Rational matrix pseudodifferential operators
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...
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| Format: | Article |
| Language: | en_US |
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Springer-Verlag
2015
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| Online Access: | http://hdl.handle.net/1721.1/92902 https://orcid.org/0000-0002-2860-7811 |
| _version_ | 1826200971023220736 |
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| author | Carpentier, Sylvain De Sole, Alberto Kac, Victor |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Carpentier, Sylvain De Sole, Alberto Kac, Victor |
| author_sort | Carpentier, Sylvain |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T11:44:35Z |
| format | Article |
| id | mit-1721.1/92902 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:44:35Z |
| publishDate | 2015 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/929022022-10-01T05:39:23Z Rational matrix pseudodifferential operators Carpentier, Sylvain De Sole, Alberto Kac, Victor Massachusetts Institute of Technology. Department of Mathematics Kac, Victor 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. National Science Foundation (U.S.) 2015-01-15T19:26:35Z 2015-01-15T19:26:35Z 2013-07 Article http://purl.org/eprint/type/JournalArticle 1022-1824 1420-9020 http://hdl.handle.net/1721.1/92902 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. https://orcid.org/0000-0002-2860-7811 en_US http://dx.doi.org/10.1007/s00029-013-0127-5 Selecta Mathematica Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer-Verlag arXiv |
| spellingShingle | Carpentier, Sylvain De Sole, Alberto Kac, Victor Rational matrix pseudodifferential operators |
| title | Rational matrix pseudodifferential operators |
| title_full | Rational matrix pseudodifferential operators |
| title_fullStr | Rational matrix pseudodifferential operators |
| title_full_unstemmed | Rational matrix pseudodifferential operators |
| title_short | Rational matrix pseudodifferential operators |
| title_sort | rational matrix pseudodifferential operators |
| url | http://hdl.handle.net/1721.1/92902 https://orcid.org/0000-0002-2860-7811 |
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