Cohomology in algebra, geometry, physicsand statistics

Structures of G(2) type in super setting and in positive characteristic, and related curvature tensors

Cartan and Killing described finite-dimensional simple Lie algebras (over fields of real or complex numbers) in terms of the distributions they preserve. The technique of root system and Dynkin (Coxeter) graphs was discovered several decades later. Two o the four series of simple infinite-dimensional Lie algebras of vector fields are Cartan prolongations of non-positive parts of simple finite-dimensional Lie algebras. For any $\mathbb{Z}$-grading of any simple finite-dimensional Lie algebra $\mathfrak{g}$ (bar the two series of examples), the Cartan prolongation of the non-positive part of $\mathfrak{g}$ returns $\mathfrak{g}$. This is not so for the exceptional Lie algebra $\mathfrak{g}_2$ in characteristic 5, whose Cartan prolongation is called Melikyan algebra. Recall that the Lie superalgebras appeared not in high energy physics in 1970s, but in topology, and either over $\mathbb{Z}$ as super Lie rings, or over finite fields. Lately, modular Lie (super)algebras became of interest due to their relation to quantum groups. I intend to tell you about two modular Lie superalgebras constructed (together with S.Bouarroudj and P.Grozman) a la the Melikyan algebra. I hope to have time to say how to compute the analogs of the curvature tensors in presence of non-integrable distribution these algebras preserve.

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