The Kernel and Image of Orbit Homomorphisms for the Witt Algebra

The Witt algebra W≥ -1 is the Lie algebra of algebraic vector fields on a line. We investigate the two-sided ideal structure of its universal enveloping algebra, by studying the orbit homomorphisms Ψₙ: U(W≥ -1) → Tₙ, an infinite family of homomorphisms to noncommutative Noetherian algebras. The orbit homomorphisms lift primitive ideals from solvable Lie algebras to U(W≥ -1), thereby playing a central role in the orbit method for the Witt algebra. We prove that the kernel of any orbit homomorphism is generated by an infinite set of differentiators as a one-sided ideal, whilst being generated by any single element of this set as a two-sided ideal. One consequence is an explicit description of primitive and semi-primitive ideals of U(W≥ -1) corresponding to one-point local functions. We also prove that the image Bₙ of the nth orbit homomorphism is both non-Noetherian and birational to the Noetherian algebra Tₙ. On the other hand, the degree zero subring of Bₙ is left and right Noetherian, and we conjecture that the same holds for U(W≥ -1).