An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators

[摘要] Let A(1) := K < x, d/dx) be the Weyl algebra and I-1 := K < x, d/dx, integral > be the algebra of polynomial integro-differential operators over a field K of characteristic zero. The Conjecture/Problem of Dixmier (1968) [still open]: is an algebra endomorphism of the Weyl algebra A(1) an automorphism? The aim of the paper is to prove that each algebra endomorphism of the algebra I-1 is an automorphism. Notice that in contrast to the Weyl algebra A(1) the algebra I-1 is a non-simple, non-Noetherian algebra which is not a domain. Moreover, it contains infinite direct sums of nonzero left and right ideals. (C) 2012 Elsevier Inc. All rights reserved.

[发布日期] 2012-12-15

[关键词] The Weyl algebra;The Conjecture/Problem of Dixmier;The algebra of polynomial integro-differential operators;The Jacobian Conjecture