On Rings of Differential Rota-Baxter Operators
aa r X i v : . [ m a t h . R A ] A ug ON RINGS OF DIFFERENTIAL ROTA-BAXTER OPERATORS
XING GAO, LI GUO, AND MARKUS ROSENKRANZ
Abstract.
Using the language of operated algebras, we construct and investigate a classof operator rings and enriched modules induced by a derivation or Rota-Baxter operator.In applying the general framework to univariate polynomials, one is led to the integro-differential analogs of the classical Weyl algebra. These are analyzed in terms of skewpolynomial rings and noncommutative Gr¨obner bases.
Mathematics Subject Classification:
Keywords:
Differential algebra; Rota-Baxter operators; generalized Weyl algebra; skewpolynomials; operator rings; universal algebra.
Contents
1. Introduction 12. Varieties of Operated Algebras 43. Operator Rings and Modules 104. Differential Rota-Baxter Operators 195. The Integro-Differential Weyl Algebra 23References 281.
Introduction
The ring of differential operators F [ ∂ ] over a given differential ring ( F , ∂ ) is a fun-damental algebraic structure a in the area of differential algebra [25, 22, 8], especially indifferential Galois theory [36] and D -module theory [11]. Building on this framework andspecializing to the case of linear ordinary differential equations (LODEs), the larger ring ofintegro-differential operators F [ ∂, r ] ⊃ F [ ∂ ] over an integro-differential ring ( F , ∂, r ) wasintroduced in [26, 27] for describing, computing and factoring the Green’s operators of reg-ular boundary problems for LODEs. As one knows from the classical theory, such Green’soperators will be integrals with the Green’s function as its nucleus b . Algebraically speak-ing, the Green’s operators are contained in the ring of integral operators F [ r ] ⊂ F [ ∂, r ]associated to the Rota-Baxter algebra ( F , r ).In the present paper we introduce the ring of differential Rota-Baxter operators F [ ∂, u ]over a given differential Rota-Baxter algebra ( F , ∂, u ). Although closely related to theintegro-differential operator ring F [ ∂, r ], this ring has a more delicate algebraic structureand a distinct range of applicability. In fact, we shall see that the ring of integro-differential Date : July 16, 2018. a See the end of this section for conventions on notation and terminology. b This is usually called the kernel k ( x, y ) of an integral operator f ( x ) r k ( x, y ) f ( y ) dy . In algebra, weprefer the less common term nucleus for avoiding confusion with the kernel of a homomorphism. operators is a quotient of F [ ∂, u ]: Loosely speaking, we may view ( F , ∂, u ) as an integro-differential algebra whose integral is initialized at a “generic point”; the passage to thequotient is then interpreted as “fixing the integration constant” (Propositions 3.4 and 5.5).For the particular case of polynomial coefficients F = k [ x ] over a field k ⊇ Q , this has beenstudied in the context of the (integro-differential) Weyl algebra [24], including the aforemen-tioned specialization isomorphism that fixes the integration constant. For this setting wenow provide also a generalization isomorphism that goes the opposite route of embeddingthe finer structure of differential Rota-Baxter operators into an integro-differential operatorring containing a generic point (Theorem 5.6).As to be expected from the quotient relation mentioned above, the ring of differentialRota-Baxter operators F [ ∂, u ] has a broader range of applicability. In particular, various classical distribution spaces from analysis can be construed as modules over F [ ∂, u ] butnot over F [ ∂, r ], taking F = C ∞ ( R ) or F = R [ x ] as coefficients (Example 3.8(c)). Thisis in stark contrast to F = C ∞ ( R ) or F = R [ x ] itself, which is both an F [ ∂, r ]-moduleand an F [ ∂, u ]-module, with r := u := r x the standard Rota-Baxter operator on F .This is so because distributions can be differentiated arbitrarily but in general they cannotbe evaluated (any point can be in the singular support of a distribution). In particular,the crucial identity r f ′ = f − f (0) for smooth functions f ∈ C ∞ ( R ) fails to hold fordistributions f ∈ D ′ ( R ). The upshot is that distributions have a sheaf structure (restrictionsto open subsets) but no evaluations (“restrictions to points”).Besides the ring of differential Rota-Baxter operators F [ ∂, u ], which is the main objectintroduced in this paper, we have already mentioned the related operator rings F [ ∂ ], F [ r ]and F [ ∂, r ]. It should be recalled [27, Prop. 17] that the latter ring is more than the sumof the two others. In fact (Proposition 3.4), we have F [ ∂, r ] = F [ ∂ ] ∔ F [ r ] \ F ∔ ( e ) as k -modules, where e := 1 F − r ◦ ∂ is the induced evaluation. Having four different operatorrings, it will be expedient to describe a universal algebraic setting that allows to generatethese four operator rings—and possibly others—in a uniform manner (Example 3.4).In fact, we shall use a slightly more special setting that is better adapted to our needs:While universal algebra applies to all varieties (categories whose objects are sets A endowedwith any number of n -operations A n → A , subject to laws in equational form), we shall onlyneed k -algebras endowed with one or several unary operations A → A , usually known as operated algebras . This leads to significant simplifications: While the algorithmic machineryof universal algebra is generally dependent on rewriting and the Knuth-Bendix algorithm,the situation of operated algebras is amenable to Gr¨obner(-Shirshov) bases [7, 4, 6]. More-over, the latter are closely related to and compatible with the skew polynomial approachused in [24] for constructing the integro-differential Weyl algebra.It should be emphasized that we allow arbitrary laws to be imposed on operated algebras,not just multilinear laws as one might be led to expect from the examples. For ground ringsof characteristic zero, we show how to transform arbitrary laws to multilinear ones, usinga suitable polarization process. This may also be the reason why the usual treatment isbased on multilinear laws. For instance (Example 2.19(b)), Rota-Baxter algebras of weightzero are normally defined through the axiom ( r f )( r g ) = r f r g + r g r f , which may beviewed as the polarized version of ( r f ) = 2 r f r f ; in characteristic zero these identitiesare equivalent. For showing that r is a Rota-Baxter operator, the latter identity may bebetter (e.g. using induction on some degree of f rather than double induction on f and g ). IFFERENTIAL ROTA-BAXTER OPERATORS 3
Since we have in mind various applications for function algebras, we restrict ourselves inthis paper to operator rings over commutative algebras . However, the construction wouldwork in essentially the same way for noncommutative algebras, writing the laws in termsof noncommutative rather than commutative decorated words (Definition 2.1). This couldbe employed for operator rings over matrix-valued functions; however, we shall not pursuethis further in the scope of the present paper.
Terminology and Notation.
We use N = { , , , . . . } for the natural numbers withzero . If M is a (multiplicative) monoid M with zero element 0 ∈ M , the subset of nonzeroelements is denoted by M × := M \{ } . If Z is any set, we write respectively M ( Z ) and C ( Z )for the free monoid and the free commutative monoid on Z ; its identity is denoted by 1.Unless specified otherwise, all rings and algebras are assumed to be associative and unital (whereas nonunital rings will be called rungs ). All modules and algebras are over a fixedcommutative ring k , which will be specialized to a field of characteristic zero in Section 5.All modules are taken to be left modules . A (commutative or noncommutative) ring withoutzero divisor is called a domain . We write Alg R and Mod R for the category of R -modules and R -algebras , respectively (suppressing the subscript R in the case R = k ). We denote the k -span of a set Z by k Z ; the ring of noncommutative polynomials is thus given by k h X i := k M ( X ) and the ring of commutative polynomials by k [ X ] := k C ( X ).Let A be a k -module with k -submodule A ′ . Then A \ A ′ denotes a linear complementof A , rather than the set-theoretic one. (We will only use this notation when such linearcomplements exist and the specific choice is irrelevant.)For ∂ and r we employ operator notation as in analysis; for example we write ( ∂f )( ∂g )rather than ∂ ( f ) ∂ ( g ). Juxtaposition has precedence over the operators so that we havefor instance ∂ fg := ∂ ( f g ) and r f r g := r ( f r g ). Moreover, we use also the customarynotation f ′ for the derivative ∂f , and by analogy f for the antiderivative r f . Structure of the Paper.
In Section 2 we start by introducing the appropriate toolsfor describing varieties and their laws in the framework of Ω-operated algebras. The mainresult in this section is the reduction of arbitrary laws to homogeneous and multilinear laws(Corollary 2.18). We end the section by introducing the four basic varieties coming fromanalysis (Example 2.19). In Section 3, the operator ring for a given variety is introduced(Definition 3.2). Modules over the operator rings are described equivalently as a special classof Ω-operated modules (Proposition 3.6). The operator rings and modules are exemplifiedin the four basic varieties (Proposition 3.4 and Example 3.8). Section 4 is devoted to oneof the four operator rings that is introduced here for the first time: the ring of differentialRota-Baxter operators. Here the main results are a left adjoint to the forgetful functor fromintegro-differential to differential Rota-Baxter algebras (Theorem 4.6) and the embeddingof the differential Rota-Baxter operator ring into a suitable integro-differential operator ring(Theorem 4.8). Finally, we turn to the important special case of polynomial coefficients inSection 5, thus considering integro-differential and differential Rota-Baxter analogs for theWeyl algebra. The most important result is that the so-called integro-differential algebraintroduced in [24] is in fact the ring of differential Rota-Baxter operators with polynomialcoefficients (Corollary 5.4), which also implies the embedding result by specializing thegeneric one (Theorem 5.6).
XING GAO, LI GUO, AND MARKUS ROSENKRANZ Varieties of Operated Algebras
Recall that an Ω -operated algebra ( A ; P ω | ω ∈ Ω) is an algebra A together with certain k -linear operators P ω : A → A . Here no restrictions are imposed on the operators P ω . Thecategory of Ω-operated algebras is denoted by Alg (Ω), the full subcategory of commutativeΩ-operated algebras by
CAlg (Ω). For S ⊆ A , we use the notation ( S ) for the operatedideal generated by S .We describe now the free object C Ω ( X ) of CAlg (Ω) over a countable set of generators X .The construction proceeds via stages C Ω ,n ( X ) that are defined recursively as follows. Westart with C Ω , ( X ) := C ( X ). Then for each ω ∈ Ω we create ⌊ C ( X ) ⌋ ω := {⌊ u ⌋ ω | u ∈ C ( X ) } as a disjoint copy of C ( X ) and define C Ω , ( X ) := C (cid:0) X ⊎ U ω ∈ Ω ⌊ C ( X ) ⌋ ω (cid:1) , where ⊎ means disjoint union. Note that elements in ⌊ C ( X ) ⌋ ω are merely symbols indexedby C ( X ); for example, ⌊ ⌋ ω is not the identity. The inclusion X ֒ → X ⊎ U ω ∈ Ω ⌊ C Ω , ( X ) ⌋ ω induces a monomorphism i , : C Ω , ( X ) = C ( X ) ֒ → C Ω , ( X ) = C (cid:0) X ⊎ U ω ∈ Ω ⌊ C Ω , ( X ) ⌋ ω (cid:1) of free commutative monoids through which we identify C Ω , ( X ) with its image in C Ω , ( X ).For n ≥
2, inductively assume that C Ω ,n − ( X ) has been defined and the embedding i n − ,n − : C Ω ,n − ( X ) ֒ → C Ω ,n − ( X )has been obtained. Then we define C Ω ,n ( X ) := C (cid:0) X ⊎ U ω ∈ Ω ⌊ C n − ( X ) ⌋ ω (cid:1) . Since C Ω ,n − ( X ) = C (cid:0) X ⊎ U ω ∈ Ω ⌊ C Ω ,n − ( X ) ⌋ ω (cid:1) is a free commutative monoid, once again theinjections ⌊ C Ω ,n − ( X ) ⌋ ω ֒ → ⌊ C Ω ,n − ( X ) ⌋ ω induce a monoid embedding C Ω ,n − ( X ) = C (cid:0) X ⊎ U ω ∈ Ω ⌊ C n − ( X ) ⌋ ω (cid:1) ֒ → C Ω ,n ( X ) = C (cid:0) X ⊎ U ω ∈ Ω ⌊ C n − ( X ) ⌋ ω (cid:1) . Finally we define the monoid C Ω ( X ) := [ n ≥ C Ω ,n ( X )whose elements are called (commutative) Ω -decorated bracket words in X . Definition 2.1.
Let X be a set, ⋆ a symbol not in X and X ⋆ := X ∪ { ⋆ } .(a) By an Ω -decorated ⋆ -bracket word on X we mean any expression in C Ω ( X ⋆ ) withexactly one occurrence of ⋆ . The set of all Ω-decorated ⋆ -bracket words on X isdenoted by C ⋆ Ω ( X ).(b) For q ∈ C ⋆ Ω ( X ) and u ∈ C Ω ( X ), we define q [ u ] := q [ ⋆ u ] to be the Ω-decoratedbracket word in C ( X ) obtained by replacing the letter ⋆ in q by u .(c) For s = P i c i u i ∈ k C Ω ( X ), where c i ∈ k , u i ∈ C Ω ( X ) and q ∈ C ⋆ Ω ( X ), we define q [ s ] := P i c i q [ u i ] , which is in k C Ω ( X ).More generally, with ⋆ , · · · ⋆ n distinct symbols not in X , set X ⋆n := X ∪ { ⋆ , . . . , ⋆ n } . IFFERENTIAL ROTA-BAXTER OPERATORS 5 (d) We define an Ω -decorated ( ⋆ , . . . , ⋆ n ) -bracket word on X to be an expression in C Ω ( X ⋆n ) with exactly one occurrence of each of ⋆ j , 1 ≤ j ≤ n . The set of allΩ-decorated ( ⋆ , . . . , ⋆ n )-bracket words on X is denoted by C ⋆n Ω ( X ).(e) For q ∈ C ⋆n Ω ( X ) and u , . . . , u n ∈ k C Ω ( X ), we define q [ u , . . . , u n ] := q [ ⋆ u , . . . , ⋆ n u n ]to be obtained by replacing the letter ⋆ j in q by u j for 1 ≤ j ≤ n .The notation q [ θ ] used above for θ = { ⋆ u } and θ = { ⋆ u , . . . , ⋆ n u n } can beextended to any substitution θ : X ⋆n ∼ → X ⋆n ; see below after Proposition 2.3.Now we describe the free object in the category CAlg (Ω). For each ω ∈ Ω we introducean operator ⌊ ⌋ ω : C Ω ( X ) → C Ω ( X ) acting as u
7→ ⌊ u ⌋ ω . Then ( C Ω ( X ); ⌊ ⌋ ω | ω ∈ Ω) is acommutative operated monoid; its linear span ( k C Ω ( X ); ⌊ ⌋ ω ∈ Ω | ω ∈ Ω) is a commutativeoperated algebra. It is moreover free in the sense of the following proposition [14, 17]. Inthe language of universal algebra, k C Ω ( X ) appears as the term algebra in the variety ofΩ-operated algebras [1]. Proposition 2.2.
The triple ( k C Ω ( X ); ⌊ ⌋ ω | ω ∈ Ω; j X ) , with j X : X ֒ → C Ω ( X ) the naturalembedding, is the free commutative Ω -operated algebra on X . In other words, for anycommutative Ω -operated algebra A and any set map f : X → A , there is a unique extensionof f to a homomorphism ¯ f : k C Ω ( X ) → A of Ω -operated algebras. In the remainder of this section, we assume that k is a Q -algebra. We first define po-larization for the non-commutative case and then induce polarization for the commutativecase via a natural homomorphism. The term polarization is adopted from Rota’s earlystudy [5, p. 928] of this normalization process (second line in the proof of Prop. 2.1).The construction of noncommutative Ω-decorated bracket words M Ω ( X ) is parallel to thecommutative case C Ω ( X ), using everywhere M ( X ) in place of C ( X ); the reader is referredto [17] for details. Clearly, k C Ω ( X ) is the quotient of k M Ω ( X ) modulo the commutators. Proposition 2.3.
The triple ( k M Ω ( X ); ⌊ ⌋ ω | ω ∈ Ω; j X ) , with j X : X ֒ → M Ω ( X ) againthe natural embedding, is the free Ω -operated algebra on X . This means for any Ω -operatedalgebra A and for any set map f : X → A , there exists a unique extension of f to a homo-morphism ¯ f : k M Ω ( X ) → A of Ω -operated algebras. Operated algebras usually satisfy additional relations, for example the aforementionedRota-Baxter axiom ( r f )( r g ) = r f r g + r g r f in the case of Rota-Baxter algebras. We modelsuch relations by decorated bracket words E ⊆ k C Ω ( Y ) or E ⊆ k M Ω ( Y ), depending onwhether we intend the commutative or noncommutative case. Note that here we use a newset of variables Y that should be distinct from the set X of generators (see Proposition 2.7below for an example combining the two sets of variables). Since relations are closed underlinear combinations, we may take E ⊆ k C Ω ( Y ) or E ⊆ k M Ω ( Y ) to be k -submodules.Elements l ∈ E will be called laws of the corresponding variety. In the following, we assumethe noncommutative case but everything can be translated easily to the commutative caseto which we shall return explicitly before Lemma 2.15.For any operated algebra A and θ : Y → A , by the universal property of k M Ω ( Y ) asthe free Ω-operated algebra on Y , there is a unique morphism of Ω-operated algebras¯ θ : k M Ω ( Y ) → A that extends θ . We use the notation l [ θ ] := ¯ θ ( l ) for the corresponding instance of an l ∈ k M Ω ( Y ); formally this is the element of A obtained from l upon replacing XING GAO, LI GUO, AND MARKUS ROSENKRANZ every y ∈ Y by θ ( y ) ∈ A , and ⌊⌋ ω by P ω for ω ∈ Ω. For the special case A = k M Ω ( Y ) thiscovers the substitutions mentioned in Definition 2.1. Definition 2.4.
Let E be a submodule of k M Ω ( Y ).(a) An E -related algebra is defined to be an Ω-operated algebra A such that l [ θ ] = 0 forany law l ∈ E and any assignment θ : Y → A .(b) The substitution closure S ( E ) ⊆ k M Ω ( Y ) of the laws E is defined to be the sub-module spanned by all instances l [ θ ] with l ∈ E and θ : Y → k M Ω ( Y ).If E = { l } , we speak of an l -algebra, and we write S ( l ) for S ( E ).Since it is usually clear from the context that E denotes a set of laws (rather than theground ring), we will often say E -algebra instead of E -related algebra. In the terminologyof universal algebra, the category of E -algebras (for a fixed set of laws E ) forms a variety ,which we write here as Alg (Ω | E ). As mentioned in the Introduction, the concept of varietyis more general since its operators need not be unary or linear. For our purposes, thisextended generality is not needed and would only complicate matters, for instance usingcongruence relations in place of operated ideals [10, § Lemma 2.5.
Let E be a submodule of k M Ω ( Y ) . Then every E -algebra is an S ( E ) -algebra,and vice versa.Proof. The sufficiency is clear since we have E ⊆ S ( E ). For showing the necessity, let A bean E -algebra, and take l [ θ ] ∈ S ( E ) with l ∈ E and θ : Y → k M Ω ( Y ). For any η : Y → A ,define ˜ η : Y → A by setting ˜ η ( y ) := θ ( y )[ η ] for any y ∈ Y , that is, ˜ η ( y ) is obtainedby replacing y in θ ( y ) by η ( y ). Then we have l [ θ ][ η ] = l [˜ η ] = 0, as A is an E -algebraand l ∈ E . (cid:3) Example 2.6.
Let Ω be a singleton and E = k {⌊ y y ⌋ − ⌊ y ⌋ y − y ⌊ y ⌋} . Then S ( E ) = k {⌊ uv ⌋ − ⌊ u ⌋ v − u ⌊ v ⌋ | u, v ∈ M Ω ( Y ) } is the substitution closure. (This describes the variety of differential algebras.)Using substitution closure, it is easy to characterize the free E -algebra . Again, this is aspecial case of a well-known result in universal algebra [10, Prop. 1.3.6]. Proposition 2.7.
For any submodule E ⊆ k M Ω ( Y ) and any set X , let S E = S E ( X ) denotethe operated ideal of k M Ω ( X ) generated by all l [ θ ] with l ∈ E and θ : Y → k M Ω ( X ) . Thenthe free E -algebra on X is the quotient F E ( X ) := k M Ω ( X ) /S E . Further exploiting the linear structure of Ω-operated algebras, it turns out that we mayactually assume that E consists of linear combinations of multilinear monomials sharing thesame variables. Let us make this precise. Given a monomial u ∈ M Ω ( Y ), we define its degreein y ∈ Y , denoted by deg y u , as the number of times that y appears in u . Its total degree isgiven by deg u := P y ∈ Y deg y u . Note that if deg y u = n there exists q ∈ M ⋆n Ω ( Y \{ y } ) suchthat u = q [ y, . . . , y ]. We call l multilinear c if deg y l = 1 for each variable y appearing in l . c In rewriting, this means one can turn the equation l = λ u + · · · + λ m u m = 0 into a rewrite rule u k → ( λ /λ k ) u + · · · + ( λ k − /λ k ) u k − + ( λ k +1 /λ k ) u k +1 + · · · + ( λ m /λ k ) u m for any leading term u k ,and the resulting rule will be linear in the sense of [1, Def. 6.3.1]. In fact, the rewriting terminology allowsvariables to be absent in terms; this is not needed for our present purposes. IFFERENTIAL ROTA-BAXTER OPERATORS 7
For any l ∈ E and y ∈ Y , let l y,n ( n ≥
0) denote the linear combination of thosemonomials of l that have degree n in y , with the convention that l y,n = δ ,n l if y does notappear in l . Then l has the unique homogeneous decomposition l = X n ≥ l y,n into its y -homogeneous parts. Definition 2.8.
Let l ∈ k M Ω ( Y ) be homogeneous in y ∈ Y with deg y l = n such thatone has l = P i ≤ k c i q i [ y, . . . , y ] with coefficients c i ∈ k × and monomials q i ∈ M ⋆n Ω ( Y \{ y } ).Then the polarization of l in y is P y ( l ) := X τ ∈ S n X i ≤ k c i q i [ τ y , . . . , τ y n ] , where the substitution variables y , . . . , y n ∈ Y are mutually distinct.Note that l can be recovered (up to a multiple) from P y ( l ) through replacing y , . . . , y n by y ; this process is called centralization . For terms containing more variables, we can alsoapply polarization so long as the terms are homogeneous in all variables. Definition 2.9.
Let l ∈ k M Ω ( Y ) be homogeneous in all its variables. Then we define its polarization P ( l ) as the result of successively polarizing all variables in l .A different order of the variables in l in the polarization process and a different choiceof the substitution variables in l amounts to a bijection of the substitution variables. Thusthe polarization of l is unique (up to bijection of variables) and multilinear . Renamingvariables if necessary, we may further assume that the number of variables not appearingin E is countably infinite; hence polarization will not run out of substitution variables. Example 2.10.
In this example, Ω is a singleton so that we may abbreviate ⌊ . . . ⌋ ω by ⌊ . . . ⌋ and M Ω ( Y ) by M ( Y ).(a) Consider l = ⌊ y ⌊ y ⌋⌋ ∈ M ( Y ) with y ∈ Y . Its polarization is given by P ( l ) = P y ( l ) = ⌊ y ⌊ y ⌋⌋ + ⌊ y ⌊ y ⌋⌋ , and we recover 2 ⌊ y ⌊ y ⌋⌋ by y , y y .(b) Let l = x y ∈ M ( Y ) with x, y ∈ Y . Then P y ( l ) = x y y + x y y and hence P ( l ) = P x ( P y ( l )) = y y y y + y y y y + y y y y + y y y y with y , y , y , y ∈ Y .(c) For l = ⌊ y ⌋ − ⌊ y ⌋ y ∈ k M ( Y ) we get P y ( l ) = ⌊ y y ⌋ + ⌊ y y ⌋ − ⌊ y ⌋ y − ⌊ y ⌋ y . Lemma 2.11.
Let E ⊆ k M Ω ( Y ) be a submodule and l ∈ S ( E ) arbitrary. If l = P i l y,i isthe homogeneous decomposition of l in y , we have l y,i ∈ S ( E ) for each i .Proof. If n is the maximal degree of l in y , clearly l y,i = 0 ∈ S ( E ) for i > n . Replacing y in l by jy for 1 ≤ j ≤ n + 1, we obtain l [ jy ] = n X i =0 j i l y,i . Regard these equations as a linear system in unknowns l y,i . Then the coefficient matrixis non-singular as a Vandermonde matrix. Thus one can solve for l y, , . . . , l y,n as Q -linearcombinations of l [ jy ] ∈ S ( E ), which shows that the l y,i are themselves in S ( E ). (cid:3) XING GAO, LI GUO, AND MARKUS ROSENKRANZ
We can use the preceding technique to conclude that the polarized form of a law isalways contained in the substitution closure. To see why this is so, consider a typicalexample l = ⌊ y ⌊ y ⌋⌋ ∈ M ( Y ) with y ∈ Y . Replacing y by y + y with y , y ∈ Y , we have ⌊ ( y + y ) ⌊ y + y ⌋⌋ = ⌊ y ⌊ y ⌋⌋ + ⌊ y ⌊ y ⌋⌋ + ⌊ y ⌊ y ⌋⌋ + ⌊ y ⌊ y ⌋⌋ ∈ S ( l )and so P y ( l ) := ⌊ y ⌊ y ⌋⌋ + ⌊ y ⌊ y ⌋⌋ ∈ S ( l ) by Lemma 2.11. Let us state the general result. Lemma 2.12.
Let E ⊆ k M Ω ( Y ) be a submodule, and assume l ∈ E is homogeneous in y .Then we have P y ( l ) ∈ S ( E ) .Proof. Since P y is a k -linear operator, we only need to consider l ∈ M Ω ( Y ). We provethe result by induction on deg y l . If deg y l = 1, we have P y ( l ) = l ∈ S ( E ). Assuming theresult for deg y l ≤ n −
1, we consider the case deg y l = n . By our assumption on l , wehave l = q [ y, . . . , y ] with q ∈ M ⋆n Ω ( Y \{ y } ). Replacing y by z + z , we obtain(1) S ( E ) ∋ q [ z + z , · · · , z + z ] = q [ z , . . . , z ] + q [ z , . . . , z ] + ˜ l. Since q [ z , . . . , z ] and q [ z , · · · , z ] are in the k -module S ( E ) we have ˜ l ∈ S ( E ). We deter-mine the homogeneous decomposition ˜ l = P n − j =1 ˜ l z ,j with respect to z . From Lemma 2.11,we know ˜ l z ,j ∈ S ( E ). We note that deg z ˜ l < n and likewise deg z ˜ l < n . By the inductionhypothesis, we have P z (˜ l z ,j ) ∈ S ( E ) for 0 < j < n . Moreover, ˜ l z ,j is homogeneous in z of degree j >
0, hence ˜ l z ,j is homogeneous in z of degree n − j < n . From the definitionof P z (˜ l z ,j ), we see that P z (˜ l z ,j ) is also homogeneous in z of degree n − j < n . By theinduction hypothesis again, we obtain now P z ( P z (˜ l z ,j )) ∈ S ( E ). Thus it suffices to prove(2) P y ( l ) = P z ( P z (˜ l z ,j ))for 0 < j < n . By its definition, P y ( l ) is a sum of n ! terms each of the form q [ τ y , . . . , τ y n ]with τ ∈ S n , so we write it as(3) P y ( l ) = X τ ∈ S n q [ τ y , . . . , τ y n ] . On the other hand, we note that q [ z + z , . . . , z + z ] = X I ⊆ [ n ] q [ I ; z , z ] , where for each subset I ⊆ [ n ], the term q [ I ; z , z ] is obtained from q by replacing ⋆ i by z for i ∈ I and by z otherwise. For I = [ n ] and I = ∅ we obtain the first two terms in (1),while for 0 < j < n we get ˜ l z ,j = X | I | = j q [ I ; z , z ] . Then we have(4) P z ( P z (˜ l z ,j )) = X | I | = j,τ ,τ q [ I ; τ , τ ] , where τ ranges over all bijections I ∼ → [ j ] and τ over all bijections [ n ] \ I ∼ → [ n ] \ [ j ], andwhere q [ I ; τ , τ ] := q [ τ y , . . . , τ y n ] is obtained from q ∈ C ⋆n Ω ( X ) via the permutation τ ∈ S n defined by τ ( i ) = τ ( i ) for i ∈ I and τ ( i ) = τ ( i ) for i ∈ [ n ] \ I . By this construction,distinct triples ( I ; τ , τ ) corresponds to distinct permutations τ ∈ S n , so distinct monomials IFFERENTIAL ROTA-BAXTER OPERATORS 9 in Eq. (4) also correspond to distinct monomials in Eq. (3). However, there are exactly (cid:0) nj (cid:1) j !( n − j )! = n ! such triples, so the sums in the two equations must agree, and the proofof (2) is complete. (cid:3) We introduce now polarization for a collection of laws E . It turns out that the resultingmodule, spanned by multilinear monomials, defines the same variety as the original E . Definition 2.13.
Let E be a submodule of k M Ω ( Y ). Then we define its polarization P ( E )as the submodule of k M Ω ( Y ) spanned by the polarizations of all homogeneous componentsof elements of E . Theorem 2.14.
For any submodule E ⊆ k M Ω ( Y ) , an Ω -operated algebra is an E -algebraif and only if it is a P ( E ) -algebra.Proof. By construction, we have E ⊆ P ( E ). By Lemma 2.5, it suffices to prove that P ( E )is contained in S ( E ). Choose a law l ∈ E and a variable y ∈ Y appearing in l . Then wehave P y ( l ) ∈ S ( E ) from Lemma 2.12; repeating the process for the other variables com-pletes the proof. (cid:3) Let us now go back to the commutative case . The concepts of degree and total degreecan of course be defined in the same way. By a straightforward induction on the depth ofbracket words, one obtains the following normalization result.
Lemma 2.15.
Every element of C Ω ( Y ) can be uniquely written as a bracket word in whichall variables of Y appear in increasing order. The lemma gives an embedding ̺ : k C Ω ( Y ) ֒ → k M Ω ( Y ) as modules. On the other hand,as algebras, we have k C Ω ( Y ) ∼ = k M Ω ( Y ) / ∼ , where ∼ is the operated ideal of k M Ω ( Y )generated by the set { uv − vu | u, v ∈ M Ω ( Y ) } . Let π : k M Ω ( Y ) → k C Ω ( Y ) be the naturalprojection. We carry over the notion of polarization from the noncommutative case, in thefollowing natural way (by abuse of notation we continue to use the same symbol P for thecommutative polarization). Definition 2.16.
Let E be a submodule of k C Ω ( Y ). If l ∈ k C Ω ( Y ) is homogeneous in allits variables, we define its polarization as P ( l ) := π ( P ( ̺ ( l ))). Similarly, the polarization ofthe module is defined as P ( E ) := π ( P ( ̺ ( E ))). Example 2.17.
As in Example 2.10, we suppress the (unique) operator labels.(a) Let l = x y ∈ k C Ω ( Y ) with x, y ∈ Y . Then its polarization is P ( l ) = 4 y y y y with y , y , y , y ∈ Y .(b) For l = ⌊ y ⌋ − ⌊ y ⌋ y ∈ k C Ω ( Y ) we have P y ( l ) = 2 ⌊ y y ⌋ − ⌊ y ⌋ y − y ⌊ y ⌋ .As an immediate corollary to Theorem 2.14, we obtain that also in the commutative caseone may polarize all laws and still describe the same variety. Corollary 2.18.
For any submodule E ⊆ k C Ω ( Y ) , an Ω -operated algebra is an E -algebraif and only if it is a P ( E ) -algebra. In this sense, it is no loss of generality if one requires that E -algebras be described by multilinear laws (but see our remarks in the Introduction). The classical examples forvarieties of operated algebras are indeed of this form. For avoiding cumbersome notation,we shall henceforth dispense with the brackets in the main examples, writing ∂f for ⌊ f ⌋ ∂ and r f for ⌊ f ⌋ r . Likewise, we shall often identify the operations P ω : A → A of an operatedalgebra ( A ; P ω | ω ∈ Ω) with their labels ω . Note also that an E -algebra is to be understoodas the corresponding k E -algebra if E is not already a k -submodule of k C Ω ( Y ). Of course,equations of the form l = r are a shorthand for l − r ∈ k C Ω ( Y ). Example 2.19.
Take Y = { f, g } for the variables. Then the four main varieties for doinganalysis are the following collections of E -algebras with operators Ω.(a) The variety Diff λ of differential k -algebras [25, 22, 8] of weight λ ∈ k :Here Ω( Diff λ ) = { ∂ } , and E ( Diff λ ) := { ∂ fg = ( ∂f ) g + f ( ∂g ) + λ ( ∂f )( ∂g ) } consistsonly of the Leibniz axiom.(b) The variety RB λ of Rota-Baxter k -algebras [3, 31, 32, 18] of weight λ ∈ k :Here Ω( RB λ ) = { r } and E ( RB λ ) := { ( r f )( r g ) = r f r g + r g r f + λ r fg } consistsof the Rota-Baxter axiom.(c) The variety DRB λ of differential Rota-Baxter k -algebras [19] of weight λ ∈ k :Now Ω( DRB λ ) = Ω( Diff λ ) ∪ Ω( RB λ ) = { ∂, r } contains both operators, and thelaws are given by E ( DRB λ ) = E ( Diff λ ) ∪ E ( RB λ ) ∪ { ∂ r f = f } . The last law isthe so-called section axiom, which specifies a “weak coupling” between ∂ and r .(d) The variety ID λ of integro-differential k -algebras [20] of weight λ ∈ k :This has the same operators Ω( ID λ ) = Ω( DRB λ ) but different laws—the weakcoupling of DRB λ is replaced by a stronger coupling [20, Thm. 2.5]: From variousequivalent formulations, we choose E ( ID λ ) = E ( Diff λ ) ∪ { f r g = r f ′ r g + r fg + λ r f ′ g, ∂ r f = f } , where the middle law describes integration by parts (which is strictly stronger thanthe Rota-Baxter axiom of RB λ ).As in [27, Def. 8] we call a differential (differential Rota-Baxter, integro-differential) algebra ordinary if ker ∂ = k . For example, ( k [ x ] , d/dx ) is ordinary but ( k [ x, y ] , ∂/∂x ) is not.The above varieties provide the basic motivation for our study of the operator rings (to bedefined in the next section), which are crucial for solving boundary problems in an algebraicsetting. While this is not the focus of the present paper, the reader may refer to the end ofthe next section for some remarks on this application (and especially on the role played bydifferential Rota-Baxter algebras).3. Operator Rings and Modules
We begin now with the description of the operator rings for a given variety of operatedalgebras. This proceeds in two steps —we introduce first a class of operator rings thatdoes not take into account any law that might be imposed on a given operated algebra(Definition 3.1). In the second step we can then impose the given laws in a suitable formonto the free operators constructed in the first step (Definition 3.2). As mentioned in theIntroduction, we work here only with commutative coefficient algebras, so from now oneverything is commutative (except of course the operator rings).
Definition 3.1.
Let ( A ; P ω | ω ∈ Ω) be a commutative Ω-operated algebra. Then we definethe induced ring of free operators as the free product A [Ω] := A ∗ k h Ω i .The obvious abuse of notation A [Ω] is harmless since confusion with the commutativepolynomial ring is unlikely. See also Remark 3.3 for further justification of this notation. IFFERENTIAL ROTA-BAXTER OPERATORS 11 If A is an E -algebra for a submodule E ⊆ k C Ω ( Y ), we would like an operator ring thatreflects the laws of E ; we will construct it as a suitable quotient of the free operators A [Ω],using the following translation from laws to operators . The operator corresponding to aspecific law shall be called the induced relator. For example, a differential ring ( A, ∂ ) isan Ω-operated algebra with Ω = { ∂ } satisfying the Leibniz law ( f g ) ′ = f ′ g + f g ′ , whichinduces the relator ∂f − f ∂ − f ′ ∈ A [Ω]; see Proposition 3.4 (a) for more details.From Corollary 2.18, we may assume that E ⊆ k C Ω ( Y ) is spanned by homogeneous andmultilinear elements. We may also assume that none of these is of total degree 0 sincesuch laws are either redundant (if l = 0) or else describe a trivial variety. Since Y iscountable, we can write its elements as y j ( j ∈ N ). Then every basis element of E havingtotal degree k + 1 can be written in the variables y , . . . , y k by a change of variables; theresulting variety remains the same by Lemma 2.5. We call such basis elements the standardlaws for the variety. For the translation process, we think of the lead variable y as the argument of the induced relator with y , . . . , y k constituting its parameters . The lattercan be instantiated by assignments , which we view as arbitrary maps a : Y ′ → A on theparameter set Y ′ := Y \ { y } . Since arguments are processed from right to left, we shalluse the order y k , . . . , y , y in the sequel.The induced relator [ l ] a ∈ A [Ω] for a standard law l under an assignment a is now definedby recursion on the depth of l . Taking l [ l ] a to be k -linear, it suffices to considermonomials l . For the base case take l ∈ C Ω , ( y k , . . . , y , y ) = C ( y k , . . . , y , y ) with l oftotal degree k + 1. By multilinearity l = y k · · · y y , and we set [ l ] a := a ( y k ) · · · a ( y ).Now assume [ . . . ] a has been defined for monomial standard laws of depth at most n andconsider l ∈ C Ω ,n +1 ( y k , . . . , y , y ). By multilinearity and the definition of C Ω ,n +1 , thereexists t ∈ C Ω ,n +1 ( y k , . . . , y ) such that either l = ty or l = t ⌊ l ′ ⌋ ω for a certain operatorlabel ω ∈ Ω and l ′ ∈ C Ω ,n ( y k , . . . , y , y ) being a monomial standard law of depth n . Weset [ l ] a := ¯ a ( t ) in the former case and use the recursion [ l ] a := ¯ a ( t ) ω [ l ′ ] a in the latter,where ¯ a : C Ω ,n +1 ( y k , . . . , y ) → A is the monoid homomorphism induced by the (restricted)assignment map a : { y k , . . . , y } → A through the universal property of C Ω ,n +1 ( y k , . . . , y ).This completes the definition of [ l ] a . We can now introduce the ring of E -operators as thequotient of the free operators modulo the translated variety laws. Definition 3.2.
Let A be an E -algebra for a submodule E ⊆ k C Ω ( Y ). Then we define the ring of E -operators as A [Ω | E ] := A [Ω] / [ E ], where [ E ] ⊂ A [Ω] is the ideal generated by [ l ] a for all standard laws l ∈ E and assignments a : Y ′ → A .Let us now look at the classical linear operator rings for the varieties of Example 2.19.Each of them comes with a noncommutative Gr¨obner basis and term order, providingtransparent canonical forms and enabling a computational treatment via the well-knownDiamond Lemma [4, Thm. 1.2]. As in the latter reference, we write the elements of theGr¨obner basis in the form m → p instead of m − p in order to emphasize the role of theleading monomial m and the tail polynomial p , suggesting their use as rewrite rules. Remark 3.3.
In the sequel, we identify identities with the varieties they define; for examplewe write F [ ∂ | Diff λ ] for F [ ∂ | E ( Diff λ )], with E ( Diff λ ) taken from Example 2.19(a).In practice, this notation is of course contracted to F [ ∂ ], further justifying the abuse ofnotation mentioned after Definition 3.1. Proposition 3.4.
Let > be any graded lexicographic term order on F [Ω] satisfying ∂ > f for all f ∈ F if ∂ ∈ Ω . Then the following four linear operator rings d can be characterizedby Gr¨obner bases as follows (primes and backprimes refer to the operations in F ): (a) Given ( F , ∂ ) ∈ Diff λ , the ring of differential operators F [ ∂ ] := F [ ∂ | Diff λ ] has theGr¨obner basis GB(
Diff λ ) = { ∂f → f ∂ + λ f ′ ∂ + f ′ | ( f ∈ F ) } . (b) Given ( F , r ) ∈ RB λ , the ring of integral operators F [ r ] := F [ r | RB λ ] has theGr¨obner basis GB( RB λ ) = { r f r → f r − r f − λ r f | f ∈ F } . (c) Given ( F , ∂, u ) ∈ DRB λ , we consider next the ring of differential Rota-Baxteroperators F [ ∂, u ] := F [ ∂, u | DRB λ ] . Its Gr¨obner basis is given by the combinedrewrite rules GB(
DRB λ ) = GB( Diff λ ) ∪ GB( RB λ ) ∪ { ∂ r → } . (d) For ( F , ∂, r ) ∈ ID λ , the ring of integro-differential operators F [ ∂, r ] := F [ ∂, r | ID λ ] has Gr¨obner basis GB( ID λ ) := GB( DRB λ ) ∪ { r f ∂ → f − r f ′ − e ( f ) e | f ∈ F } ,provided e the shift f f + λf ′ has an inverse f f .We have F [ ∂, r ] = F [ ∂ ] ∔ F [ r ] \ F ∔ ( e ) as k -modules, where F [ ∂, r ] contains both F [ ∂ ] and F [ r ] as subalgebras. Moreover, if I is the ideal generated by { e f − e ( f ) e | f ∈ F } , wehave F [ ∂, r ] ∼ = F [ ∂, u ] /I , where u := r is viewed as part of ( F , ∂, u ) ∈ DRB λ .Proof. Let us first prove the four items stated in the proposition (viewing all axioms in themain variable g and using arbitrary assignments a with a ( f ) ∈ F shortened to f ):(a) Clearly, the only relators are [ l ] a = ∂f − f ′ − f ∂ − λ f ′ ∂ , corresponding to the Leibnizaxiom l := ∂fg − ( ∂f ) g − f ( ∂g ) − λ ( ∂f )( ∂g ) = 0. From ∂ > f one sees that theleading monomial is ∂f . There is just one S-polynomial coming from the overlapambiguity ∂fg between the rule ∂f → f ∂ + λ f ′ ∂ + f ′ and the (tacit) rule fg → f ∗ F g .Using the Leibniz rule in F , one checks immediately that the S-polynomial reducesto zero, so GB( Diff λ ) is indeed a Gr¨obner basis.(b) Here the Rota-Baxter axiom l := ( r f )( r g ) − r f r g − r g r f − λ r fg = 0 yields therelators [ l ] a = f r − r f r − r f − λ r f whose leading monomial is r f r because theterm order is graded. One obtains an S-polynomial from the self-overlap r f r ¯ f r ofthe rule r f r → f r − r f − λ r f . Again one checks that this S-polynomial reducesto zero, and GB( RB λ ) is thus a Gr¨obner basis.(c) The relators are those of (a) and (b), and additionally ∂ r − ∂ r → ∂ r f r , and again its S-polynomial immediatelyreduces to zero so that GB( DRB λ ) is a Gr¨obner basis.(d) From the definition e := 1 − r ∂ and the Leibniz rule we have the tautological relation f g − e ( f g ) = r f ′ g + r f g ′ + λ r f ′ g ′ . Let us start by recalling [20, Thm. 2.5(b)] that the well-known integration-by-partsaxiom g r f − r g ′ r f − r fg − λ r g ′ f = 0 characterizing ID λ is equivalent to themultiplicativity condition e ( f g ) = e ( f ) e ( g ). Indeed, by the definition of e , the d Note that we rely on the context to disambiguate the notations F [ ∂, u ] and F [ ∂, r ]. In the frame of thispaper, a Rota-Baxter operator will always be denoted by u when it comes from a differential Rota-Baxteralgebra, and by r when it comes from an integro-differential algebra. e This is of course always satisfied in the zero weight case (with trivial shift). But it is also satisfied in theclassical example with weight λ = ±
1: the sequence space M Z over a k -module M with forward/backwarddifference as derivation. This has increment/decrement as mutually inverse shifts. IFFERENTIAL ROTA-BAXTER OPERATORS 13 condition is r ( f g ) ′ + r f ′ · r g ′ = f r g ′ + g r f ′ . From this one obtains the axiom byexpanding ( f g ) ′ according to the (weighted) Leibniz rule and substituting f r f .Conversely, the axiom of ID λ implies that im r is an ideal of F . From the definitionof e one has the identity f g = e f e g + e f r g ′ + e g r f ′ + r f ′ · r g ′ , which yieldsmultiplicativity upon applying e since e projects onto ker ∂ so that e f e g ∈ ker ∂ is left invariant (note that ker ∂ is a subalgebra of F because of the Leibniz rule).The three remaining terms are all in the ideal r f , which is the complement of ker ∂ under the projection, hence they vanish under e .We exploit the equivalent characterization of ID λ in terms e ( f g ) = e ( f ) e ( g ) bysubstituting the tautological relation from above to obtain the equivalent law l := r f g ′ − f g + r f ′ g + λ r f ′ g ′ + e ( f ) e ( g ) = 0 . This axiom gives rise to the new relator [ l ] a = r ( f + λf ′ ) ∂ − f + r f ′ + e ( f ) e , whichyields the rule r f ∂ → f − r f ′ − e ( f ) e upon replacing f by f and picking r f ∂ as the leading monomial due to the grading. Since the plain Rota-Baxter axiom isimplied by the integration-by-parts axiom [20, Lem. 2.3(b)], the relator constructedin Item (c) is also contained in the current relator ideal, hence the correspondingrule is admissible in GB( ID λ ). For seeing that this is again a Gr¨obner basis, werefer to the proof of [27, Prop. 13]. The latter assumes that k is a field and usesa k -basis of F but this is only a convenience tuned to the algorithmic treatment.As pointed out after [28, Prop. 26], choosing a basis is avoided by factoring outthe linear ideal (this happens in the formation of the free operators A [Ω] in thecurrent setup). Note also that here we take Φ = { e } , which means all rules of [27,Table 1] with characters ϕ, ψ ∈ Φ on the right-hand side are absent. f With thisunderstanding, the above definition of F [ ∂, r ] coincides with the one in [27], whichtherefore establishes GB( ID λ ) as a Gr¨obner basis.We prove now the k -module decomposition(5) F [ ∂, r | ID λ ] = F [ ∂ | Diff λ ] ∔ F [ r | RB λ ] \F ∔ ( e )with ( e ) ⊂ F [ ∂, r | ID λ ] being the two-sided ideal generated by e . Note that here and in therest of this proof, we renounce the abbreviation of F [Ω | E ] by F [Ω] used in the statement ofthe proposition. This is because we need to distinguish the free operator ring from various E -operator rings. Furthermore, we write F [Ω] E for the k -submodule of normal formsin F [Ω] with respect to the reduction system induced by E and the given term order on F [Ω].By the well-known Diamond Lemma [4, Thm. 1.2], we have F [ ∂, r ] = F [ ∂, r ] ID λ ∔ [ ID λ ].We claim that it suffices to prove(6) F [ ∂, r ] ID λ = F [ ∂ ] Diff λ ∔ F [ r ] RB λ \F ∔ ( e ) ID λ . f The character e ∈ Φ is not part of the operator set Ω, and its appearance on the right-hand side is tobe understood merely as an abbreviation e := 1 − r ∂ . Moreover, the corresponding rules with e on theleft-hand side are not required in E since they follow from the other rules. Indeed, substituting the decomposition (6) into the Diamond-Lemma decomposition andthen taking the quotient by [ ID λ ] yields g (7) F [ ∂, r | ID λ ] = F [ ∂ ] Diff λ + [ ID λ ][ ID λ ] ∔ F [ r ] RB λ \F + [ ID λ ][ ID λ ] ∔ ( e ) ID λ + [ ID λ ][ ID λ ] . Since
Diff λ ⊂ ID λ , we may replace the first denominator on the right-hand side of (7)by (cid:0) F [ ∂ ] Diff λ + [ Diff λ ] (cid:1) + [ ID λ ] = F [ ∂ ] + [ ID λ ], using now the Diamond Lemma for Diff λ .In the same way, the second denominator is given by F [ r ] \ F + [ ID λ ]. For the thirddenominator we get ( e ) directly from the Diamond Lemma. Applying the second isomor-phism theorem to the first and second summand yields (5) since [ ID λ ] ∩ F [ ∂ ] = [ Diff λ ]and [ ID λ ] ∩ (cid:0) F [ r ] \ F (cid:1) = [ RB λ ], noting that ( e ) / [ ID λ ] is just ( e ) ⊂ F [ ∂, r | ID λ ] in (5).We give now a proof of (6), which follows closely the more general argument h givenin [28], specifically Lemma 23 as well as Propositions 25 and 26 therein. Let us start byanalyzing the irreducible monomials. We claim that each monomial w ∈ F [ ∂, r ] ID λ iseither of the form w = f ∂ i ∈ F [ ∂ ] Diff λ ( f ∈ F , i ≥
0) or w = f r g ∈ F [ r ] RB λ ( f, g ∈ F )or f r ∂ i +1 ( f ∈ F , i ≥ w contains any occurrences of ∂ , they must be in thetail of w since ∂f ( f ∈ F ) is reducible relative to [ Diff λ ] ⊂ ID λ and also ∂ r relativeto [ ∂ r − ⊂ [ ID λ ]. This means we have w = v∂ i with prefix monomial v ∈ F [ r ] and i ≥ v can have at most one occurrence of r since r f r ( f ∈ F ) is reducible relativeto [ RB λ ] ⊂ [ ID λ ]. Hence we have either w = g r f ∂ i or w = f ∂ i for some f, g ∈ F . In thelatter case, we obtain w ∈ F [ ∂ ] Diff λ and are done. In the former case, we can must have i = 0or f = 1 since otherwise w is reducible relative to relative to [ r f ∂ − f + r f ′ + e ( f ) e ] ⊂ [ ID λ ].Hence we have either the case w = g r f ∈ F [ r ] RB λ , where f = 1 is possible. Or else wehave the irreducible monomial w = g r ∂ i with i > e ) ID λ ; unlike those of F [ ∂ ] Diff λ and F [ r ] Diff λ ,these are not monomials. Since any element of ( e ) ID λ can be written as a k -linear com-bination of w e ˜ w = 0 with monomials w, ˜ w , it suffices to analyze those. As we have seenabove, if w contains any occurrences of ∂ , they must be at its tail. But since ∂ e = 0, therecan in fact be no ∂ in w . By the above analysis of normal forms for w , the only remainingpossibilities are w = f and w = f r g for some f, g ∈ F . But the latter is also excludedsince r g e = r g − r g r ∂ is reducible relative to [ RB λ ] ⊂ ID λ . Hence we conclude w = f .Regarding the monomial ˜ m , we it cannot start with any g ∈ F since e g = g − r ∂g is re-ducible relative to [ Diff λ ] ⊂ ID λ . Furthermore, ˜ w cannot start with r since e r = 0. By ouranalysis of irreducible monomials, this leaves with the only remaining possibility ˜ w = ∂ i .Altogether this show that w e ˜ w = f e ∂ i . We may thus conclude that all three k -moduleson the right-hand side of (6) are in fact left F -modules with the following generators:While F [ ∂ ] Diff λ is generated by ∂ i ( i ≥ F [ r ] RB λ \ F by r f ( f ∈ F ), the normalforms in ( e ) ID λ are generated by e ∂ i .For establishing (6), it is sufficient to show that each U ∈ F [ ∂, r ] ID λ splits uniquelyas U = U ∂ + U r + U e , containing a part U ∂ ∈ F [ ∂ ] Diff λ , a part U r ∈ F [ r ] RB λ \F , and finally apart U e ∈ ( e ) ID λ . Each irreducible monomial f ∂ i of U is put into U ∂ , and each irreduciblemonomial f r g into U r . For irreducible monomials of the form f r ∂ i +1 = f ∂ i − f e ∂ i , g Note that M = A ∔ B ∔ C ∔ Z implies M/Z = ( A + Z ) /Z ∔ ( B + Z ) /Z ∔ ( C + Z ) /Z for arbitrarysubmodules A, B, C, Z of some module M . h The proofs in [28] use only ring-theoretic properties of k ; no field or zero characteristic is required.They are more general in that they allow character sets Φ ) { e } . IFFERENTIAL ROTA-BAXTER OPERATORS 15 we put f ∂ i into U ∂ and − f e ∂ i into U e . Thus we have U = U ∂ + U r + U e ; let us nowprove uniqueness. Hence assume P i a i ∂ i + P i b i r c i + P i d i e ∂ i = 0, each sum havingfinitely many nonzero coefficients a i , b i , c j , d i ∈ F . By the definition of e , this is the sameas P i ( a i + d i ) ∂ i + P i b i r c i − P i d i r ∂ i +1 = 0. By the definition of the free operatorring F [ ∂, r ], all monomials are linearly independent over k , hence a i + d i = b i = d i = 0and then also a i = 0. This completes the uniqueness proof for splitting U . We have nowestablished the k -module decomposition (6) and therefore also (5). Since F [ ∂ ] and F [ r ]are both closed under multiplication, they are subalgebras of F [ ∂, r ].Finally, let us prove the quotient statement F [ ∂, u | ID λ ] ∼ = F [ ∂, u | DRB λ ] /I , wherefor once we use the same symbol for the Rota-Baxter operator in ID λ and DRB λ . (Recallthat the notational distinction between r and u is purely a convenience that allows us tosuppress the laws to be factored out.) Writing out the definitions, we must thus prove F [ ∂, u ][ ID λ ] ∼ = F [ ∂, u ][ DRB λ ] . [ e f − e ( f ) e | f ∈ F ] , which reduces to showing [ ID λ ] / [ DRB λ ] = [ e f − e ( f ) e | f ∈ F ] by the third isomorphismtheorem. Hence it suffices to show [ ID λ ] = [ DRB λ ] ∔ [ e f − e ( f ) e | f ∈ F ] as k -modules,where the directness of the sum is obvious. For the inclusion from left to right, we mustshow that every r f ∂ − f + r f ′ + e ( f ) e is in [ ID ′ λ ] := [ DRB λ ] + [ e f − e ( f ) e | f ∈ F ].Substituting f + λf ′ for f , we may also show that every r f := r ( f + λf ′ ) ∂ − f + r f ′ + e ( f ) e is in [ ID ′ λ ]. But we have indeed r f = r (cid:16) f ∂ + λ f ′ ∂ + f ′ − ∂f (cid:17) − (cid:16) e f − e ( f ) e (cid:17) ∈ [ ID ′ λ ]since the first summand is in Diff λ ⊂ DRB λ and the second in [ e f − e ( f ) e | f ∈ F ]. Forthe inclusion from right to left, it suffices to show that every e f − e ( f ) e is in ID λ . But wehave just proved that r f + e f − e ( f ) e ∈ [ Diff λ ] ⊂ [ ID λ ]. Since we have also r f ∈ ID λ , theproof is completed. (cid:3) As the name suggests, there is another important aspect to E -operators that we shouldconsider here—they operate on suitable domains. These domains are a special class of mod-ules that we shall now introduce. Recall first that an Ω -operated module ( M ; p ω | ω ∈ Ω) overa commutative ring A is an A -module M with A -linear operators p ω : M → M . As in thecase of Ω-operated algebras, no restrictions are imposed on the operators p ω . An operatedmorphism ϕ : ( M ; p ω | ω ∈ Ω) → ( M ′ ; p ′ ω | ω ∈ Ω) is an A -linear homomorphism ϕ : M → M ′ such that ϕ ◦ p ω = p ′ ω ◦ ϕ for all ω ∈ Ω; the resulting category of Ω-operated modules over A is denoted by Mod A (Ω).Now assume that A ∈ CAlg (Ω) is an operated algebra. Then the free operators T ∈ A [Ω] act naturally on the Ω-operated A -module M . Since T is a k -linear combination ofnoncommutative monomials t ∈ M (Ω ⊎ A ), it suffices to define t · m for m ∈ M . By theuniversal property of M (Ω ⊎ A ), we obtain a unique monoid action by setting ω · m := p ω ( m )for ω ∈ Ω and a · m := am for a ∈ A . Thus M becomes an A [Ω]-module, and we can nowintroduce the module-theoretic analog of E -algebras. Definition 3.5.
Fix an operated algebra A ∈ CAlg (Ω) and a submodule E ⊆ k C Ω ( Y ) ofstandard laws. Then an E -related module over A is an operated module M ∈ Mod A (Ω)with L · m = 0 for all relators L ∈ [ E ] and m ∈ M . Again we will briefly speak of E -modules (since the context will make it clear that E isa set of laws). They form a full subcategory of Mod A (Ω) denoted by Mod A (Ω | E ). Therole of the E -operator ring becomes clear now: Operators correspond to the natural actiondefined above if A is an E -algebra. This can be made precise by the following statement. Proposition 3.6.
Let A be an E -algebra for a submodule E ⊆ k C Ω ( Y ) . Then we have theisomorphism of categories Mod A (Ω | E ) ∼ = Mod A [Ω | E ] .Proof. As noted above, an E -module M ∈ Mod A (Ω | E ) ⊆ Mod A (Ω) can also be viewedas an A [Ω]-module under the natural action, and as such it satisfies [ E ] · M = 0. Butthen the action of A [Ω | E ] with ( T + [ E ]) · m := T · m is well-defined and gives M thestructure of an A [Ω | E ]-module. Conversely, every such module restricts to an operatedmodule M ∈ Mod A (Ω) with [ E ] · M = 0.Of course, every morphism of Mod A [Ω | E ] is also a morphism of Mod A (Ω | E ). For the otherdirection, let ϕ be a morphism of E -modules. For showing ϕ (cid:0) ( T +[ E ]) · m (cid:1) = (cid:0) T +[ E ] (cid:1) · ϕ ( m )for T ∈ A [Ω] and m ∈ M , it suffices to show ϕ ( T · m ) = T · ϕ ( m ). Since ϕ is k -linear,we may assume a monomial T ∈ M (Ω ⊎ A ) and use induction on the degree of T . Thebase case T = 1 is trivial, hence assume the claim for monomials of degree n and let T have degree n + 1. Then there exists T ′ ∈ M (Ω ⊎ A ) of degree n such that either T = aT ′ for a ∈ A or T = ωT ′ with ω ∈ Ω. In the former case the claim follows because ϕ is A -linear, in the latter case because it is a morphism of Ω-operated modules. This completesthe proof that ϕ is also a morphism of Mod A [Ω | E ] . (cid:3) Fact 3.7.
Some standard constructions for creating new modules also work in the operatedsetting. Let us mention a few that are also relevant for the examples to be given afterwards.(a) If A is an E -algebra and S an arbitrary set, the free module A S is an E -module. Theaction of p ω ( ω ∈ Ω) on a module element f ∈ A S is defined by ( p ω f )( s ) := P ω ( f s )for s ∈ S . It is easy to see that for any free operator L ∈ A [Ω] and f ∈ A S onehas ( L · f )( s ) = L · f ( s ) for all s ∈ S , where the left action takes place in A S andthe right action in A . Hence one obtains L · f = 0 for all relators L ∈ [ E ] ⊂ A [Ω]and all f ∈ A S , which confirms that A S is an E -module. Note that A S is free as an A -module but generally not as an A [Ω | E ]-module (see Example 3.8(a) below).(b) If M , . . . , M k are E -modules over A , their direct product M × · · · × M k is an E -module with operators p ω ( ω ∈ Ω) acting component-wise. If M = · · · = M k = M ,this gives the free module M S over the finite set S = { , . . . , k } .(c) Whenever M is an E -module over A , the dual module M ∗ is naturally an E ∗ -modulewith operators p ∗ ω ( ω ∈ Ω); here p ∗ ω : M ∗ → M ∗ is defined as the dual map of the A -linear map p ω : M → M . If L ∈ A [Ω] is any free operator and f ∈ M ∗ one checksimmediately that ( L ∗ · f )( m ) = f ( L · m ) for all m ∈ M . In other words, the actionon M ∗ is the dual of the action on M . In particular, one sees that L ∗ · f = 0 for allrelators L ∈ [ E ] ⊂ A [Ω] and all f ∈ M ∗ , confirming that M ∗ is E ∗ -related (meaningit satisfies the transpose of all relators induced by E ). Since any E -algebra A is alsoan E -module over itself, A ∗ is also an E ∗ -module.(d) If M is an E -module with a submodule M ′ ⊆ M that is closed under all all oper-ators p ω ( ω ∈ Ω), their restrictions to the submodule M ′ make the latter into an E -module or, more precisely, an E -related submodule of M . IFFERENTIAL ROTA-BAXTER OPERATORS 17
Example 3.8.
Let us now exemplify the concept of E -module for the four standard varieties given in Example 2.19, corresponding to the four operator rings of Proposition 3.4. We makeagain use of the convention stated in Remark 3.3.(a) The Diff λ -modules are commonly known as differential modules [33, Def. 1.2.4(iii)],usually taken with weight λ = 0 over a differential field ( F , ∂ ). Their equivalentformulation as F [ ∂ ]-modules is often used as an alternative definition [36, Def. 2.5].Differential modules are crucial for differential Galois theory as they provide anabstract way of formulating linear differential equations. In the important specialcase when the underlying differential ring is F = k [ x ], the operator ring is the Weylalgebra A ( k ), and the corresponding A ( k )-modules are known as D -modules [11]since D := A ( k ) = k [ x ][ ∂ ] is the underlying differential operator ring. For example, k [ x ] n is a differential module by Fact 3.7(b). If k is a field, any k -basis e , . . . , e n of k n is of course a k [ x ]-basis for k [ x ] n but since ∂e , . . . , ∂e n = 0 it is not an A ( k )-basis.In other words, k [ x ] n is free as a k [ x ]-module but not as a k [ x ][ ∂ ]-module.Another important class of examples with k = R is concerned with vector fields on a manifold M . In detail, each vector field V ∈ X ( M ) induces a covariant deriva-tive ∇ V : X ( M ) → X ( M ) with characteristic property ∇ V ( f W ) = f ′ W + f ∇ V ( W )for f ∈ C ∞ ( M ) and W ∈ X ( M ). The vector fields X ( M ) thus form a differentialmodule over the differential algebra C ∞ ( M ).(b) The category of RB λ -modules has been introduced in [15, Def. 2.1(a)] under thename of Rota-Baxter module for a given Rota-Baxter algebra ( F , r ). Their equiva-lent description in terms of F [ r ]-modules is elaborated in [15, § DRB λ -modules, which we may also call differential Rota-Baxter modules . In D -module theory i , it is often pointed out thatvarious spaces of (real or complex valued) distributions are differential modules (forweight λ = 0 and ground field k = R or k = C ) and hence D -modules since dis-tributions can be multiplied by smooth functions so they are in particular modulesover F := k [ x ]. It is seldom appreciated that some of these distribution spaces arein fact differential Rota-Baxter modules over F and hence F [ ∂, u ]-modules. Forexample, let D ′ ( R ) + ⊂ D ′ ( R ) be the space of all distributions T with left -boundedsupport, meaning supp( T ) ⊆ [ a, ∞ [ for some a ∈ R . Analogously, we write D ( R ) − forthe space of test functions with right -bounded support; it is clear that this is a (non-unital!) differential Rota-Baxter algebra with standard derivation ∂ and Rota-Baxteroperator u := − r ∞ x . In fact, D ( R ) − is a degenerate (nonunital) integro-differentialalgebra since the induced evaluation e := 1 D ( R ) + − u ◦ ∂ = 0 is trivially multi-plicative. In other words, ∂ is bijective with u as its inverse, and the strong Rota-Baxter axiom f u g = u f ′ u g + u f g immediately follows from f ′ u g = ( f u g ) ′ − f g .If H ∈ D ′ ( R ) + is the Heaviside function, the operator u : D ′ ( R ) + → D ′ ( R ) + definedby the convolution u T := H ⋆ T is known to be a two-sided inverse [12, § ∂ . One checks that u : D ′ ( R ) + → D ′ ( R ) + is the transpose of u : D ( R ) − → D ( R ) − , just as the distributional derivative ∂ : D ′ ( R ) + → D ′ ( R ) + is (bydefinition) the transpose of the standard derivation ∂ : D ( R ) − → D ( R ) − . Thus we i The two occurrences of D in “ D -modules” and in the distribution space D ′ ( R ) are unrelated. In fact, D ′ ( R ) is the dual of the differentiable class D ( R ) := C ∞ ( R ) of smooth functions with compact support. obtain a Rota-Baxter module ( D ′ ( R ) + , ∂, u ), which is actually a degenerate integro-differential module over the nonunital Rota-Baxter algebra D ( R ) − . Of course, onemay apply a similar construction to endow the space D ′ ( R ) − of right -bounded distri-butions with the structure of a differential Rota-Baxter module over the nonunitaldifferential Rota-Baxter algebra of left -bounded test functions.(d) Finally, let us consider the category of ID λ -modules, which we also call integro-differential modules . Again we shall give an important example from distributiontheory. Endowing E ( R ) := C ∞ ( R ) with the usual derivation ∂ = d/dx and the Rota-Baxter operator u f := r x f ( ξ ) dξ yields a “dually integro-differential” module E ∗ ( R )by Fact 3.7(c), as the dual of the integro-differential algebra ( E ( R ) , ∂, u ). Just as theone-sided distribution spaces of Item (c), this is in fact a differential module as wellas a Rota-Baxter module since the relators ∂f → f ∂ + f ′ and u f u → f u − u f areskew-symmetric under transposition. Hence we should take the negated transposesof ∂, r : E ( R ) → E ( R ); this is of course standard practice in defining the distributionalderivative [12, § transposed section law u ◦ ∂ = 1 E ′ ( R ) .One checks that both ∂ and u restrict to the topological dual E ′ ( R ) ⊂ E ∗ ( R )consisting of all continuous functionals E ( R ) → k , relative to the well-known locallyconvex topology of E ( R ); see for example [35, (7.8)]. Therefore E ′ ( R ) is a differentialRota-Baxter submodule of E ∗ ( R ) by Fact 3.7(d), except that the section law is trans-posed. In analysis, E ′ ( R ) is known as the space of compactly supported distributions .It may seem surprising that ∂ is injective and u surjective on E ′ ( R ). In fact, onechecks that ker u = R δ and im( ∂ ) = { T ∈ E ′ ( R ) | T (1) = 0 } . It is known [35, (10.4)]that in D ′ ( R ) ⊃ E ′ ( R ), the kernel of ∂ is given by the constant distributions; butsince their support is R , they are not in E ′ ( R ). Conversely, the image of ∂ on D ′ ( R )is full [35, Cor. § T (1) = 0 is void. As we have seen, ∂ is not surjective on E ′ ( R ), whichis of course well-known [35, Ex. 11.2].Is ( E ′ ( R ) , ∂, u ) an integro-differential module (with transposed section law)? Onemust check if the (transposed) induced evaluation e := 1 E ′ ( R ) − ∂ ◦ u : E ′ ( R ) → E ′ ( R )is multiplicative in the sense that e ( f T ) = e ( f ) e ( T ) for all T ∈ E ′ ( R ) and f ∈ E ( R ).Since e : E ′ ( R ) → E ′ ( R ) is the transpose of e : E ( R ) → E ( R ), one has e ( T ) = T (1) δ for any T ∈ E ′ ( R ). But then one sees that e ( e x δ ) = δ ( e x ) · δ = e · δ = e ( e x ) e ( δ ) = 1 · δ , which shows that E ′ ( R ) is in fact not an integro-differential module. The problem isthat the corresponding relator e f → e ( f ) e gets transposed to f e → e ( f ) e , whichyields the true identity f e ( T ) = f (0) e ( T ) or f T (1) δ = f (0) T (1) δ . If T (1) = 0,this is trivially valid; otherwise division by T (1) yields the familiar sifting property of the Dirac distribution [12, p. 38]. Of course we may replace δ by δ c for any c ∈ R if we use the Rota-Baxter operator r xa on E ( R ) instead of r x .For seeing an honest integro-differential module, we refer to [30], where the algebraic distribution module ( DF , ð , (cid:30) r ) over a given ordinary shifted integro-differential algebra F ,such as the classical example F = C ∞ ( R ), is constructed and investigated. This providesa purely algebraic structure (involving no topology, in particular taking F only as anintegro-differential algebra with shift maps such as f ( x ) f ( x − c ) for c ∈ R in the IFFERENTIAL ROTA-BAXTER OPERATORS 19 classical example), providing just piecewise functions and Dirac distributions on top of F .In the classical example, this gives rise to the Heaviside function H a = H ( x − a ) and theirderivatives δ a . Compared to the analytic distribution spaces of Example 3.8 (c), (d), thisis a very small module. However, it contains exactly what is needed for specifying andcomputing the Green’s operator G ∈ F [ ∂, r ] of a LODE boundary problem [34, §§ G : F → F , it can be assigned a
Green’s function g ( x, ξ ). This is a (bivariate)function involving Heavisides and—for ill-posed problems— also Diracs, characterized by adistributional differential equation. For a comprehensive presentation, we refer the readerto [30], specifically Theorems 26 and 29 therein. The actual computation of the Green’soperator G on the basis of a given fundamental system is detailed in [27], the extraction ofthe Green’s function g ( x, ξ ) from G in [29].4. Differential Rota-Baxter Operators
As pointed out earlier, the operator rings in Proposition 3.4(a), (b), (d) are known anddefined elsewhere, but the ring in (c) is introduced here for the first time. In the rest of thispaper, we will therefore concentrate on the ring F [ ∂, u ]. As a first step, let us analyze itscanonical forms, in a way similar to [28, Prop. 25] and the above k -module decompositionfor F [ ∂, r ]. In the following, recall that F [ u ] \F denotes a linear complement rather thanthe set-theoretic one. Lemma 4.1.
Let ( F , ∂, u ) ∈ DRB λ . Then we have F [ ∂, u ] = F [ ∂ ] ∔ F [ u ] \ F ∔ [ e ] ,where [ e ] := k { f u g ∂ k | f, g ∈ F ; k > } is a rung that we call the evaluation rung .Proof. The proof of the direct sum is completely analogous to that of the correspondingstatement in Proposition 3.4, with (6) being replaced by(8) F [ ∂, u ] DRB λ = F [ ∂ ] Diff λ ∔ F [ u ] RB λ \ F ∔ [ e ] DRB λ . The analysis of irreducible monomials w ∈ F [ ∂, u ] DRB λ is also the same, except that theremaining case w = f u g∂ k with f, g ∈ F and k ≥ w ∈ [ e ] if k > w ∈ F [ u ] RB λ otherwise. The direct sum (8) now followsimmediately since the evaluation rung [ e ], unlike the evaluation ideal ( e ), is generated byirreducible monomials.It remains to prove that [ e ] is multiplicatively closed, meaning ( f u g ∂ k )( ˜ f u ˜ g ∂ ˜ k ) ∈ [ e ].It suffices to ensure w k := u g ∂ k ˜ f u ˜ g ∂ ∈ [ e ], and for that we use induction over k > k = 1 we have ∂ ˜ f = ˜ f ′ + ˜ f ∂ + λ ˜ f ′ ∂ and w k = u ˜ f ′ g u ˜ g ∂ + u ˜ f g ˜ g ∂ + λ u ˜ f ′ g ˜ g∂ ∈ [ e ]after applying the Rota-Baxter rule in the first summand. For the induction step weconsider w k +1 , assuming the claim holds for k . We obtain w k +1 = u g∂ k ˜ f ′ u ˜ g ∂ + u g∂ k ˜ f ˜ g∂ + λ u g∂ k ˜ f ′ ˜ g∂, where the first summand is contained in [ e ] by the induction hypothesis and the secondexpands into a linear combination of terms having the shape ˜ w k ∂ l (1 ≤ l ≤ k + 1), whichare clearly contained in [ e ] as well. (cid:3) Note that both F [ u ] + := F [ u ] \F and F [ ∂ ] + := F [ ∂ ] \F are rungs, which feature in thealternative k -module decomposition F [ ∂, u ] = F ∔ F [ ∂ ] + ∔ F [ u ] + ∔ [ e ] . Moreover, one checks immediately that [ e ] is actually an ( F [ u ] + , F [ ∂ ] + )-bimodule. Accord-ing to the subsequent lemma, the evaluation rung is also closely related to the evaluation(hence its name). Note that we continue to call the projector e := 1 F − u ∂ the evaluation ofthe differential Rota-Baxter algebra ( F , ∂, u ) although it is not multiplicative (unless F isin fact an integro-differential algebra). However, it is still a projector onto ker ∂ along im r .By abuse of language, the corresponding e ∈ F [ ∂, u ] will also be referred to as evaluation. Lemma 4.2.
The evaluation rung [ e ] is a bimodule over k [ e ] , with e as right annihilator.Proof. Since e = e , the ring k [ e ] is the k -span of 1 and e . Therefore it suffices to verifythe inclusion e [ e ] ⊆ [ e ] and [ e ] e = 0. The latter is immediate from the definition of [ e ],the former follows from e f u g ∂ k = [ e ( f ) , u ] g∂ k ∈ [ e ] via the Rota-Baxter axiom; here thebracket denotes the commutator in F [ ∂, u ]. (cid:3) Before we study further properties of differential Rota-Baxter algebras and their operatorrings, let us give two simple examples (the weight is zero for both).
Example 4.3.
Let k have characteristic zero. The most basic example of a differentialRota-Baxter algebra is clearly the polynomial ring k [ x ], with standard derivation ∂ = d/dx and Rota-Baxter operator r = r x or more generally r xa for any initialization point a ∈ k .Here we think of r xa : k [ x ] → k [ x ] in purely algebraic terms, as the k -linear map definedby x k ( x k +1 − a k +1 ) / ( k + 1). This example will play a great role in Section 5 althoughit is not a genuine example (in the sense that it is also an integro-differential algebra). Example 4.4.
For seeing a natural example of a differential Rota-Baxter algebra that is notan integro-differential algebra, we call on analysis. Of course, the primordial example of anintegro-differential algebra consists of the (real or complex valued) smooth functions C ∞ ( R )or C ∞ [ a, b ]; see [27, Ex. 5]. Here ∂ and r = r xξ ( ξ ∈ R or ξ ∈ [ a, b ]) are defined analytically.A slight variation of this example leads to a differential Rota-Baxter algebra, namely the(real or complex valued) piecewise smooth functions P C ∞ ( R ) or P C ∞ [ a, b ]. For example,we take all functions that are smooth on the whole domain minus finitely many points. Theoperations are defined as before except that ∂f and r f is undefined at the points where f isso. (The ring operations + , − , ∗ have to be defined carefully since singularities may cancel;the result is always to be taken with all removable singularities actually removed. Thisprocess is also well-known in complex analysis where meromorphic functions can be definedin a similar way.)The piecewise smooth functions are clearly a differential Rota-Baxter algebra . However,they are not an integro-differential algebra for if they were, the evaluation 1 − ∂ r wouldbe multiplicative—which it cannot be for functions undefined on the initialization point ξ .For a more explicit example, let us take P C ∞ [0 ,
1] with initialization point ξ = 0. TheHeaviside function h ( x ) := H ( x − / ∈ P C ∞ [0 ,
1] is the characteristic function of thesubinterval [1 / , r h = r x / dx = x − / h · r H ( x − / x .This means we have r ( h · = h · r h ∈ ker ∂ , and [20, Rem. 2.6(c)] showsthat ( P C ∞ [0 , , ∂, r ) is not an integro-differential algebra.In Proposition 3.4 the relation between the operator rings F [ ∂, u ] and F [ ∂, r ] is illumi-nated in one direction only: It shows the differential Rota-Baxter operators F [ ∂, u ] to havea finer structure from which one obtains the integro-differential operator ring F [ ∂, r ] asa quotient. However, we shall see below (Proposition 4.8) that the finer ring F [ ∂, u ] can IFFERENTIAL ROTA-BAXTER OPERATORS 21 also be embedded into a suitably “generic” integro-differential operator ring. Applying thisto the special case of polynomial coefficients will enable us to give an operator-theoreticinterpretation to the integro-differential Weyl algebra (Section 5).As a preparation to this construction, let us first determine the free integro-differentialalgebra ( ˜ F , ∂, u ) over a given differential Rota-Baxter algebra ( F , ∂, r ). In other words,we want to “extend” F just enough to build an integro-differential structure. Categoricallyspeaking, the association F 7→ ˜ F is the left adjoint of the forgetful functor ID λ → DRB λ .However, note that r is not an extension of u . Proposition 4.5.
Given ( F , ∂, u ) ∈ DRB λ , construct ˜ F = F ⊗ K F over K := ker ∂ ,extending the derivation to ∂ : ˜ F → ˜ F , f ⊗ f ε ( ∂f ) ⊗ f ε and defining r : ˜ F → ˜ F via r f ⊗ f ε := ( u f ) ⊗ f ε − ⊗ ( f ε u f ) . Then one obtains ( ˜ F , ∂, r ) ∈ ID λ with evaluation e ( f ⊗ f ε ) = 1 ⊗ f f ε , and an embedding ι : F → ˜ F , f f ⊗ of differential algebras.Proof. Let us first reassure ourselves that ∂ : ˜ F → ˜ F is well-defined. It suffices to provethat P i f ( i ) ⊗ f ε ( i ) = 0 implies P i f ( i ) ′ ⊗ f ε ( i ) = 0 for finite families f ( i ) , f ε ( i ) ∈ F .Hence assume P i f ( i ) ⊗ f ε ( i ) = 0. By [13, Lem. 6.4], there are c ( i, j ) ∈ K and g ( i ) ∈ F satisfying the relations P j c ( i, j ) g ( j ) = f ( i ) for all i and P i c ( i, j ) f ε ( i ) = 0 for all j ,which yields P i f ( i ) ′ ⊗ f ε ( i ) = P ij c ( i, j ) g ( j ) ′ ⊗ f ε ( i ) = P j g ( j ) ′ ⊗ (cid:0) P i c ( i, j ) f ε ( i ) (cid:1) = 0since c ( i, j ) ′ = 0. Next we note that ι : F → ˜ F is injective since its image is F ⊗ K K ∼ = F ;it is a morphism of Diff λ because ∂ ( f ⊗
1) = f ′ ⊗ r : ˜ F → ˜ F is well-defined. Henceassume P i f ( i ) ⊗ f ε ( i ) = 0 as before; we show P i (cid:0) u f ( i ) (cid:1) ⊗ f ε ( i ) = P i ⊗ (cid:0) f ε ( i ) u f ( i ) (cid:1) .Since ˜ c ( i, j ) := u c ( i, j ) g ( j ) − c ( i, j ) u g ( j ) ∈ K , we get ˜ c ( i, j ) ⊗ f ε ( i ) = 1 ⊗ ˜ c ( i, j ) f ε ( i ) andtherefore X i (cid:0) u f ( i ) (cid:1) ⊗ f ε ( i ) = X i,j ˜ c ( i, j ) ⊗ f ε ( i ) = X i,j ⊗ ˜ c ( i, j ) f ε ( i ) = X i ⊗ (cid:0) f ε ( i ) u f ( i ) (cid:1) , where the first equality uses P i,j (cid:0) c ( i, j ) u g ( j ) (cid:1) ⊗ f ε ( i ) = P j (cid:0) u g ( j ) (cid:1) ⊗ (cid:0) P i c ( i, j ) f ε ( i ) (cid:1) = 0and the last P i,j ⊗ (cid:0) c ( i, j ) u g ( j ) (cid:1) f ε ( i ) = P j ⊗ (cid:0) P i c ( i, j ) f ε ( i ) (cid:1) u g j = 0.Using now the fact that u : F → F is a Rota-Baxter operator, a short calculation revealsthat r : ˜ F → ˜ F is as well. Moreover, it is immediate that ∂ r = 1 ˜ F , so ( ˜ F , ∂, r ) is at leasta differential Rota-Baxter algebra. Its evaluation is given by e ( f ⊗ f ε ) = f ⊗ f ε − r f ′ ⊗ f ε = f ⊗ f ε − ( u f ′ ) ⊗ f ε + 1 ⊗ ( f ε u f ′ )= e F ( f ) ⊗ f ε + 1 ⊗ ( f ε u f ′ ) = 1 ⊗ f ε (cid:0) e F ( f ) + u f ′ (cid:1) = 1 ⊗ f ε f, where in the third step we have used the definition of the evaluation on F and in the fourththe fact that all tensors are over K = ker ∂ = im e . From this we see that the evaluationon ˜ F is multiplicative, which implies ( ˜ F , ∂, r ) ∈ ID λ by [20, Thm. 2.5(b)]. (cid:3) Theorem 4.6.
The integro-differential algebra ˜ F defined in Proposition 4.5 is free over F .In other words, any DRB λ -morphism ϕ : F → G to an integro-differential algebra G factorsas ϕ = ˜ ϕ ◦ ι for a unique ID λ -morphism ˜ ϕ : ˜ F → G .Proof.
Let us first prove uniqueness of ˜ ϕ . Assuming ϕ = ˜ ϕ ◦ ι , we have ˜ ϕ ( f ⊗
1) = ϕ ( f ).Moreover, 1 ⊗ f ε = e ( f ε ⊗
1) implies ˜ ϕ (1 ⊗ f ε ) = e G (cid:0) ˜ ϕ ( f ε ⊗ (cid:1) = e G (cid:0) ϕ ( f ε ) (cid:1) since ˜ ϕ is an ID λ -morphism and thus commutes with the evaluation. As ˜ ϕ is a morphism of k -algebraswe obtain ˜ ϕ ( f ⊗ f ε ) = ϕ ( f ) e G (cid:0) ϕ ( f ε ) (cid:1) , which determines ˜ ϕ : ˜ F → G uniquely.For proving existence, it suffices to show that defining ˜ ϕ ( f ⊗ f ε ) := ϕ ( f ) e G (cid:0) ϕ ( f ε ) (cid:1) yieldsan ID λ -morphism ˜ ϕ . Indeed, it is a k -algebra homomorphism since ϕ and e G are; onesees immediately that it respects the derivation. Let us now check that ˜ ϕ also respects theRota-Baxter structure, meaning ˜ ϕ (cid:0) r ( f ⊗ f ε ) (cid:1) = r G ˜ ϕ ( f ⊗ f ε ). For the left-hand side, weapply ˜ ϕ to r f ⊗ f ε = ( u f ) ⊗ f ε − ⊗ ( f ε u f ) to obtain ϕ ( u f ) e G (cid:0) ϕ ( f ε ) (cid:1) − e G (cid:0) ϕ ( f ε u f ) (cid:1) = e G (cid:0) ϕ ( f ε ) (cid:1) (cid:16) ϕ ( u f ) − e G (cid:0) ϕ ( u f ) (cid:1)(cid:17) using the multiplicativity of ϕ and e G on the second term. Since by definition 1 G − e G = r G ∂ G ,the parenthesized expression above is r G ∂ G ϕ ( u f ) = r G ϕ (cid:0) ∂ u f ) = r G ϕ ( f ). For the right-hand side, using r G on ˜ ϕ ( f ⊗ f ε ) = ϕ ( f ) e G (cid:0) ϕ ( f ε ) (cid:1) yields e G (cid:0) ϕ ( f ε ) (cid:1) r G ϕ ( f ) since ( G , ∂ G , r G ) isan integro-differential algebra and r G is linear over ker ∂ G = im e G by [20, Rem. 2.6(d)]. (cid:3) The crucial point of the embedding of F [ ∂, u ] into a ring of integro-differential operatorsis that r f ∂ k , though a normal form of F [ ∂, u ], splits when viewed as an integro-differentialoperator. Its reduction to normal forms can be computed as follows. Lemma 4.7.
Let ( F , ∂, r ) be an integro-differential algebra. Then we have (9) r f ∂ k = k − X i =0 ( − i (cid:0) f ( i ) − e ( f ( i ) ) e (cid:1) ∂ k − i − + ( − k r f ( k ) for all k > .Proof. We use induction on k >
0. The base case k = 1 follows from the ID λ -relator ofProposition 3.4(d). Assume now the claim holds for some k >
0. Then we have r f ∂ k +1 = k − X i =0 ( − i (cid:0) f ( i ) − e ( f ( i ) ) e (cid:1) ∂ k − i + ( − k r f ( k ) ∂, and the last term yields ( − k f ( k ) + ( − k +1 e ( f ( k ) ) e + ( − k +1 r f ( k +1) by the case k = 1.Incorporating the first two summands into the summation, one obtains (9) with k + 1 inplace of k , which completes the induction. (cid:3) We can now provide the embedding of differential Rota-Baxter operators into a ring ofintegro-differential operators with “generic” integral. The punch line is that one must passto the free integro-differential algebra introduced in Proposition 4.6. Since its Rota-Baxteroperator introduces new integration constants, one may view it as being initialized at ageneric point; this will become clearer in Section 5.
Theorem 4.8.
Let ( F , ∂, u ) be an ordinary differential Rota-Baxter algebra, and ( ˜ F , ∂, r ) the free integro-differential algebra defined in Proposition 4.6. Then the assignment (10) f ∂ k f ∂ k , f u ˜ f f r ˜ f , f u ˜ f ∂ k f r ˜ f ∂ k defines an algebra monomorphism ψ : F [ ∂, u ] → ˜ F [ ∂, r ] .Proof. From Proposition 3.4 we know that F [ ∂, u ] = F [ ∂ ] ∔ F [ u ] \F ∔ [ e ], where the threecomponents have normal forms f ∂ k , f u ˜ f and f u ˜ f ∂ k , respectively. Hence the map ψ iswell-defined, and it is clearly k -linear. We can also describe ψ in a different but equivalent IFFERENTIAL ROTA-BAXTER OPERATORS 23 way: Recall that
F ∗ k h ∂, u i is a coproduct in the category of (noncommutative) algebras,with canonical injections i : F → F ∗ k h ∂, u i and i : k h ∂, u i → F ∗ k h ∂, u i . Similarly,˜ F ∗ k h ∂, r i is a coproduct with canonical injections ˜ı and ˜ı . Then by the universal propertyfor the coproduct F ∗ k h ∂, u i , there is an algebra morphism j : F ∗ k h ∂, u i → ˜ F ∗ k h ∂, r i suchthat j ◦ i = ˜ı ◦ ι and j ◦ i = ˜ı ◦ i , where i : k h ∂, u i → k h ∂, r i is the (trivial) isomorphism thatrenames u into r . Writing [ RB λ ] ⊂ F ∗ k h ∂, u i and [ ID λ ] ⊂ ˜ F ∗ k h ∂, r i for the relator idealsof F [ ∂, u ] and ˜ F [ ∂, r ], respectively, we have j [ DRB λ ] ⊂ [ ID λ ] and hence ˜ pj [ DRB λ ] = 0for the canonical projection ˜ p : ˜ F ∗ k h ∂, r i → ˜ F [ ∂, r ] = (cid:0) ˜ F ∗ k h ∂, r i (cid:1)(cid:14) [ ID λ ]. Writing p : F ∗ k h ∂, u i → F [ ∂, u ] = (cid:0) F ∗ k h ∂, u i (cid:1)(cid:14) [ DRB λ ] for the other projection, we concludethat ˜ pj : F ∗ k h ∂, u i → ˜ F [ ∂, r ] descends to an algebra morphism F [ ∂, u ] → ˜ F [ ∂, r ], whichis easily recognized as ψ so that ˜ pj = ψp .It remains to prove that ψ is injective. Recall that although f u ˜ f ∂ k ∈ F [ ∂, u ] is a normalform, this is not the case for its image f r ˜ f ∂ k ∈ ˜ F [ ∂, r ]. In fact, we will apply Lemma 4.7for rewriting the latter as a k -linear combination of ˜ F [ ∂, r ]-normal forms. Now to showthat ψ is injective, assume ψ ( P j w j ) = 0 with w j = 0. Since ˜ F [ ∂, r ] = ˜ F [ ∂ ] ∔ ˜ F [ r ] \ ˜ F ∔ ( e ),those ψ ( w j ) ∈ ˜ F [ ∂, r ] in the sum P j ψ ( w j ) = 0 that belong to F [ ∂ ] and F [ r ] \F must cancelwith corresponding contributions in the expansion (9) of the other ψ ( w j ) ∈ ˜ F [ ∂, r ]. Hencewe are left with a sum of the form P k,l w k,l = 0 of evaluation terms coming from ψ ( f l u g l ∂ k ),which are given by w k,l = k − X i =0 ( − i +1 f l e ( g ( i ) l ) e ∂ k − i − = k − X i =0 ( − i +1 ( f l ⊗ g ( i ) l ) e ∂ k − i − ∈ ( e ) ⊂ ˜ F [ ∂, r ] . Let ¯ k be the highest exponent k occurring among the ψ ( f l u g l ∂ k ), and set w l := w l, ¯ k . Sincethe e ∂ i are k -linearly independent, extracting the highest-order terms e ∂ ¯ k − , correspond-ing to i = 0 in the above sum, yields the relation P l f l ⊗ g l = 0. Applying the crite-rion [13, Lem. 6.4] there exist a lm ∈ k and h m ∈ F such that P m a lm h m = f l for each l ,and P l a lm g l = 0 for each m . This implies P l f l u g l ∂ k = P m h m u (cid:0) P l a lm g l (cid:1) ∂ k = 0,which means that there are no k -th order terms w j ∈ [ e ] ⊂ F [ ∂, u ] in the originalsum ψ ( P j w j ) = 0. By induction on k , we conclude that P j w j has in fact no term w j ∈ [ e ].But then there are no terms w j ∈ F [ ∂ ] or w j ∈ F [ u ] \ F since their images in ˜ F [ ∂, r ] wouldhave nothing to cancel. Hence P j w j = 0, completing the proof that ψ is injective. (cid:3) Since the ring of integro-differential operators is rather well-understood [27, 28], it isadvantageous to have a description of the less familiar and somewhat more subtle differentialRota-Baxter operator ring F [ ∂, u ] as a subring of ˜ F [ ∂, r ]. However, there is a price to pay—the expansion of integration constants from k ⊂ F to k ⊗ k F ⊂ ˜ F . This becomes evenmore transparent in the case of polynomial coefficients, which we describe next.5. The Integro-Differential Weyl Algebra
It is most efficient for our purposes to view the classical
Weyl algebra A ( k ) in thelanguage of skew polynomial rings [10, § k is a field of characteristic zero. If A is a k -algebra j with derivation δ : A → A and ξ an indeterminate, the skew polyno-mial ring A [ ξ ; δ ] is the free left A -module L n ≥ Aξ n with k -basis ξ n . The multiplicationon A [ ξ ; δ ] extends the one on A through the rule ξa = aξ + δ ( a ), subject to the obviousidentifications ξ = 1 , ξ n +1 = ξξ n . Extending also the homotheties in the obvious way, oneobtains a k -algebra A [ ξ ; δ ] that contains A as subalgebra.Using A = k [ x ] with the standard derivation δ = d/dx and indeterminate ξ = ∂ yields theone-dimensional Weyl algebra A ( k ) = k [ x ][ ∂ ; δ ]. One can also introduce the n -dimensionalWeyl algebra in a similar way, starting with the algebra A = k [ x , . . . , x n ] and the deriva-tions δ k = ∂/∂x k and adjoining the indeterminates ξ k = ∂ k to obtain the skew poly-nomial ring A n ( k ) = k [ x , . . . , x n ][ ∂ , δ ] · · · [ ∂ n , δ n ]. Here we restrict ourselves to the one-dimensional case, for which we shall henceforth use the alternative notation A ( ∂ ) := A ( k )as in [24]. In view of the upcoming Rota-Baxter analogs, we refer to A ( ∂ ) as the differentialWeyl algebra .To be more precise, we actually employ the opposite route of defining A ( ∂ ) := k [ ∂ ][ x ; δ ]where δ : k [ ∂ ] → k [ ∂ ] is now defined as the negative of the standard derivation, so that δ ( ∂ n ) := − n ∂ n − . One sees immediately that both definitions are equivalent since theWeyl algebra enjoys the well-known automorphism x ↔ − ∂ . The reason for this unusualdefinition is that, for the Rota-Baxter counterpart of A ( ∂ ), only the second definitionwill work. k Indeed, we introduce [24] the integro Weyl algebra A ( ℓ ) := k [ ℓ ][ x ; δ ] with thederivation δ ( ℓ n ) := + n ℓ n +1 . Note that here as in [24] we use ℓ rather than r for the Rota-Baxter operator; this improves the readability of iterated integrals and emphasizes the dualnature of ∂ and ℓ .Both derivations are fully determined by their action on the generators, namely δ ( ∂ ) = − δ ( ℓ ) = ℓ . The former encodes the Leibniz axiom in the commutator form [ x, ∂ ] = 1,the latter the analogous Rota-Baxter axiom [ x, ℓ ] = ℓ . Let us now make this precise bycomparing those Weyl algebras with the corresponding linear operator rings of Section 3.From now on, all weights are zero; we shall suppress the subscript λ = 0 for the standardvarieties in Diff , RB , DRB and ID . Lemma 5.1.
We have A ( ∂ ) ∼ = k [ x ][ ∂ | Diff ] and A ( ℓ ) ∼ = k [ x ][ ℓ | RB ] as k -algebras.Proof. By Proposition 3.4 we know that k [ x ][ ∂ | Diff ] and k [ x ][ ℓ | RB ] are respectivelydefined by the Leibniz relation ∂f = f ∂ + f ′ and the Rota-Baxter relation ℓf ℓ = f ℓ − ℓf ;the latter employs the notation ℓ instead of r for uniformity. Clearly, it is enough torequire the relations on the k -basis x n of k [ x ]. But the Leibniz relation for f = x n followimmediately by a simple induction argument from the special case f = x , which is justthe commutator relation [ ∂, x ] = −
1. For the Rota-Baxter relation, we show now that itsuffices to take the special case f = 1, embodied in the commutation [ x, ℓ ] = ℓ . We useinduction on n > n ℓx n − ℓ ≡ x n ℓ − ℓx n modulo the two-sided ideal ( xℓ − ℓx − ℓ ). j In contrast to [10, § A be a domain though if it is then A [ ξ ; δ ] is as well.One sees easily [23, § k -algebra A , and this will indeed be crucialfor our definition of the integro-differential Weyl algebra. k Following the standard definition of the differential case, one would need δ ( x ) = − ℓ , which does notyield a derivation δ : k [ x ] → k [ x ]. Algorithmically speaking, the problem is that while the degrees of ∂ aredecreasing, the ones of ℓ are increasing; see the remark before [24, Def. 2]. IFFERENTIAL ROTA-BAXTER OPERATORS 25
The base case n = 1 being trivial, assume the claim for n ≥
1. Then we have ℓx n ℓ ≡ n ℓ x n − ℓ + ℓ x n ≡ n ( xℓ − ℓx ) x n − ℓ + ( xℓ − ℓx ) x n or ( n + 1) ℓx n ℓ ≡ n xℓx n − ℓ + xℓx n − ℓx n +1 ≡ x n +1 ℓ − ℓx n +1 , where the last step uses the induction hypothesis. Hence we obtain(11) k [ x ][ ∂ | Diff ] ∼ = k h x, ∂ i / ( ∂x − x∂ −
1) and k [ x ][ ℓ | RB ] ∼ = k h x, ℓ i / ( xℓ − ℓx − ℓ ) . Using the reductions x∂ → ∂x − xℓ → ℓx + ℓ for the ideals in (11), this corresponds tothe multiplication in the skew polynomial rings A ( ∂ ) = k [ ∂ ][ x ; δ ] and A ( ℓ ) = k [ ℓ ][ x ; δ ]. (cid:3) The integro Weyl algebra shares certain common features with its differential counterpartbut also exhibits some striking differences [24]. While both are Noetherian integral domains,the differential Weyl algebra is a simple ring but the integro Weyl algebra is not. Onthe other hand, the latter comes with a natural grading whereas the former only enjoysfiltration. In this paper, we are not concerned with their further study. Let us just mentionthe following noteworthy commutations.
Lemma 5.2.
We have the commutations [ x i , ℓ ] = i ℓx i − ℓ and [ x, ℓ j ] = j ℓ j +1 in A ( ℓ ) .Proof. The first commutation is [24, Lem. 11] and follows also from the proof of Lemma 5.1above. The second commutation is the defining property of A ( ℓ ) = k [ ℓ ][ x ; δ ] as a skewpolynomial ring. (cid:3) For introducing the integro-differential Weyl algebra A ( ∂, ℓ ) one needs a coefficient do-main k [ ∂, ℓ ] that contains k [ ∂ ] as well as k [ ℓ ], subject to the natural requirement ∂ℓ = 1.In other words, we set k [ ∂, ℓ ] = k h D, L i / ( DL −
1) where ∂ and ℓ are the residue classesof D and L , respectively. This ring has been studied extensively; see for example [21, 16].The derivation δ on k [ ∂, ℓ ] is determined uniquely as an extension of the derivations on k [ ∂ ]and k [ ℓ ]. Defining now the integro-differential Weyl algebra by A ( ∂, ℓ ) := k [ ∂, ℓ ][ x ; δ ], it isimmediately clear that A ( ∂, ℓ ) contains A ( ∂ ) and A ( ℓ ) as subalgebras.We refer again to [24] for some basic algebraic properties of A ( ∂, ℓ ); for deeper andmore general results one may consult [2] and the references therein. Let us only mentionthat A ( ∂, ℓ ) is neither Noetherian nor free of zero divisors. Writing e := 1 − ℓ∂ ∈ A ( ∂, ℓ )for what we call again the evaluation , we have A ( ∂, ℓ ) = A ( ∂ ) ∔ A ( ℓ ) \ k [ x ] ∔ ( e ) as k -modules [24, (18)]. The resemblance with the decomposition of Lemma 4.1 is no coinci-dence as can be seen in Corollary 5.4 below. Lemma 5.3.
For A ( ∂, ℓ ) one may choose the k -bases B i := D ∪ R ∪ E i ( i = 1 , , containing the subbases D = { x i ∂ k | i, k ≥ } and R = { x i ℓ j | i ≥ j > } together with (a) E := { x i ℓ j e ∂ k | i, j, k ≥ } , (b) E := { x i ℓ j ∂ k | i ≥ j, k > } , (c) E := { x i ℓx j ∂ k | i, j ≥ k > } .Hence one has B = { x i ℓ j ∂ k | i, j, k ≥ } for the case (b). Moreover, one may also use thesubbasis R ′ := { x i ℓx j | i, j ≥ } in place of R .Proof. The basis B has already been derived; see the observation before [24, Lem. 19].Both R and R ′ are known to be k -bases of A ( ℓ ) \ k [ x ] ≤ A ( ∂, ℓ ), called the right basis andthe mid basis; see [24, Cor. 12] and the remark before [24, Lem. 11]. Hence the subbases R and R ′ are interchangeable. Let us next prove that B is also a basis. We write D ( n ) := { x i ∂ k | ≤ i, k ≤ n } and R ( n ) := { x i ℓ j | ≤ i ≤ n, < j ≤ n } for the truncations of D and R . Likewise we have E ( n ) := { x i ℓ j e ∂ k | ≤ i ≤ n ; 0 ≤ j, k < n } and E ( n ) := { x i ℓ j ∂ k | ≤ i ≤ n ; 0 < j, k ≤ n } for the truncated complements. Now set B i ( n ) := D ∪ R ∪ E i for i = 1 and i = 2. Clearlywe have | B ( n ) | = | B ( n ) | and lim −→ B i ( n ) = B i for i = 1 and i = 2. Since ℓ j e ∂ k = ℓ j (1 − ℓ∂ ) ∂ k = ℓ j ∂ k − ℓ j +1 ∂ k +1 we see that B ( n ) generates k B ( n ). But the latter has B ( n ) for a basis since it is a subsetof the k -basis B of A ( ∂, ℓ ). Since B ( n ) is thus a generating set of the same cardinality,we conclude that B ( n ) is also a k -basis of k B ( n ) and hence linearly independent, and soare those of B = lim −→ B ( n ). It follows that B is a k -basis of A ( ∂, ℓ ).In fact, one can easily exhibit an explicit basis transformation between B and B . Wedefine ψ : k B → k B by fixing D and R while setting ψ ( x i ℓ j ∂ k ) = ( x i ( ℓ j − k − P km =1 ℓ j − m e ∂ k − m ) if j ≥ k,x i ( ∂ k − j − P jm =1 ℓ j − m e ∂ k − m ) if j < k. Similarly, we define ϕ : k B → k B by fixing again D and R , and sending x i ℓ j e ∂ k to x i ( ℓ j ∂ k − ℓ j +1 ∂ k +1 ). Let us now show that ψ ◦ ϕ = 1 and ϕ ◦ ψ = 1. Obviously it sufficesnow to consider E and E . For j ≥ k one has( ϕ ◦ ψ )( x i ℓ j ∂ k ) = ϕ ( x i ℓ j − k − k X m =1 x i ℓ j − m e ∂ k − m )= x i ℓ j − k − k X m =1 x i ( ℓ j − m ∂ k − m − ℓ j − m +1 ∂ k − m +1 ) = x i ℓ j ∂ k ;and for j < k again( ϕ ◦ ψ )( x i ℓ j ∂ k ) = ϕ ( x i ∂ k − j − j X m =1 x i ℓ j − m e ∂ k − m )= x i ∂ k − j − j X m =1 x i ( ℓ j − m ∂ k − m − ℓ j − m +1 ∂ k − m +1 ) = x i ℓ j ∂ k . For the other direction, in case j ≥ k one obtains( ψ ◦ ϕ )( x i ℓ j e ∂ k ) = ψ ( x i ℓ j ∂ k − x i ℓ j +1 ∂ k +1 )= x i (cid:16) k X m =0 ℓ j − m e ∂ k − m − k X m =1 ℓ j − m e ∂ k − m (cid:17) = x i ℓ j e ∂ k ;and in case j < k likewise( ψ ◦ ϕ )( x i ℓ j e ∂ k ) = ψ ( x i ℓ j ∂ k − x i ℓ j +1 ∂ k +1 )= x i (cid:16) j X m =0 ℓ j − m e ∂ k − m − j X m =1 ℓ j − m e ∂ k − m (cid:17) = x i ℓ j e ∂ k . IFFERENTIAL ROTA-BAXTER OPERATORS 27
Since B is a k -basis of A ( ∂, ℓ ) we have the isomorphism A ( ∂, ℓ ) ∼ = k B , which togetherwith the isomorphism ϕ : k B ∼ = k B yields A ( ∂, ℓ ) ∼ = k B , and this implies that B isalso a k -basis of A ( ∂, ℓ ) as already proved above.For proving that B is a k -basis of A ( ∂, ℓ ), one proceeds similarly using the transitionmaps ϕ : k B → k B and ψ : k B → k B defined by ϕ ( x i ℓ j ∂ k ) = j − X m =0 ( − m m ! ( j − m − x i + j − m − ℓx m ∂ k ,ψ ( x i ℓx j ∂ k ) = j X m =0 ( − j − m j ! m ! x i + m ℓ j − m +1 ∂ k in view of the identities [24, (17)/(16)]. Alternatively, one may use the two truncatedbases B ′ ( n ) := D ( n ) ∪ R ( n ) ∪ E ′ ( n ) and B ( n ) := D ( n ) ∪ R ( n ) ∪ E ( n ) convergingto B = lim −→ B ′ ( n ) and B = lim −→ B ( n ), where one defines E ′ ( n ) := { x i ℓ j ∂ k | ≤ i < n ; 0 < j, k ≤ n ; i + j ≤ n } , E ( n ) := { x i ℓx j ∂ k | ≤ i, j < n ; 0 < k ≤ n ; i + j < n } . The rest of the argument is then as above, with B ′ in place of B , and B in place of B . (cid:3) The three bases correspond to direct decompositions A ( ∂, ℓ ) = k D ∔ k R ∔ k E i ( i =1 , ,
3) with standard components k D = A ( ∂ ) and k R = A ( ℓ ) \ k [ x ]. The extra componentis either the evaluation ideal k E = ( e ), the left k [ x ]-submodule k E , or the evaluationrung [ e ] = k E . Corollary 5.4.
We have A ( ∂, ℓ ) ∼ = k [ x ][ ∂, u | DRB ] as k -algebras.Proof. In view of the decomposition in Lemma 4.1 and the isomorphisms of Lemma 5.1,this follows immediately from Lemma 5.3 since the evaluation rung [ e ] ≤ k [ x ][ ∂, u ] has the k -basis { x i u x j ∂ k | i, j ≥ , k > } , which corresponds to E . (cid:3) It is now easy to derive the following specialization isomorphism [24, Thm. 20] from thegeneral quotient result on the differential Rota-Baxter operator rings.
Proposition 5.5.
We have A ( ∂, ℓ ) / ( e x ) ∼ = k [ x ][ ∂, r | ID ] as k -algebras.Proof. Using the isomorphism of Corollary 5.4, this follows from Proposition 3.4. (cid:3)
Note that here we have used the standard Rota-Baxter operator r : x k x k +1 / ( k + 1)for the integro-differential Weyl algebra and the corresponding integro-differential operatorring k [ x ][ ∂, r ]. As can be seen from [24, Thm. 20], one can also start from any otherintegro-differential structure ( ∂, r ) on k [ x ] for obtaining a similar isomorphism except thatone factors out the ideal ( e x − c e ) where c := e ( x ) ∈ k is the integration constant associatedwith the integral operator r .The specialization isomorphism (Proposition 5.5) can be interpreted as “simulating”integro-differential operators by differential Rota-Baxter operators (in the important caseof polynomial coefficients). Since the structure of the latter is finer, this is in principlenot surprising. However, we can also derive a corresponding generalization isomorphism that identifies the finer ring of differential Rota-Baxter operators as a subalgebra in anoverarching integro-differential operator ring. To this end, we take our earlier result of thegeneral theory (Theorem 4.8) and interpret it in the more concrete polynomial setting. Theorem 5.6.
For ε transcendental over k , endow ˜ k [ x ] = k [ x, ε ] with derivation ∂ = ∂/∂x and integral r = r xε . Then there is a unique k -algebra monomorphism ι : A ( ∂, ℓ ) ֒ → ˜ k [ x ][ ∂, r ] that sends ℓ to r while fixing x and ∂ .Proof. Applying Theorem 4.8 to F := k [ x ] we observe that ˜ F = k [ x ] ⊗ k k [ x ] ∼ = k [ x, ε ],defined by Proposition 4.5, has the derivation and integral as described in the currenttheorem. Indeed, ∂ ( x i ⊗ x j ) = ∂ ( x i ) ⊗ x j means ∂ ( x i ε j ) = ( ∂/∂x ) x i ε j for the derivationwhile r ( x i ⊗ x j ) = ( u x i ) ⊗ x j − ⊗ ( x j u x i ), where u denotes the standard Rota-Baxteroperator on k [ x ], translates to r x i ε j = x i +1 i +1 ε j − ε j ε i +1 i +1 = r xε x i ε j for the integral. Hence ˜ F coincides with ˜ k [ x ] as an integro-differential algebra.Let us now consider the map ι : A ( ∂, ℓ ) → ˜ k [ x ][ ∂, r ]. Since x, ∂ and ℓ generate A ( ∂, ℓ ),the uniqueness claim follows. But we know from Corollary 5.4 that A ( ∂, ℓ ) ∼ = F [ ∂, u ], andwith this identification the map ι is clearly the same as the k -algebra monomorphism givenin Theorem 4.8. (cid:3) The intuitive idea behind the generalization isomorphism is that one adjoins a genericinitialization point ε for the integral r . The associated (multiplicative) evaluation e = 1 − r ∂ sends f ( x, ε ) ∈ ˜ k [ x ] to f ( ε, ε ) ∈ ˜ k := k [ ε ]. This yields an isomorphic copy ι (cid:0) A ( ∂, ℓ ) (cid:1) of theintegro-differential Weyl algebra in ˜ k [ x ][ ∂, r ]. However, one should observe that ι (cid:0) A ( ∂, ℓ ) (cid:1) by itself is only a differential Rota-Baxter operator ring and not an integro-differentialoperator ring: The evaluation f ( x, ε ) f ( ε, ε ) does not restrict to a map on its coefficientdomain k [ x ].One may also derive a k - basis of ι (cid:0) A ( ∂, ℓ ) (cid:1) ≤ ˜ k [ x ][ ∂, r ]. For any integro-differentialoperator ring one has the relation r f e = ( r f ) e ; see [27, Table 1] and Footnote f in theproof of Proposition 3.4. Setting f = 1 and iterating j times the integral r = r xε oneobtains the relation r · · · r e = ( x − ε ) j /j ! e . Hence ι maps the basis elements x i ℓ j e ∂ k ∈ E of Lemma 5.3 to (1 /j !) x i ( x − ε ) j e ∂ k while “fixing” those of D and R ′ . Acknowledgments : This work was supported by the National Natural Science Foundationof China (Grant No. 11371177 and 11371178), Fundamental Research Funds for the CentralUniversities (Grant No. lzujbky-2017-162), the National Science Foundation of US (GrantNo. DMS 1001855), the Engineering and Physical Sciences Research Council of UK (GrantNo. EP/I037474/1), and the Austrian Science Fund (FWF Grant No. P30052).We thank the anonymous referee for valuable suggestions helping to improve the paper.
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