Generalizations of local bijectivity of Keller maps and a proof of 22-dimensional Jacobian conjecture

We give a proof of two-dimensional Jacobian conjecture.Comment: This FINAL version is prepared based on the author's two days' lectures (on May 22, 2022 and May 29,2022; the video of the lectures can be found at https://m.koushare.com/video/videodetail/27869

Classification of Quasifinite Modules over the Lie Algebras of Weyl Type

For a nondegenerate additive subgroup $G$ of the $n$-dimensional vector space $F^n$ over an algebraically closed field $F$ of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type $W(G,n)$ spanned by all differential operators $u D_1^{m_1}... D_n^{m_n}$ for $u\in F[G]$ (the group algebra), and $m_1,...,m_n \ge 0$, where $D_1, ...,D_n$ are degree operators. In this paper, it is proved that an irreducible quasifinite $W(\Z,1)$-module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded $W(\Z,1)$-modules is completely given. It is also proved that an irreducible quasifinite $W(G,n)$-module is a module of the intermediate series and a complete classification of quasifinite $W(G,n)$-modules is also given, if $G$ is not isomorphic to $\Z$.Comment: 11 pages, LaTe