[Paper Review] Automorphism groups of Witt algebras

Jianzhi Han, Yucai Su|arXiv (Cornell University)|Feb 5, 2015

Advanced Differential Equations and Dynamical Systems15 references3 citations

TL;DR

This paper establishes an isomorphism between the automorphism groups of the polynomial algebra $A_n$ and the Witt algebra $W_n$ for all $n \geq 1$ by characterizing nonzero endomorphisms of $W_n$ via Jacobi tuples—n-tuples of polynomials with constant non-zero Jacobian determinant. The key result is that ${\rm Aut}\,W_2 \cong {\rm Aut}\,A_2$, which is fully determined as the group generated by permutations, scalings, and triangular automorphisms, resolving the automorphism group of $W_2$ explicitly.

ABSTRACT

The automorphism groups ${ m Aut\,}A_n$ and ${ m Aut\,}W_n$ of the polynomial algebra $A_n=C[x_1,x_2,\cdots, x_n]$ and the rank $n$ Witt algebra $W_n={ m Der\,}A_n$ are studied in this paper. It is well-known that ${ m Aut\,}A_n$ for $n\ge3$ and ${ m Aut\,}W_n$ for $n\ge2$ are open. In the present paper, by characterizing the semigroup ${ m End\,}W_n\setminus\{0\}$ of nonzero endomorphisms of $W_n$ via the semigroup of the so-called Jacobi tuples, we establish an isomorphism between ${ m Aut}\,A_n$ and ${ m Aut\,}W_n$ for any positive integer $n$. In particular, this enables us to work out the automorphism group ${ m Aut\,}W_2$ of $W_2$.

Motivation & Objective

• To determine the automorphism group ${\rm Aut}\,W_n$ for the Witt algebra $W_n = {\rm Der}\,A_n$, a long-standing open problem in infinite-dimensional Lie algebras.
• To establish a structural link between the automorphism groups of the polynomial algebra $A_n$ and the Witt algebra $W_n$ via endomorphisms of $W_n$.
• To resolve the case $n=2$ by proving ${\rm Aut}\,W_2 \cong {\rm Aut}\,A_2$, which is explicitly described.
• To show that the Jacobi conjecture (on polynomial automorphisms with constant Jacobian) is equivalent to the Witt algebra’s conjecture (that every nonzero endomorphism of $W_n$ is an automorphism).

Proposed method

• The authors embed $W_n$ into the derivation algebra $\bar{W}_n = {\rm Der}\,\bar{A}_n$ of the field of rational functions $\bar{A}_n = \mathbb{C}(x_1,\dots,x_n)$, enabling a geometric interpretation of endomorphisms.
• They define a semigroup structure on the set ${\rm JT}_n$ of Jacobi tuples—n-tuples $(f_1,\dots,f_n) \in A_n^n$ with $J(f_1,\dots,f_n) \in \mathbb{C}^\times$—using composition via substitution: $f \cdot g = (g_1(f), \dots, g_n(f))$.
• They construct a map $\zeta: {\rm Aut}\,A_n \to {\rm End}\,W_n \setminus \{0\}$ by associating each automorphism $\tau \in {\rm Aut}\,A_n$ to a unique endomorphism $\sigma_{f_\tau} \in {\rm End}\,W_n$ via the induced action on derivations.
• They prove that $\zeta$ is a well-defined isomorphism from ${\rm Aut}\,A_n$ onto ${\rm Aut}\,W_n$, using the inverse matrix of the Jacobian matrix to define the dual derivation basis.
• They show that any nonzero endomorphism of $W_n$ corresponds to a Jacobi tuple, and under the Jacobi conjecture, such endomorphisms are automorphisms, thus proving the Witt algebra’s conjecture implies the Jacobi conjecture.
• For $n=2$, they use the known structure of ${\rm Aut}\,A_2$ (generated by permutations, scalings, and triangular automorphisms) to fully determine ${\rm Aut}\,W_2$ via the isomorphism.

Experimental results

Research questions

• RQ1 Is there a structural isomorphism between ${\rm Aut}\,A_n$ and ${\rm Aut}\,W_n$ for all $n \geq 1$?
• RQ2 Can the automorphism group of $W_2$ be fully determined, given the difficulty of the general case?
• RQ3 Does the Jacobi conjecture (that constant Jacobian implies polynomial automorphism) imply the Witt algebra’s conjecture (that every nonzero endomorphism of $W_n$ is an automorphism)?
• RQ4 Can the semigroup of nonzero endomorphisms of $W_n$ be parametrized by Jacobi tuples, and does this parametrization yield an isomorphism with ${\rm Aut}\,A_n$?

Key findings

• The paper establishes a canonical isomorphism $\zeta: {\rm Aut}\,A_n \to {\rm Aut}\,W_n$ for all $n \geq 1$, showing that the automorphism groups of $A_n$ and $W_n$ are isomorphic.
• For $n=2$, the automorphism group ${\rm Aut}\,W_2$ is completely determined as the group generated by the permutation $s_1$, scalings $\tau_a$ for $a \in \mathbb{C}^\times$, and triangular automorphisms $\psi_p$ for $p \in \mathbb{Z}^{\geq 0}$, matching the known structure of ${\rm Aut}\,A_2$.
• The authors prove that the Jacobi conjecture is equivalent to the Witt algebra’s conjecture: every nonzero endomorphism of $W_n$ is an automorphism if and only if every Jacobi tuple generates $A_n$.
• The semigroup of nonzero endomorphisms of $W_n$ is parametrized by the set ${\rm JT}_n$ of Jacobi tuples under composition via substitution, with the multiplication rule $f \cdot g = (g_1(f), \dots, g_n(f))$.
• The map $\sigma_{f_\tau} \in {\rm End}\,W_n$ induced by $\tau \in {\rm Aut}\,A_n$ is surjective if and only if the Jacobian of $\tau$ is a nonzero constant, which is guaranteed under the Jacobi conjecture.
• The proof shows that $\sigma_{f_\tau}$ is surjective because the image contains both $\partial_j$ and $x_i^2 \partial_j$ for all $i,j$, which generate $W_n$ when $n \geq 2$, thus forcing $\sigma_{f_\tau} = {\rm Aut}\,W_n$.