The Kernel and Image of Orbit Homomorphisms for the Witt Algebra
Tuan Anh Pham, James Timmins
TL;DR
The paper analyzes the universal enveloping algebra $U(W_{\ge -1})$ of the one-sided Witt algebra via an infinite family of orbit homomorphisms $\Psi_n$ into noncommutative Noetherian algebras $T_n = A_1 \otimes U(\mathfrak{g}_n)$ and their images $B_n$. It proves that each kernel $\ker \Psi_n$ is a principal two-sided ideal generated by the differentiator $\Omega^{(2n+2)}_{2n+1,-1}$, while these kernels are not finitely generated as left or right ideals; it also shows that each $B_n$ is non-Noetherian yet birational to $T_n$, and that the degree-zero subring $(B_n)_0$ is (surprisingly) Noetherian. These results yield explicit descriptions of primitive and semi-primitive annihilators linked to one-point local functionals and demonstrate a universal lifting mechanism for such annihilators through the orbit method. Collectively, the work illuminates the large and intricate ideal structure of $U(W_{\ge -1})$, provides concrete generators for a broad class of kernels, and supports conjectures about Noetherianity of Witt-type degree-zero subrings and the global behavior of two-sided ideals.
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 $Ψ_n: U(W_{\geq -1}) \rightarrow T_n$, 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 $U(W_{\geq -1})$.