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Matrices carrées sur l’algèbre de Weyl. (Square matrices over the Weyl algebra). (French) Zbl 0655.16010
The Weyl algebra \(W_ n\) is the algebra of all linear differential operators on \(C^ n\) with polynomial coefficients. Two square matrices \(a\in M_{\ell}(W_ n)\), \(a'\in M_{\ell '}(W_ n)\) are said to be equivalent if for some \(r\geq 1\) there exist \(b,c\in GL_ r(W_ n)\) such that \(b(a\oplus I_{r-\ell})=(a'\oplus I_{r-\ell '})c\) (here \(I_ p\) is the identity matrix of order p). For example, any invertible matrix is equivalent to \(1\in W_ n\). A matrix \(a\in M_{\ell}(W_ n)\) is said to be deterministic if it is not a zero divisor. The main result of the paper claims that the equivalence class of a deterministic matrix always contains a \(2\times 2\) matrix, but not, in general, a triangular matrix (of any order).
MSC:
16S50 Endomorphism rings; matrix rings
16W20 Automorphisms and endomorphisms
32C38 Sheaves of differential operators and their modules, \(D\)-modules
58J15 Relations of PDEs on manifolds with hyperfunctions
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