Rational matrix pseudodifferential operators

• Sylvain Carpentier ^[1] ; Alberto De Sole ^[2] ; Victor G. Kac ^[3]
  1. [1] École Normale Supérieure

     École Normale Supérieure

     Francia

     
  2. [2] Università de Roma La Sapienza

     Università de Roma La Sapienza

     Roma Capitale, Italia

     
  3. [3] M.I.T

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 20, Nº. 2, 2014, págs. 403-419
• Idioma: inglés
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• Resumen

  • The skewfield K(∂)K(∂) of rational pseudodifferential operators over a differential field KK is the skewfield of fractions of the algebra of differential operators K[∂]K[∂]. In our previous paper, we showed that any H∈K(∂)H∈K(∂) has a minimal fractional decomposition H=AB−1H=AB−1, where A,B∈K[∂],B≠0A,B∈K[∂],B≠0, and any common right divisor of AA and BB is a non-zero element of KK. Moreover, any right fractional decomposition of HH is obtained by multiplying AA and BB on the right by the same non-zero element of K[∂]K[∂]. In the present paper, we study the ring Mn(K(∂))Mn(K(∂)) of n×nn×n matrices over the skewfield K(∂)K(∂). We show that similarly, any H∈Mn(K(∂))H∈Mn(K(∂)) has a minimal fractional decomposition H=AB−1H=AB−1, where A,B∈Mn(K[∂]),BA,B∈Mn(K[∂]),B is non-degenerate, and any common right divisor of AA and BB is an invertible element of the ring Mn(K[∂])Mn(K[∂]). Moreover, any right fractional decomposition of HH is obtained by multiplying AA and BB on the right by the same non-degenerate element of Mn(K[∂])Mn(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.