Journal of Mathematics
Volume 2019 (2019), Article ID 2409193, 9 pages
https://doi.org/10.1155/2019/2409193

Research Article

Relations of the Algebra of Polynomial Integrodifferential Operators

Andre S. E. Mialebama Bouesso and G. Mobouale Wamba

Universite Marien Ngouabi, Faculté des Sciences et Technique, Parcours Mathematiques, BP: 69, Brazzaville, Congo

Correspondence should be addressed to Andre S. E. Mialebama Bouesso; sainteudes@gmail.com

Received 7 January 2019; Accepted 19 February 2019; Published 18 March 2019

Academic Editor: Andrei V. Kelarev

Copyright © 2019 Andre S. E. Mialebama Bouesso and G. Mobouale Wamba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We revisit the algebra of polynomial integrodifferential operators and we give a generalization of its relations. Generators of prime ideals of height 1, of the maximal ideal, and of the smallest ideal of are discussed. A technique for obtaining a generating set for a given two-sided ideal of is also discussed.

1. Introduction

The algebra of polynomial integrodifferential operators was introduced in [1] and was studied extensively in [2–6]. Early in 2009, the author of [1] introduced the Jacobian algebra (see [7]) which has similar properties to the algebra of polynomial integrodifferential operators and both algebras are ideally equivalent. In 2010, the same author introduced the algebra (see [8]) of one-sided inverses of a polynomial algebra which is a generalized Weyl algebra as well as the Jacobian algebra and the algebra of polynomial integrodifferential operators . As applications, the algebra of polynomial integrodifferential operators has interesting applications in computer algebra as described in [9, 10]. Our goal in this work is to firstly study new relations of the algebra of polynomial integrodifferential operators in order to facilitate the use of this algebra in a computational point of view. Our starting point is to prove some relations introduced in [1] in Proposition 4. What is more, we give a generalization of these relations in Corollary 7 and then we give new other relations in Lemma 10. In the second part of this paper we study generators of the ideals of . Prime ideals of height 1 listed in [1] are revisited in Theorem 11, sets of generators of the only maximal ideal and of the smallest ideal of are studied, and a given two-sided ideal of is also discussed. The relations discussed in this paper can be used to revisit the special case of , and some results were obtained in the general case of Generalized Weyl algebras especially those described in [11–16].

2. Relations of

Let be a field and be a polynomial ring over . Denote .

Definition 1. By a Weyl algebra we mean the free associative algebra satisfying and where .

Definition 2. By the algebra of polynomial integro-differential operators we mean the algebra generated by the Weyl algebra and endomorphisms

i.e

Remark 3. Observe that for

Set ; regarding as the identity map of , we have the endomorphism

Denoting , we see that ; that is, . From here, we recall Proposition 2.2 introduced in [1].

Proposition 4 (see Proposition 2.2, [1]).

(1)The algebra is generated by the elements that satisfy the following defining relations: ,

(a);(b);(c);(d);(e)For , we have .

(2)The algebra is a Generalized Weyl algebra where and . Moreover, the algebra is graded where , where and such that

(3)Each element of is a unique finite sum for unique elements .

Proposition 5 proves the first item of Proposition 4 left in [1].

Proposition 5. The algebra satisfies the following relations:

(1);(2);(3);(4)(5);(6)For , we have .

Proof.

(1).(2)By definition

 Thus, .(3).(4).(5). Observe that

 since .(6)Applying Remark 3, where and for all , then we have . Using now Definition 1, we see that

Setting , we easily see that where .

Example 6 (construction of ). Let us consider the polynomial ring in two indeterminates. By a monomial in we mean an element of the form where , and by a polynomial we mean a finite linear combination of monomials. Consider the following operators:

O1

O2

O3

O4

O5

O6

The algebra of polynomial integrodifferential operators is the free associative algebra generated by the operators defined in O1,...,O6 with coefficients in and multiplication given by the composition of applications. The relations between operators are given by Proposition 5.

With notations as above we have the following result.

Corollary 7. For we have

(1);(2);(3);(4);(5);(6);(7).

Proof. (1) For we have

By Proposition 5 we have . Replacing this in the previous expression we have

Observe that

Replacing this in the previous expression we have the following.

By induction using the same technique we see that

For we have

We have seen that , and using this we have

By induction we easily see that

(2)

Using Proposition 5 we see that . Applying the same techniques as in the previous item we see that

(3) By Proposition 5 we see that

and then

Observe that

and replacing this in we see that

By we see that

and combining and we have

(4) Using the same technique as in the previous item and the fact that

we see that

(5)

since we have shown above that . Observe that

Replacing all this in we have the following result:

Now let us prove that . Observe that

Observe that , and using this in we have the following result:

Now it becomes easy to see that

(6) The same techniques as in the previous item can be easily applied.

Example 8 (application). Let be a univariate polynomial ring and let us consider the algebra constructed from ; we have . Setting , one can study the noncommutative polynomial ring .

(1) Monomials in : in our case monomials are of the form where . Note that are not monomials but polynomials for since by Corollary 7 we have

(i);(ii).

(2) Polynomials in : a polynomial in is a linear combination of monomials. As a specific case, let us consider the polynomial . According to the order we have imposed to our variables, this polynomial is not well-organized. Observe that

 after considering (1) in Proposition 5 and (1) in Corollary 7. Replacing in we get

 Using (5) in Corollary 7 we get

We have seen in [1] that where and the only proper ideal of is .

Corollary 9. (1) The algebra is principal and its only proper ideal is ;
(2) is an module.

Proof. Consider the notation above.
(1) We have seen in [1] that the only proper ideal of is . Let us now prove that .
Let ; then

Observe that each term of belongs in as a two-sided ideal; therefore .
The converse is straightforward since .
(2) can be regarded as an module. Moreover,

We easily see that the external multiplication is only concern with .

Lemma 10. For and ,

(1);(2);(3);(4);(5).

Proof.

(1)Seeing that algebra is associative and the fact that (by Proposition 4), then

 Furthermore .(2). Observe that

 But

(3).(4).(5).

(a)If then and , since then .(b)If , then and , since .

3. Two-Sided Ideals of

The theory of ideals of is already studied in [1]. The goal of this section is to study generating sets of those ideals and to study the behavior of a given ideal.

3.1. Generators of Primes Ideals of Height 1

Corollary 3.3 in [1] proved that the only prime ideals of of height 1 are

where .

With notations as above we have the following result.

Theorem 11. For we have

for some .

Proof. By definition where

If , then where and .
Observe that ; that is,

where for . By Proposition 2.2 in [1], we see that

where and .
It is easy to see as a two-sided ideal of . Observe that

We then have
since ; i.e., if

we have

This means that for ; thus

implies straightforwardly

Using the same technique as above, we have

Thus

Corollary 12.

(1)The only maximal ideal of is

 where .(2)The smallest nonzero ideal of is .

Proof. It is well known (see [1]) that

(1)the only maximal ideal of is

 that is,

(2)the smallest nonzero ideal of is

It is straightforward to see that . We prove the converse for since the general case can be obtained by induction. Let ; then . That is,

for some . That is,

for , and .
We have the following.

Let us expand (58):

since different variable commutes. Observe that
for ; this leads to . To avoid this, let us take ; in this case we have the following.

Observe that
for some . This is because

(i) and;(ii) since .

Observe also that
for some (using Corollary 7).
Replacing this in we get

Since different variable commutes, we have by Proposition 5(6), and this leads directly to . To avoid this case, we take ; in this case becomes

Again since different variable commutes, we have

Replacing this in we get

Observe the following.

In any of these cases we finish with an expression of the form

for some and ; thus

Let us expand (59). Applying the same technique as in (58) we get

where and .
In (68) we see that and in (69) ; thus

Example 13. Let us consider again the algebra . The maximal ideal of is where and for some polynomials and . Thus . The smallest ideal is .

3.2. Generators of Ideals of

Let be a two-sided ideal of ; then can be written as .

For , let ; then

It is proven in [1], Proposition 2.2, that where and

For we have

where and . Thus we have the following.

(1) If each , then

Observe that

and ; this implies the following.

If

note that

and then

(2) Assume without loss of generalities that there exist such that

and in this case

Remark 14. Obtaining a set of generators of a given two-sided ideal of is very important. According to the theory of ideals studied in [1], we see that any two-sided ideal of can be written as a product or intersection of primes ideals of . A very interesting result will be to propose an algorithm for finding these prime ideals from a given ideal. This work proposes the first step of tools necessary to deal with this problem, but to propose a decomposition algorithm, we will need to study the theory of two-sided Groebner bases (see [17]) on as well as some applications such as the two-sided module of syzygies, since it is very important for the computation of generating sets of intersection of ideals as described in [18]. Our next project will develop computer algebra in and solve this problem.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

V. V. Bavula, “The algebra of integro-differential operators on a polynomial algebra,” Journal Of The London Mathematical Society-Second Series, vol. 83, no. 2, pp. 517–543, 2011.

V. V. Bavula, “Generalized Weyl algebras and their representations,” Algebra i Analiz, vol. 4, no. 1, pp. 75–97, 1992.

V. V. Bavula, “The algebra of integro-differential operators on an affine line and its modules,” Journal of Pure and Applied Algebra, vol. 217, no. 3, pp. 495–529, 2013.

V. V. Bavula, “An analogue of the conjecture of Dixmier is true for the algebra of polynomial integro-differential operators,” Journal of Algebra, vol. 372, pp. 237–250, 2012.

V. V. Bavula, “The group of automorphisms of the algebra of polynomial integro-differential operators,” Journal of Algebra, vol. 348, pp. 233–263, 2011.

V. Bavula, V. Bekkert, and V. Futorny, “Indecomposable generalized weight modules over the algebra of polynomial integro-differential operators,” Proceedings of the American Mathematical Society, vol. 146, no. 6, pp. 2373–2380, 2018.

V. V. Bavula, “The Jacobian algebras,” Journal of Pure and Applied Algebra, vol. 213, no. 5, pp. 664–685, 2009.

V. V. Bavula, “The algebra of one-sided inverses of a polynomial algebra,” Journal of Pure and Applied Algebra, vol. 214, no. 10, pp. 1874–1897, 2010.

A. Quadrat and G. Regensburger, “Polynomial solutions and annihilators of ordinary integro-differential operators,” IFAC Proceedings Volumes, vol. 46, no. 2, pp. 308–313, 2013.

G. Regensburger, “SYMbolic computation with integro-differential operators,” in Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp. 397–404, New York, NY, USA, 2016.

V. Bekkert, G. Benkart, and V. Futorny, “Weight modules for Weyl algebras,” in Kac-Moody Lie algebras and related topics, vol. 343 of Contemp. Math., pp. 17–42, Amer. Math. Soc., Providence, RI, 2004.

M. A. Farinati, A. Solotar, and M. Surez-Ivarez, “Hochschild homology and cohomology of generalized Weyl algebras,” Annales de l'institut Fourier (Grenoble), vol. 53, no. 2, pp. 465–488, 2003.

J. T. Hartwig, “Locally finite simple weight modules over twisted generalized Weyl algebras,” Journal of Algebra, vol. 303, no. 1, pp. 42–76, 2006.

M. Prest and G. Puninski, “Pure injective envelopes of finite length modules over a generalized Weyl Algebras,” Journal of Algebra, vol. 251, pp. 150–177, 2002.

L. Richard and A. Solotar, “Isomorphisms between quantum generalized Weyl algebras,” Journal of Algebra and Its Applications, vol. 5, no. 3, pp. 271–285, 2006.

J.-E. Roos, “Determination de la dimension homologique des algèbres de Weyl,” Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, vol. 271, pp. 23–26, 1972.

V. Levandovskyy, Non-Commutative Computer Algebra for Polynomial Algebras: Groebner bases, applications and implementation [Ph.D. thesis], University of Kaisersalautern, 2005.

R. F. Babindamana and A. S. Mialebama Bouesso, “Intersection of ideals in a polynomial ring over a dual valuation domain,” Journal of Mathematics, vol. 2018, Article ID 9316901, 11 pages, 2018.