + λ d A ( x )) , ∀ x ∈ A , v ∈ V . Proof.
Given x , y ∈ A and v ∈ V , we have x ⊢ λ ( y ⊢ λ v ) = ( x + λ d A ( x ))(( y + λ d A ( y )) v ) = ( xy + λ ( xd A ( y ) + d A ( x ) y + λ d A ( x ) d A ( y ))) v = ( xy + λ d A ( xy )) v = ( xy ) ⊢ λ v . Similarly, ( v ⊣ λ x ) ⊣ λ y = v ⊣ λ ( xy ). Thus ( V , ⊢ λ , ⊣ λ ) is a bimodule over the associative algebra A .For x ∈ A and v ∈ V , d V ( x ⊢ λ v ) = d V (( x + λ d A ( x )) v ) = d V ( xv ) + λ d V ( d A ( x ) v ) = d A ( x ) v + xd V ( v ) + λ d A ( x ) d V ( v ) + λ ( d A ( d A ( x )) v + d A ( x ) d V ( v ) + λ d A ( d A ( x )) d V ( v )) = d A ( x ) v + λ d A ( d A ( x )) v + xd V ( v ) + λ d A ( x ) d V ( v ) + λ ( d A ( x ) d V ( v ) + λ d A ( d A ( x )) d V ( v )) = d A ( x ) ⊢ λ v + x ⊢ λ d V ( v ) + λ d A ( x ) ⊢ λ d V ( v ) . Similarly, one shows the equality d V ( v ⊣ λ x ) = v ⊣ λ d A ( x ) + d V ( v ) ⊣ λ x + λ d V ( v ) ⊣ λ d A ( x ) . Also, it is obvious that ( x ⊢ λ v ) ⊣ λ y = x ⊢ λ ( v ⊣ λ y ). Thus, ( V , ⊢ λ , ⊣ λ , d V ) is a bimodule over thedi ff erential algebra ( A , d A ). (cid:3) For distinction, we let V λ denote the new bimodule structure over ( A , d A ) given in Lemma 3.1.Denote by C n DO λ ( d A , d V ) = Hom( ⊗ n A , V ), which is called the space of n -chains of the di ff erentialoperator d A with coe ffi cients in the bimodule ( V , d V ). LI GUO, YUNNAN LI, YUNHE SHENG, AND GUODONG ZHOU
Definition 3.2.
The cohomology of the cochain complex ( C ∗ DO λ ( d A , d V ) = ⊕ ∞ n = C n DO λ ( d A , d V ) , ∂ λ ),denoted by H ∗ DO λ ( d A , d V ), is called the cohomology of the di ff erential operator d A with coe ffi -cients in the bimodule ( V , d V ), where ∂ λ : C n DO λ ( d A , d V ) −→ C n + DO λ ( d A , d V )is the Hochschild coboundary operator of the associative algebra A with the coe ffi cients in thebimodule V λ given in Lemma 3.1. More precisely, we have ∂ λ f ( x , . . . , x n + ) = x ⊢ λ f ( x , . . . , x n + ) + n X i = ( − i f ( x , . . . , x i x i + , . . . , x n + ) + ( − n + f ( x , . . . , x n ) ⊣ λ x n + for all f ∈ C n DO λ ( d A , d V ) , x , . . . , x n + ∈ A .3.2. Cohomology of di ff erential algebras. We now combine the classical Hochschild cohomol-ogy of associative algebras and the newly defined cohomology of di ff erential operators to definethe cohomology of the di ff erential algebra ( A , d A ) with coe ffi cients in the bimodule ( V , d V ).Define the set of n -cochains by(3) C n DA λ ( A , V ) : = C n Alg ( A , V ) ⊕ C n − DO λ ( d A , d V ) , n ≥ , C Alg ( A , V ) = V , n = . Define a linear map ∂ DA λ : C n DA λ ( A , V ) → C n + DA λ ( A , V ) by ∂ DA λ ( f , g ) : = ( ∂ f , ∂ λ g + ( − n δ f ) , ∀ f ∈ C n Alg ( A , V ) , g ∈ C n − DO λ ( d A , d V ) , n ≥ , (4) ∂ DA λ v : = ( ∂ v , δ v ) , ∀ v ∈ C Alg ( A , V ) = V , (5)where the linear map δ : C n Alg ( A , V ) → C n DO λ ( d A , d V ) is defined by δ f ( x , . . . , x n ) : = n X k = λ k − X ≤ i < ··· < i k ≤ n f ( x , . . . , d A ( x i ) , . . . , d A ( x i k ) , . . . , x n ) − d V f ( x , . . . , x n ) , for any f ∈ C n Alg ( A , V ), n ≥ δ v = − d V ( v ) , ∀ v ∈ C Alg ( A , V ) = V . Proposition 3.3.
The linear map δ is a cochain map from the cochain complex ( C ∗ Alg ( A , V ) , ∂ ) to ( C ∗ DO λ ( d A , d V ) , ∂ λ ) . The rather long and technical proof of this result is is postponed to the appendix in order not tointerrupt the flow of the presentation.
Remark 3.4.
Note that C n DO λ ( d A , d V ) equals to C n Alg ( A , V ) as linear spaces but they are equal ascochain complexes only when λ =
0. When λ is not zero, a new bimodule structure is needed todefine ∂ λ which eventually leads to the rather long and technical argument in order to establishthe cochain map in Proposition 3.3. Theorem 3.5.
The pair ( C ∗ DA λ ( A , V ) , ∂ DA λ ) is a cochain complex. So ∂ DA λ = . Proof.
For any v ∈ C Alg ( A , V ), by Proposition 3.3 we have ∂ DA λ v = ∂ DA λ ( ∂ v , δ v ) = ( ∂ v , ∂ λ δ v − δ∂ v ) = . IFFERENTIAL ALGEBRAS 9
Given any f ∈ C n Alg ( A , V ) , g ∈ C n − DO λ ( d A , d V ) with n ≥
1, also by Proposition 3.3 we have ∂ DA λ ( f , g ) = ∂ DA λ ( ∂ f , ∂ λ g + ( − n δ f ) = ( ∂ f , ∂ λ ( ∂ λ g + ( − n δ f ) + ( − n + δ∂ f ) = . Therefore, ( C ∗ DA λ ( A , V ) , ∂ DA λ ) is a cochain complex. (cid:3) Definition 3.6.
The cohomology of the cochain complex ( C ∗ DA λ ( A , V ) , ∂ DA λ ), denoted by H ∗ DA λ ( A , V ),is called the cohomology of the di ff erential algebra ( A , d A ) with coe ffi cients in the bimodule( V , d V ).To end this subsection, we compute 0-cocycles, 1-cocycles and 2-cocycles of the cochain com-plex ( C ∗ DA λ ( A , V ) , ∂ DA λ ).It is obvious that for all v ∈ V , ∂ DA λ v = ∂ v = , d V ( v ) = . For all ( f , v ) ∈ Hom( A , V ) ⊕ V , ∂ DA λ ( f , v ) = ∂ f = , and x ⊢ λ v − v ⊣ λ x = f ( d A ( x )) − d V ( f ( x )) , ∀ x ∈ A . For all ( f , g ) ∈ Hom( ⊗ A , V ) ⊕ Hom( A , V ), ∂ DA λ ( f , g ) = ∂ f = , and x ⊢ λ g ( y ) − g ( xy ) + g ( x ) ⊣ λ y = − λ f ( d A ( x ) , d ( y )) − f ( d A ( x ) , y ) − f ( x , d A ( y )) + d V ( f ( x , y )) , for all x , y ∈ A . In the next two sections, we shall need a subcomplex of the cochain complex C ∗ DA λ ( A , V ). Let(6) ˜ C n DA λ ( A , V ) : = C n Alg ( A , V ) ⊕ C n − DO λ ( d A , d V ) , n ≥ , C Alg ( A , V ) , n = , , n = . Then it is obvious that ( ˜ C ∗ DA λ ( A , V ) = ⊕ ∞ n = ˜ C n DA λ ( A , V ) , ∂ DA λ ) is a subcomplex of the cochain com-plex ( C ∗ DA λ ( A , V ) , ∂ DA λ ). We denote its cohomology by ˜ H ∗ DA λ ( A , V ). Obviously, ˜ H n DA λ ( A , V ) = H n DA λ ( A , V ) for n > Relationship among the cohomologies.
The coboundary operator ∂ DA λ can be illustratedby the following diagram: · · · −→ C n Alg ( A , V ) ( − n δ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ∂ / / C n + Alg ( A , V ) ( − n + δ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ ∂ / / C n + Alg ( A , V ) −→ · · ·· · · −→ C n − DO λ ( d A , d V ) ∂ λ / / C n DO λ ( d A , d V ) ∂ λ / / C n + DO λ ( d A , d V ) −→ · · · . Thus we have
Proposition 3.7.
There exists an exact sequence of cochain complexes, → C ∗− DO λ ( d A , d V ) ι → C ∗ DA λ ( A , V ) π → C ∗ Alg ( A , V ) → , where ι and π are the inclusion and the projection respectively. The relations among the various cohomology groups are given by the following theorem.
Theorem 3.8.
We have the following long exact sequence of cohomology groups, · · · → H n − DO λ ( d A , d V ) ¯ ι → H n DA λ ( A , V ) ¯ π → HH n Alg ( A , V ) ( − n ¯ δ → H n DO λ ( d A , d V ) → · · · , where ¯ δ : HH n Alg ( A , V ) → H n DO λ ( d A , d V ) is given by ¯ δ [ f ] = [ δ f ] . Here [ f ] and [ δ f ] denote thecohomological classes of f ∈ C n Alg ( A , V ) and δ f ∈ C n DO λ ( d A , d V ) . Thus the linear maps ¯ π establish a relationship between the cohomology groups of the di ff er-ential algebra and those of the underlying algebra, with the error terms controlled by the coho-mology groups of the di ff erential operator. This is resemblance of the Mayer-Vietoris sequence. Proof.
By Proposition 3.7 and the Snake Lemma, we have the long exact sequence · · · → H n − DO λ ( d A , d V ) ¯ ι → H n DA λ ( A , V ) ¯ π → HH n Alg ( A , V ) ∆ n → H n DO λ ( d A , d V ) → · · · . It remains to prove that the connecting homomorphism ∆ n : HH n Alg ( A , V ) → H n DO λ ( d A , d V ) areexactly ( − n ¯ δ . Indeed, by the construction of ∆ n and Eq. (4), for any [ f ] ∈ HH n Alg ( A , V ), we have¯ ι ∆ n ([ f ]) = [ ∂ DA λ ( f , = [(0 , ( − n δ f )] , which implies that ∆ n = ( − n ¯ δ . (cid:3)
4. A belian extensions of differential algebras
In this section, we study abelian extensions of di ff erential algebras and show that they areclassified by the second cohomology, as one would expect of a good cohomology theory. Definition 4.1. An abelian extension of di ff erential algebras is a short exact sequence of homo-morphisms of di ff erential algebras0 −−−−−→ V i −−−−−→ ˆ A p −−−−−→ A −−−−−→ d V y d ¯ A y d A y −−−−−→ V i −−−−−→ ˆ A p −−−−−→ A −−−−−→ uv = u , v ∈ V . We will call ( ˆ A , d ˆ A ) an abelian extension of ( A , d A ) by ( V , d V ). Definition 4.2.
Let ( ˆ A , d ˆ A ) and ( ˆ A , d ˆ A ) be two abelian extensions of ( A , d A ) by ( V , d V ). Theyare said to be isomorphic if there exists an isomorphism of di ff erential algebras ζ : ( ˆ A , d ˆ A ) → ( ˆ A , d ˆ A ) such that the following commutative diagram holds:0 −−−−−→ ( V , d V ) i −−−−−→ ( ˆ A , d ˆ A ) p −−−−−→ ( A , d A ) −−−−−→ (cid:13)(cid:13)(cid:13)(cid:13) ζ y (cid:13)(cid:13)(cid:13)(cid:13) −−−−−→ ( V , d V ) i −−−−−→ ( ˆ A , d ˆ A ) p −−−−−→ ( A , d A ) −−−−−→ . A section of an abelian extension ( ˆ A , d ˆ A ) of ( A , d A ) by ( V , d V ) is a linear map s : A → ˆ A suchthat p ◦ s = Id A .Now for an abelian extension ( ˆ A , d ˆ A ) of ( A , d A ) by ( V , d V ) with a section s : A → ˆ A , we definelinear maps ρ l : A → End k ( V ) , x ( v xv ) and ρ r : A → End k ( V ) , x ( v vx ) respectivelyby xv : = s ( x ) v , vx : = vs ( x ) , ∀ x ∈ A , v ∈ V . IFFERENTIAL ALGEBRAS 11
Proposition 4.3.
With the above notations, ( V , ρ l , ρ r , d V ) is a bimodule over the di ff erential alge-bra ( A , d A ) .Proof. For any x , y ∈ A , v ∈ V , since s ( xy ) − s ( x ) s ( y ) ∈ V implies s ( xy ) v = s ( x ) s ( y ) v , we have ρ l ( xy )( v ) = s ( xy ) v = s ( x ) s ( y ) v = ρ l ( x ) ◦ ρ l ( y )( v ) . Hence, ρ l is an algebra homomorphism. Similarly, ρ r is an algebra anti-homomorphism. More-over, d ˆ A ( s ( x )) − s ( d A ( x )) ∈ V means that d ˆ A ( s ( x )) v = s ( d A ( x )) v . Thus we have d V ( xv ) = d V ( s ( x ) v ) = d ˆ A ( s ( x ) v ) = d ˆ A ( s ( x )) v + s ( x ) d ˆ A ( v ) + λ d ˆ A ( s ( x )) d ˆ A ( v ) = s ( d A ( x )) v + s ( x ) d V ( v ) + λ s ( d A ( x )) d V ( v ) = d A ( x ) v + xd V ( v ) + λ d A ( x ) d V ( v ) . Hence, ( V , ρ l , ρ r , d V ) is a bimodule over ( A , d A ). (cid:3) We further define linear maps ψ : A ⊗ A → V and χ : A → V respectively by ψ ( x , y ) = s ( x ) s ( y ) − s ( xy ) , ∀ x , y ∈ A ,χ ( x ) = d ˆ A ( s ( x )) − s ( d A ( x )) , ∀ x ∈ A . We transfer the di ff erential algebra structure on ˆ A to A ⊕ V by endowing A ⊕ V with a multiplication · ψ and a di ff erential operator d χ defined by( x , u ) · ψ ( y , v ) = ( xy , xv + uy + ψ ( x , y )) , ∀ x , y ∈ A , u , v ∈ V , (7) d χ ( x , v ) = ( d A ( x ) , χ ( x ) + d V ( v )) , ∀ x ∈ A , v ∈ V . (8) Proposition 4.4.
The triple ( A ⊕ V , · ψ , d χ ) is a di ff erential algebra if and only if ( ψ, χ ) is a 2-cocycleof the di ff erential algebra ( A , d A ) with the coe ffi cient in ( V , d V ) .Proof. If ( A ⊕ V , · ψ , d χ ) is a di ff erential algebra, then the associativity of · ψ implies(9) x ψ ( y , z ) − ψ ( xy , z ) + ψ ( x , yz ) − ψ ( x , y ) z = . Since d χ satisfies (1), we deduce that(10) χ ( xy ) − x ⊢ λ χ ( y ) − χ ( x ) ⊣ λ y + d V ( ψ ( x , y )) − ψ ( d A ( x ) , y ) − ψ ( x , d A ( y )) − λψ ( d A ( x ) , d A ( y )) = . Hence, ( ψ, χ ) is a 2-cocycle.Conversely, if ( ψ, χ ) satisfies equalities (9) and (10), one can easily check that ( A ⊕ V , · ψ , d χ ) isa di ff erential algebra. (cid:3) Now we are ready to classify abelian extensions of a di ff erential algebra. Theorem 4.5.
Let V be a vector space and d V ∈ End k ( V ) . Then abelian extensions of a di ff eren-tial algebra ( A , d A ) by ( V , d V ) are classified by the second cohomology group ˜ H DA λ ( A , V ) of ( A , d A ) with coe ffi cients in the bimodule ( V , d V ) .Proof. Let ( ˆ A , d ˆ A ) be an abelian extension of ( A , d A ) by ( V , d V ). We choose a section s : A → ˆ A toobtain a 2-cocycle ( ψ, χ ) by Proposition 4.4. We first show that the cohomological class of ( ψ, χ )does not depend on the choice of sections. Indeed, let s and s be two distinct sections providing2-cocycles ( ψ , χ ) and ( ψ , χ ) respectively. We define φ : A → V by φ ( x ) = s ( x ) − s ( x ). Then ψ ( x , y ) = s ( x ) s ( y ) − s ( xy ) = ( s ( x ) + φ ( x ))( s ( y ) + φ ( y )) − ( s ( xy ) + φ ( xy )) = ( s ( x ) s ( y ) − s ( xy )) + s ( x ) φ ( y ) + φ ( x ) s ( y ) − φ ( xy ) = ( s ( x ) s ( y ) − s ( xy )) + x φ ( y ) + φ ( x ) y − φ ( xy ) = ψ ( x , y ) + ∂φ ( x , y )and χ ( x ) = d ˆ A ( s ( x )) − s ( d A ( x )) = d ˆ A ( s ( x ) + φ ( x )) − ( s ( d A ( x )) + φ ( d A ( x ))) = ( d ˆ A ( s ( x )) − s ( d A ( x ))) + d ˆ A ( φ ( x )) − φ ( d A ( x )) = χ ( x ) + d V ( φ ( x )) − φ ( d A ( x )) = χ ( x ) − δφ ( x ) . That is, ( ψ , χ ) = ( ψ , χ ) + ∂ DA λ ( φ ). Thus ( ψ , χ ) and ( ψ , χ ) are in the same cohomologicalclass in ˜ H DA λ ( A , V ).Next we prove that isomorphic abelian extensions give rise to the same element in ˜ H DA λ ( A , V ).Assume that ( ˆ A , d ˆ A ) and ( ˆ A , d ˆ A ) are two isomorphic abelian extensions of ( A , d A ) by ( V , d V )with the associated homomorphism ζ : ( ˆ A , d ˆ A ) → ( ˆ A , d ˆ A ). Let s be a section of ( ˆ A , d ˆ A ). As p ◦ ζ = p , we have p ◦ ( ζ ◦ s ) = p ◦ s = Id A . Therefore, ζ ◦ s is a section of ( ˆ A , d ˆ A ). Denote s : = ζ ◦ s . Since ζ is a homomorphism ofdi ff erential algebras such that ζ | V = Id V , we have ψ ( x , y ) = s ( x ) s ( y ) − s ( xy ) = ζ ( s ( x )) ζ ( s ( y )) − ζ ( s ( xy )) = ζ ( s ( x ) s ( y ) − s ( xy )) = ζ ( ψ ( x , y )) = ψ ( x , y )and χ ( x ) = d ˆ A ( s ( x )) − s ( d A ( x )) = d ˆ A ( ζ ( s ( x ))) − ζ ( s ( d A ( x ))) = ζ ( d ˆ A ( s ( x )) − s ( d A ( x ))) = ζ ( χ ( x )) = χ ( x ) . Consequently, all isomorphic abelian extensions give rise to the same element in ˜ H DA λ ( A , V ).Conversely, given two 2-cocycles ( ψ , χ ) and ( ψ , χ ), we can construct two abelian exten-sions ( A ⊕ V , · ψ , d χ ) and ( A ⊕ V , · ψ , d χ ) via equalities (7) and (8). If they represent the samecohomological class in ˜ H DA λ ( A , V ), then there exists a linear map φ : A → V such that( ψ , χ ) = ( ψ , χ ) + ∂ DA λ ( φ ) . Define ζ : A ⊕ V → A ⊕ V by ζ ( x , v ) : = ( x , φ ( x ) + v ) . Then ζ is an isomorphism of these two abelian extensions. (cid:3) Remark 4.6.
In particular, any vector space V with linear endomorphism d V can serve as a trivialbimodule of ( A , d A ). In this situation, central extensions of ( A , d A ) by ( V , d V ) are classified bythe second cohomology group H DA λ ( A , V ) of ( A , d A ) with the coe ffi cient in the trivial bimodule( V , d V ). Note that for a trivial bimodule ( V , d V ), since ∂ λ v = v ∈ V , we have H DA λ ( A , V ) = ˜ H DA λ ( A , V ) . IFFERENTIAL ALGEBRAS 13
5. D eformations of differential algebras
In this section, we study formal deformations of a di ff erential algebra. In particular, we showthat if the second cohomology group ˜ H DA λ ( A , A ) =
0, then the di ff erential algebra ( A , d A ) is rigid.Let ( A , d A ) be a di ff erential algebra. Denote by µ A the multiplication of A . Consider the 1-parameterized family µ t = ∞ X i = µ i t i , µ i ∈ C DA λ ( A , A ) , d t = ∞ X i = d i t i , d i ∈ C DO λ ( d A , d A ) . Definition 5.1. A of a di ff erential algebra ( A , d A ) is a pair( µ t , d t ) which endows the k [[ t ]]-module ( A [[ t ]] , µ t , d t ) with the di ff erential algebra structure over k [[ t ]] such that ( µ , d ) = ( µ A , d A ).Given any di ff erential algebra ( A , d A ), interpret µ A and d A as the formal power series µ t and d t with µ i = δ i , µ A and d i = δ i , d A respectively for all i ≥
0. Then ( A [[ t ]] , µ A , d A ) is a 1-parameterformal deformation of ( A , d A ).The pair ( µ t , d t ) generates a 1-parameter formal deformation of the di ff erential algebra ( A , d A )if and only if for all x , y , z ∈ A , the following equalities hold: µ t ( µ t ( x , y ) , z ) = µ t ( x , µ t ( y , z )) , (11) d t ( µ t ( x , y )) = µ t ( d t ( x ) , y ) + µ t ( x , d t ( y )) + λµ t ( d t ( x ) , d t ( y )) . (12)Expanding these equations and collecting coe ffi cients of t n , we see that Eqs. (11) and (12) areequivalent to the systems of equations: n X i µ i ( µ n − i ( x , y ) , z ) = n X i µ i ( x , µ n − i ( y , z )) , (13) X k , l ≥ k + l = n d l µ k ( x , y ) = X k , l ≥ k + l = n ( µ k ( d l ( x ) , y ) + µ k ( x , d l ( y ))) + λ X k , l , m ≥ k + l + m = n µ k ( d l ( x ) , d m ( y )) . (14) Remark 5.2.
For n =
0, Eq. (13) is equivalent to the associativity of µ A , and Eq. (14) is equivalentto the fact that d A is a λ -derivation. Proposition 5.3.
Let ( A [[ t ]] , µ t , d t ) be a -parameter formal deformation of a di ff erential algebra ( A , d A ) . Then ( µ , d ) is a 2-cocycle of the di ff erential algebra ( A , d A ) with the coe ffi cient in theregular bimodule ( A , d A ) .Proof. For n =
1, Eq. (13) is equivalent to ∂µ =
0, and Eq. (14) is equivalent to ∂ λ d + δµ = . Thus for n =
1, Eqs. (13) and (14) imply that ( µ , d ) is a 2-cocycle. (cid:3) If µ t = µ A in the above 1-parameter formal deformation of the di ff erential algebra ( A , d A ), weobtain a 1-parameter formal deformation of the di ff erential operator d A . Consequently, we have Corollary 5.4.
Let d t be a -parameter formal deformation of the di ff erential operator d A . Thend is a 1-cocycle of the di ff erential operator d A with coe ffi cients in the regular bimodule ( A , d A ) .Proof. In the special case when n =
1, Eq. (14) is equivalent to ∂ λ d =
0, which implies that d isa 1-cocycle of the di ff erential operator d A with coe ffi cients in the regular bimodule ( A , d A ). (cid:3) Definition 5.5.
The 2-cocycle ( µ , d ) is called the infinitesimal of the 1-parameter formal defor-mation ( A [[ t ]] , µ t , d t ) of ( A , d A ). Definition 5.6.
Let ( A [[ t ]] , µ t , d t ) and ( A [[ t ]] , ¯ µ t , ¯ d t ) be 1-parameter formal deformations of ( A , d A ).A formal isomorphism from ( A [[ t ]] , ¯ µ t , ¯ d t ) to ( A [[ t ]] , µ t , d t ) is a power series Φ t = P i ≥ φ i t i : A [[ t ]] → A [[ t ]], where φ i : A → A are linear maps with φ = Id A , such that Φ t ◦ ¯ µ t = µ t ◦ ( Φ t × Φ t ) , (15) Φ t ◦ ¯ d t = d t ◦ Φ t . (16)Two 1-parameter formal deformations ( A [[ t ]] , µ t , d t ) and ( A [[ t ]] , ¯ µ t , ¯ d t ) are said to be equivalent if there exists a formal isomorphism Φ t : ( A [[ t ]] , ¯ µ t , ¯ d t ) → ( A [[ t ]] , µ t , d t ). Theorem 5.7.
The infinitesimals of two equivalent -parameter formal deformations of ( A , d A ) are in the same cohomology class in ˜ H DA λ ( A , A ) .Proof. Let Φ t : ( A [[ t ]] , ¯ µ t , ¯ d t ) → ( A [[ t ]] , µ t , d t ) be a formal isomorphism. For all x , y ∈ A , wehave Φ t ◦ ¯ µ t ( x , y ) = µ t ◦ ( Φ t × Φ t )( x , y ) , Φ t ◦ ¯ d t ( x ) = d t ◦ Φ t ( x ) . Expanding the above identities and comparing the coe ffi cients of t , we obtain¯ µ ( x , y ) = µ ( x , y ) + φ ( x ) y + x φ ( y ) − φ ( xy ) , ¯ d ( x ) = d ( x ) + d A ( φ ( x )) − φ ( d A ( x )) . Thus, we have ( ¯ µ , ¯ d ) = ( µ , d ) + ∂ DA λ ( φ ) , which implies that [( ¯ µ , ¯ d )] = [( µ , d )] in ˜ H DA λ ( A , A ). (cid:3) Definition 5.8.
A 1-parameter formal deformation ( A [[ t ]] , µ t , d t ) of ( A , d A ) is said to be trivial ifit is equivalent to the deformation ( A [[ t ]] , µ A , d A ), that is, there exists Φ t = P i ≥ φ i t i : A [[ t ]] → A [[ t ]], where φ i : A → A are linear maps with φ = Id A , such that Φ t ◦ µ t = µ A ◦ ( Φ t × Φ t ) , (17) Φ t ◦ d t = d A ◦ Φ t . (18) Definition 5.9.
A di ff erential algebra ( A , d A ) is said to be rigid if every 1-parameter formal de-formation is trivial. Theorem 5.10.
Regarding ( A , d A ) as the regular bimodule over itself, if ˜ H DA λ ( A , A ) = , thedi ff erential algebra ( A , d A ) is rigid.Proof. Let ( A [[ t ]] , µ t , d t ) be a 1-parameter formal deformation of ( A , d A ). By Proposition 5.3,( µ , d ) is a 2-cocycle. By ˜ H DA λ ( A , A ) =
0, there exists a 1-cochain φ ∈ C Alg ( A , A ) such that( µ , d ) = − ∂ DA λ ( φ ) . (19)Then setting Φ t = Id A + φ t , we have a deformation ( A [[ t ]] , ¯ µ t , ¯ d t ), where¯ µ t ( x , y ) = (cid:0) Φ − t ◦ µ t ◦ ( Φ t × Φ t ) (cid:1) ( x , y ) , ¯ d t ( x ) = (cid:0) Φ − t ◦ d t ◦ Φ t (cid:1) ( x ) . IFFERENTIAL ALGEBRAS 15
Thus, ( A [[ t ]] , ¯ µ t , ¯ d t ) is equivalent to ( A [[ t ]] , µ t , d t ). Moreover, we have¯ µ t ( x , y ) = (Id A − φ t + φ t + · · · + ( − i φ i t i + · · · )( µ t ( x + φ ( x ) t , y + φ ( y ) t )) , ¯ d t ( x ) = (Id A − φ t + φ t + · · · + ( − i φ i t i + · · · )( d t ( x + φ ( x ) t )) . Therefore, ¯ µ t ( x , y ) = xy + ( µ ( x , y ) + x φ ( y ) + φ ( x ) y − φ ( xy )) t + ¯ µ ( x , y ) t + · · · , ¯ d t ( x ) = d A ( x ) + ( d A ( φ ( x )) + d ( x ) − φ ( d A ( x ))) t + ¯ d ( x ) t + · · · . By Eq. (19), we have ¯ µ t ( x , y ) = xy + ¯ µ ( x , y ) t + · · · , ¯ d t ( x ) = d A ( x ) + ¯ d ( x ) t + · · · . Then by repeating the argument, we can show that ( A [[ t ]] , µ t , d t ) is equivalent to ( A [[ t ]] , µ A , d A ).Thus, ( A , d A ) is rigid. (cid:3) A ppendix : P roof of P roposition x i , j : = x i , . . . , x j , i ≤ j , with the convention x i , j = i > j . For any 1 ≤ i < · · · < i k ≤ n and f ∈ C n Alg ( A , V ), define a function f ( i ,..., i k ) by f ( i ,..., i k ) ( x , . . . , x n ): = f ( x , . . . , x i − , d A ( x i ) , x i + . . . , x i − , d A ( x i ) , x i + . . . x i k − , d A ( x i k ) , x i k + , . . . , x n ) . In preparation for the proof of Proposition 3.3, we first give two technical lemmas.
Lemma 5.11.
For any f ∈ C n Alg ( A , V ) , x , . . . , x n + ∈ A with n ≥ , we have n X k = λ k − X ≤ i < ··· < i k ≤ n n X j = ( − j f ( i ,..., i k ) ( x , j − , x j x j + , x j + , n + ) = n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ j ≤ nj , ir − , ir , ∀ r ( − j f ( x , . . . , d A ( x i ) , . . . , x j x j + , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ kir − , ir − ( − i r − f ( x , . . . , d A ( x i ) , . . . , x i r − d A ( x i r ) , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + , ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) x i r + , . . . , d A ( x i k ) , . . . , x n + ) + n + X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + = ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) d A ( x i r + ) , . . . , d A ( x i k ) , . . . , x n + ) Proof.
In the second line of this proof, by convention i = , i k + = n +
2. By Eq. (1), we have n X k = λ k − X ≤ i < ··· < i k ≤ n n X j = ( − j f ( i ,..., i k ) ( x , j − , x j x j + , x j + , n + ) = n X k = λ k − X ≤ i < ··· < i k ≤ n X ≤ j ≤ nis < j < is + ≤ s ≤ k ( − j f ( . . . , d A ( x i ) , . . . , d A ( x i s ) , . . . , x j x j + , . . . , d A ( x i s + + ) , . . . , d A ( x i k + ) , . . . ) + n X k = λ k − X ≤ i < ··· < i k ≤ n k X r = ( − i r f ( . . . , d A ( x i ) , . . . , d A ( x i r x i r + ) , . . . , d A ( x i k + ) , . . . ) = n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ j ≤ nj , ir , ir − , ∀ r ( − j f ( · · · , d A ( x i ) , . . . , x j x j + , . . . , d A ( x i k ) , . . . ) + n X k = λ k − X ≤ i < ··· < i k ≤ n k X r = ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) x i r + , . . . , d A ( x i k + ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n k X r = ( − i r f ( x , . . . , d A ( x i ) , . . . , x i r d A ( x i r + ) , . . . , d A ( x i k + ) , . . . , x n + ) + n X k = λ k X ≤ i < ··· < i k ≤ n k X r = ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) d A ( x i r + ) , . . . , d A ( x i k + ) , . . . , x n + ) = n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ j ≤ nj , ir , ir − , ∀ r ( − j f ( . . . , d A ( x i ) , . . . , x j x j + , . . . , d A ( x i k ) , . . . ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ kir − , ir − ( − i r − f ( x , . . . , d A ( x i ) , . . . , x i r − d A ( x i r ) , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + , ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) x i r + , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k + ≤ n + X ≤ r ≤ kir + = ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) d A ( x i r + ) , . . . , d A ( x i k + ) , . . . , x n + ) = n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ j ≤ nj , ir , ir − , ∀ r ( − j f ( . . . , d A ( x i ) , . . . , x j x j + , . . . , d A ( x i k ) , . . . ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ kir − , ir − ( − i r − f ( x , . . . , d A ( x i ) , . . . , x i r − d A ( x i r ) , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + , ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) x i r + , . . . , d A ( x i k ) , . . . , x n + ) + n + X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + = ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) d A ( x i r + ) , . . . , d A ( x i k ) , . . . , x n + ) . (cid:3) IFFERENTIAL ALGEBRAS 17
Lemma 5.12.
For any f ∈ C n Alg ( A , V ) , x , . . . , x n + ∈ A with n ≥ , n + X k = λ k − X ≤ i < ··· < i k ≤ n + ( ∂ f ) ( i ,..., i k ) ( x , n + ) − d V ( ∂ f ( x , n + ))(20) = n X k = λ k − X ≤ i < ··· < i k ≤ n x ⊢ λ f ( i ,..., i k ) ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n n X j = ( − j f ( i ,..., i k ) ( x , j − , x j x j + , x j + , n + ) + ( − n + n X k = λ k − X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) ⊣ λ x n + − x ⊢ λ d V ( f ( x , n + )) + n X j = ( − j − d V ( f ( x , j − , x j x j + , x j + , n + )) + ( − n d V ( f ( x , n )) ⊣ λ x n + . Proof. As d V ( xv ) = d A ( x ) v + x ⊢ λ d V ( v ) , d V ( vx ) = vd A ( x ) + d V ( v ) ⊣ λ x for any v ∈ V , x ∈ A , wehave d V ( ∂ f ( x , n + )) = d A ( x ) f ( x , n + ) + x ⊢ λ d V ( f ( x , n + )) + n X j = ( − j d V ( f ( x , j − , x j x j + , x j + , n + )) + ( − n + f ( x , n ) d A ( x n + ) + ( − n + d V ( f ( x , n )) ⊣ λ x n + . Hence, we only need to check Eq. (20) as follows. By Lemma 5.11, we have n + X k = λ k − X ≤ i < ··· < i k ≤ n + ( ∂ f ) ( i ,..., i k ) ( x , n + ) = λ n ∂ f ( d A ( x ) , . . . , d A ( x n + )) + n X k = λ k − X ≤ i < ··· < i k ≤ n + ( ∂ f ) ( i ,..., i k ) ( x , n + ) = λ n d A ( x ) f ( d A ( x ) , . . . , d A ( x n + )) + λ n n X i = ( − i f ( d A ( x ) , . . . , d A ( x i ) d A ( x i + ) , . . . , d A ( x n + )) + ( − n + λ n f ( d A ( x ) , . . . , d A ( x n )) d A ( x n + ) + d A ( x ) f ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + d A ( x ) f ( i − ,..., i k − ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + x f ( i − ,..., i k − ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ j ≤ nj , ir − , ir , ∀ r ( − j f ( x , . . . , d A ( x i ) , . . . , x j x j + , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ kir − , ir − ( − i r − f ( x , . . . , d A ( x i ) , . . . , x i r − d A ( x i r ) , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + , ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) x i r + , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + = ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) d A ( x i r + ) , . . . , d A ( x i k ) , . . . , x n + ) + ( − n + f ( x , n ) d A ( x n + ) + n X k = λ k − ( − n + X ≤ i < ··· < i k − ≤ n f ( i ,..., i k − ) ( x , n ) d A ( x n + ) + n X k = λ k − ( − n + X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) x n + = λ n d A ( x ) f ( d A ( x ) , . . . , d A ( x n + )) + ( − n + λ n f ( d A ( x ) , . . . , d A ( x n )) d A ( x n + ) + d A ( x ) f ( x , n + ) + n − X k = λ k X ≤ i < ··· < i k ≤ n d A ( x ) f ( i ,..., i k ) ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n x f ( i ,..., i k ) ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ j ≤ nj , ir − , ir , ∀ r ( − j f ( x , . . . , d A ( x i ) , . . . , x j x j + , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ kir − , ir − ( − i r − f ( x , . . . , d A ( x i ) , . . . , x i r − d A ( x i r ) , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + , ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) x i r + , . . . , d A ( x i k ) , . . . , x n + ) + n + X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + = ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) d A ( x i r + ) , . . . , d A ( x i k ) , . . . , x n + ) + ( − n + f ( x , n ) d A ( x n + ) + n − X k = λ k ( − n + X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) d A ( x n + ) + n X k = λ k − ( − n + X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) x n + = d A ( x ) f ( x , n + ) + ( − n + f ( x , n ) d A ( x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n x f ( i ,..., i k ) ( x , n + ) IFFERENTIAL ALGEBRAS 19 + n X k = λ k X ≤ i < ··· < i k ≤ n d A ( x ) f ( i ,..., i k ) ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ j ≤ nj , ir − , ir , ∀ r ( − j f ( x , . . . , d A ( x i ) , . . . , x j x j + , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ kir − , ir − ( − i r − f ( x , . . . , d A ( x i ) , . . . , x i r − d A ( x i r ) , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + , ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) x i r + , . . . , d A ( x i k ) , . . . , x n + ) + n + X k = λ k − X ≤ i < ··· < i k ≤ n + X ≤ r ≤ k − ir + = ir + ( − i r f ( x , . . . , d A ( x i ) , . . . , d A ( x i r ) d A ( x i r + ) , . . . , d A ( x i k ) , . . . , x n + ) + n X k = λ k ( − n + X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) d A ( x n + ) + n X k = λ k − ( − n + X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) x n + = d A ( x ) f ( x , n + ) + ( − n + f ( x , n ) d A ( x n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n x ⊢ λ f ( i ,..., i k ) ( x , n + ) + n X k = λ k − X ≤ i < ··· < i k ≤ n n X j = ( − j f ( i ,..., i k ) ( x , j − , x j x j + , x j + , n + ) + ( − n + n X k = λ k − X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) ⊣ λ x n + . (cid:3) Proof of Proposition 3.3.
For any v ∈ C Alg ( A , V ) = V and x ∈ A , we have δ ( ∂ v )( x ) = ∂ v ( d A ( x )) − d V ( ∂ v ( x )) = d A ( x ) v − vd A ( x ) − d V ( xv − vx ) = d A ( x ) v − vd A ( x ) − d A ( x ) v − xd V ( v ) − λ d A ( x ) d V ( v ) + d V ( v ) x + vd A ( x ) + λ d V ( v ) d A ( x ) = − xd V ( v ) − λ d A ( x ) d V ( v ) + d V ( v ) x + λ d V ( v ) d A ( x ) = − x ⊢ λ d V ( v ) + d V ( v ) ⊣ λ x = x ⊢ λ δ v − δ v ⊣ λ x = ∂ λ ( δ v )( x ) . For any f ∈ C n Alg ( A , V ) , x , . . . , x n + ∈ A with n ≥ ∂ λ ( δ f )( x , n + ) = x ⊢ λ δ f ( x , n + ) + n X j = ( − j δ f ( x , j − , x j x j + , x j + , n + ) + ( − n + δ f ( x , n ) ⊣ λ x n + = n X k = λ k − X ≤ i < ··· < i k ≤ n x ⊢ λ f ( i ,..., i k ) ( x , n + ) + n X j = n X k = λ k − X ≤ i < ··· < i k ≤ n ( − j f ( i ,..., i k ) ( x , j − , x j x j + , x j + , n + ) + ( − n + n X k = λ k − X ≤ i < ··· < i k ≤ n f ( i ,..., i k ) ( x , n ) ⊣ λ x n + − x ⊢ λ d V ( f ( x , n + )) + n X j = ( − j − d V ( f ( x , j − , x j x j + , x j + , n + )) + ( − n d V ( f ( x , n )) ⊣ λ x n + . On the other hand, we have δ ( ∂ f )( x , n + ) = n + X k = λ k − X ≤ i < ··· < i k ≤ n + ( ∂ f ) ( i ,..., i k ) ( x , n + ) − d V ( ∂ f ( x , n + )) . Hence, by Eq. (20), ∂ λ δ = δ∂ . Acknowledgments.
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