Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras

V. V. Bavula

Izvestiya. Mathematics, Tome 77 (2013) no. 6, pp. 1067-1104

Résumé

We make a detailed study of the Lie algebras $\mathfrak{u}_n$, $n\geqslant 2$, of triangular polynomial derivations, their injective limit $\mathfrak{u}_\infty$, and their completion $\widehat{\mathfrak{u}}_\infty$. We classify the ideals of $\mathfrak{u}_n$ (all of which are characteristic ideals) and use this classification to give an explicit criterion for Lie factor algebras of $\mathfrak{u}_n$ and $\mathfrak{u}_m$ to be isomorphic. We introduce two new dimensions for (Lie) algebras and their modules: the central dimension $\operatorname{c.dim}$ and the uniserial dimension $\operatorname{u.dim}$, and show that $\operatorname{c.dim}(\mathfrak{u}_n)=\operatorname{u.dim}(\mathfrak{u}_n) =\omega^{n-1}+\omega^{n-2}+\dots+\omega +1$ for all $n\geqslant 2$, where $\omega$ is the first infinite ordinal. Similar results are proved for the Lie algebras $\mathfrak{u}_\infty$ and $\widehat{\mathfrak{u}}_\infty$. In particular, $\operatorname{u.dim}(\mathfrak{u}_\infty)=\omega^\omega$ and $\operatorname{c.dim}(\mathfrak{u}_\infty)=0$.

Détail

Keywords: Lie algebra, triangular polynomial derivations, isomorphism problem, the derived series and lower central series, locally nilpotent derivations, locally nilpotent and locally finite-dimensional Lie algebras.
Mots-clés : automorphism

Bibliographie

[1] V. V. Bavula, The groups of automorphisms of the Lie algebras of triangular polynomial derivations, arXiv: 1304.4225

[2] V. V. Bavula, “Every monomorphism of the Lie algebra of triangular polynomial derivations is an automorphism”, C. R. Math. Acad. Sci. Paris, 350:11–12 (2012), 553–556 | DOI | MR | Zbl

[3] V. V. Bavula, “Finite dimensionality of $\operatorname{Ext}^n$ and $\operatorname{Tor}_n$ of simple modules over one class of algebras”, Funct. Anal. Appl., 25:3 (1991), 229–230 | DOI | MR | Zbl

[4] V. V. Bavula, “Generalized Weyl algebras and their representations”, St. Petersburg Math. J., 4:1 (1993), 71–92 | MR | Zbl

[5] B. Rotman, G. T. Kneebone, The theory of sets and transfinite numbers, Oldbourne, London, 1966 | MR | Zbl