Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 095, 11 pages      arXiv:1901.07532      https://doi.org/10.3842/SIGMA.2019.095
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Cohomology of Restricted Filiform Lie Algebras ${\mathfrak m}_2^\lambda(p)$

Tyler J. Evans ^a and Alice Fialowski ^bc
a) Department of Mathematics, Humboldt State University, Arcata, CA 95521, USA
^b) Institute of Mathematics, University of Pécs, Pécs, Hungary
^c) Institute of Mathematics Eötvös Loránd University, Budapest, Hungary

Received August 19, 2019, in final form November 24, 2019; Published online December 01, 2019

Abstract
For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i] = e_{i+1}$ for $1$<$i$<$p$ and $[e_2,e_i]=e_{i+2}$ for $2$<$i$<$p-1$, we show that there is a family ${\mathfrak m}_2^{\lambda}(p)$ of restricted Lie algebra structures parameterized by elements $\lambda \in {\mathbb F}^p$. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and $[p]$-operations in the corresponding restricted one-dimensional central extensions.

Key words: restricted Lie algebra; central extension; cohomology; filiform Lie algebra.

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