The algebra of integro-differential operators on a polynomial algebra

作者：Bavula V V^*

来源：Journal of the London Mathematical Society-Second Series, 2011, 83: 517-543.

DOI：10.1112/jlms/jdq081

We prove that the algebra I(n) := K < x(1), ... , x(n), ... x(1), partial derivative/partial derivative x(1), ... , partial derivative/partial derivative x(n), integral(1), ... , integral(n) > of integro-differential operators on a polynomial algebra is a prime, central, catenary, self-dual, non-Noetherian algebra of classical Krull dimension n and of Gelfand-Kirillov dimension 2n. Its weak homological dimension is n, and n <= gldim(I(n)) <= 2n. All the ideals of I(n) are found explicitly, there are only finitely many of them (at most 2(2n)), they commute (ab - ba) and are idempotent ideals (a(2) = a). The number of ideals of I(n) is equal to the Dedekind number partial derivative(n). An analogue of Hilbert's Syzygy Theorem is proved for I(n). The group of units of the algebra In is described (it is a huge group). A canonical form is found for each integro-differential operator (by proving that the algebra I(n) is a generalized Weyl algebra). All the mentioned results hold for the Jacobian algebra A(n) (but GK(A(n)) = 3n, note that I(n) subset of A(n)). It is proved that the algebras I(n) and A(n) are ideal equivalent.

• 出版日期2011-4