Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : k[X] → k[X] and d ₂ : k[Y] → k[Y] be k-derivations, where k[X] ? k[x ₁,…,x _n ], k[Y] ? k[y ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of k[X, Y] such that d| _k[X] = d ₁ and d| _k[Y] = d ₂. We prove that if d ₁ and d ₂ are positively homogeneous and if d ₁ has no nontrivial Darboux polynomials, then every Darboux polynomial of d ₁ ⊕ d ₂ belongs to k[Y] and is a Darboux polynomial of d ₂. We prove a similar fact for the algebra of constants of d ₁ ⊕ d ₂ and present several applications of our results.     

DERIVATIONS OF THE 3-LIE ALGEBRA REALIZED BY gl(n, ℂ)

This paper studies structures of the 3-Lie algebra M realized by the general linear Lie algebra gl(n, ?). We show that M has only one nonzero proper ideal. We then give explicit expressions of both derivations and inner derivations of M. Finally, we investigate substructures of the (inner) derivation algebra of M.     

Skew Derivations and Engel Conditions

It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x ^{k ₀} ), x ^{k ₁} ], x ^{k ₂} ],…], x ^{k _n} ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ?)-derivation.     

A remark on normal derivations

Given a Hilbert space , let be operators on . Anderson has proved that if is normal and , then for all operators . Using this inequality, Du Hong-Ke has recently shown that if (instead) , then for all operators . In this note we improve the Du Hong-Ke inequality to for all operators . Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.