Resumen de Rational matrix pseudodifferential operators

Sylvain Carpentier, Alberto De Sole, Victor G. Kac

• 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.