2024/06/24 by Dimitry Leites, Leites, Dimitry, Irina Shchepochkina +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2406.16760
We overview classifications of simple infinite-dimensional complex ℤ-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose elements are vector fields with polynomial, or formal power series, or divided power coefficients. A given vectorial Lie (super)algebra with a (Weisfeiler) filtration corresponding to a maximal subalgebra of finite codimension is called W-filtered; the associated graded algebra is called W-graded. Here, we correct several published results: (1) prove our old claim "the superization of É. Cartan's problem (classify primitive Lie algebras) is wild", (2) solve a tame problem: classify simple W-graded and W-filtered vectorial Lie superalgebras, (3) describe the supervariety of deformation parameters for the serial W-graded simple vectorial superalgebras, (4) conjecture that the exceptional simple vectorial superalgebras are rigid. We conjecture usefulness of our method in classification of simple infinite-dimensional vectorial Lie (super)algebras over fields of positive characteristic.