A note on invariant operators of the Weyl algebra (Russian)

Abstract

Let L be a finite-dimensional complex Lie algebra with a basis X1,…,Xn and L∗ the dual space with a dual basis x1,…,xn. Suppose that [Xi,Xj]=∑kckijXk. Then there exists a (co)representation
Xi↦∑k,jckijxk∂∂xj
of L in the space of analytic functions on L∗. A function F is invariant if Xi∘F(x1,…,xn)=∑k,jckijxk∂∂xjF(x1,…,xn).
In the case of a pseudo-orthogonal algebra Iso(p,q) the author finds a maximal algebraically independent system of invariants C1,…,Cm consisting of Casimir operators where m=[p+q−12]. It is shown that invariants of the Weyl algebra W(p,q) have the form IJ−1, where I and J are invariants for Iso(p,q).