Rational matrix pseudodifferential operators
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Kac_Rational matrix.pdf
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Author(s)
Carpentier, Sylvain
De Sole, Alberto
Kac, Victor
Date Issued
July 2013
Journal
Selecta Mathematica
Publisher
Springer-Verlag
Citation
Carpentier, Sylvain, Alberto De Sole, and Victor G. Kac. “Rational Matrix Pseudodifferential Operators.” Sel. Math. New Ser. 20, no. 2 (July 4, 2013): 403–419.
Version
Original manuscript
Abstract
The skewfield K(∂) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[∂]. In our previous paper, we showed that any H ∈ K(∂) has a minimal fractional decomposition H = AB[superscript −1] , where A,B ∈ K[∂], B ≠ 0, and any common right divisor of A and B is a non-zero element of K . Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-zero element of K[∂] . In the present paper, we study the ring M[subscript n](K(∂)) of n × n matrices over the skewfield K(∂). We show that similarly, any H ∈ M[subscript n](K(∂)) has a minimal fractional decomposition H = AB[superscript −1], where A,B ∈ M[subscript n](K[∂]), B is non-degenerate, and any common right divisor of A and B is an invertible element of the ring M[subscript n](K[∂]). Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-degenerate element of M[subscript n](K[∂]). We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00029-013-0127-5
