Some remarks on non-commutative principal ideal rings

Carpentier, Sylvain; Sole, Alberto De; Kač, Victor G.

doi:10.1016/j.crma.2013.01.006

Cited by 7 publications

(11 citation statements)

References 8 publications

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“…, B N (see Theorem 2.11). This result (which was proved for N = 2 in [3]) plays an important role in our theory of minimal rational expressions. A rational matrix pseudodifferential operator usually comes in the form of a rational expression…”

Section: Introductionmentioning

confidence: 53%

“…In the present paper we continue the study of the algebra Mat ℓ×ℓ K(∂) of ℓ × ℓ rational matrix pseudodifferential operators that we began in [2,3,4].…”

Section: Introductionmentioning

confidence: 89%

“…invertible in Mat ℓ×ℓ K(∂), and A and B are right coprime. Moreover, for any other (right) fractional decomposition H = A B −1 one has A = AD, B = BD, where D ∈ Mat ℓ×ℓ K[∂] is non degenerate, see [3,4]. In these papers we establish several equivalent properties of a minimal fractional decomposition H = AB −1 .…”

Section: Introductionmentioning

confidence: 99%

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Singular Degree of a Rational Matrix Pseudodifferential Operator

Carpentier

Sole

Kač

2014

Int Math Res Notices

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In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H = AB −1 , where A and B are matrix differential operators, and B is non-degenerate of minimal possible degree deg (B). In the present paper we introduce the singular degree sdeg(H) = deg(B), and show that for an arbitrary rational expressionIf the equality holds, we call such an expression minimal. We study the properties of the singular degree and of minimal rational expressions. These results are important for the computations involved in the Lenard-Magri scheme of integrability.

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Section: Introductionmentioning

confidence: 53%

“…In the present paper we continue the study of the algebra Mat ℓ×ℓ K(∂) of ℓ × ℓ rational matrix pseudodifferential operators that we began in [2,3,4].…”

Section: Introductionmentioning

confidence: 89%

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Singular Degree of a Rational Matrix Pseudodifferential Operator

Carpentier

Sole

Kač

2014

Int Math Res Notices

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“…We denote by and the rings of matrices over the ring and skew field respectively. Since is a principal ideal ring, then the ring is also a principal ideal ring (see proof in [ 24 ], as well as the short and useful review of non-commutative principal ideal rings [ 25 ]).…”

Section: Algebraic Properties Of Difference Operatorsmentioning

confidence: 99%

“… or ). A principal ideal ring satisfies the right (and left) Ore property (Theorem 2.2 (c) in [ 25 ]). Namely, for any there exist (resp.…”

Section: Appendix a Basic Concepts For A Unital Associative Principamentioning

confidence: 99%

Rational Recursion Operators for Integrable Differential–Difference Equations

Carpentier

Михайлов

Wang

2019

Commun. Math. Phys.

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In this paper we introduce preHamiltonian pairs of difference operators and study their connections with Nijenhuis operators and the existence of weakly non-local inverse recursion operators for differential-difference equations. We begin with a rigorous setup of the problem in terms of the skew field Q of rational (pseudo-difference) operators over a difference field F with a zero characteristic subfield of constants k ⊂ F and the principal ideal ring M n (Q) of matrix rational (pseudo-difference) operators. In particular, we give a criteria for a rational operator to be weakly non-local. A difference operator H is called preHamiltonian, if its image is a Lie k-subalgebra with respect the the Lie bracket on F. Two preHamiltonian operators form a preHamiltonian pair if any k-linear combination of them is preHamiltonian. Then we show that a preHamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a preHamiltonian pair. This provides a systematical method to check whether a rational operator is Nijenhuis. As an application, we construct a preHamiltonian pair and thus a Nijenhuis recursion operator for the differential-difference equation recently discovered by Adler & Postnikov. The Nijenhuis operator obtained is not weakly non-local. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well known examples including the Toda, the Ablowitz-Ladik and the Kaup-Newell differential-difference equations.

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