Central Extensions of Restricted Affine Nilpotent Lie Algebras $n_{+}(A_1^{(1)})(p)$

Abstract

Consider the maximal nilpotent subalgebra $n_{+}(A_1^{(1)})$ of the simplest affine algebra $A_1^{(1)}$ which is one of the $\mathbb{N}$-graded Lie algebras with minimal number of generators. We show truncated versions of this algebra in positive characteristic admit the structure of a family of restricted Lie algebras. We compute the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients by giving bases. With these we explicitly describe the restricted 1-dimensional central extensions.

20 pages, 9 page appendix

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