New simple Lie superalgebras as queerified associative algebras

Abstract

Over $\mathbb{C}$, Montgomery superized Herstein's construction of simple Lie algebras from finite-dimensional associative algebras, found obstructions to the procedure and applied it to $\mathbb{Z}/2$-graded associative algebra of differential operators with polynomial coefficients. Since the 1990s, Vasiliev and Konstein with their co-authors constructed (via the Herstein--Montgomery method, having rediscovered it) simple Lie (super)algebras from the associative (super)algebra such as Vasiliev's higher spin algebras (a.k.a. algebras of observables of the rational Calogero model) and algebras of symplectic reflections. The "queerification" is another method for cooking a~simple Lie superalgebra from the simple associative (super)algebra. The above examples of associative (super)algebras, and Lie (super)algebras of "matrices of complex size" can be "queerified" by adding new elements resembling Faddeev--Popov ghosts. Conjectures: 1) a "queerified" Hamiltonian describes a version of the Calogero model with $1\vert 1$-dimensional time; 2) metabelean algebras and inhomogeneous subalgebras of Lie superalgebras naturally widen supersymmetries in future theories; 3) only graded-commutative algebras can imitate algebras of functions in a reasonably rich non-commutative Geometry.