The Kernel and Image of Orbit Homomorphisms for the Witt Algebra

Abstract

The Witt algebra $W_{\geq -1}$ 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 $\Psi_n: U(W_{\geq -1}) \rightarrow T_n$, an infinite family of homomorphisms to noncommutative Noetherian algebras. The orbit homomorphisms lift primitive ideals from solvable Lie algebras to $U(W_{\geq -1})$, 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 $U(W_{\geq -1})$ corresponding to one-point local functions. We also prove that the image $B_n$ of the nth orbit homomorphism is both non-Noetherian and birational to the Noetherian algebra $T_n$. On the other hand, the degree zero subring of $B_n$ is left and right Noetherian, and we conjecture that the same holds for $U(W_{\geq -1})$.