Filtered deformations of the Frank algebras (Q1037075)

A filtered deformation of a graded Lie algebra \(L\) is a filtered Lie algebra \({\mathcal L}\) such that the associated graded Lie algebra of \({\mathcal L}\) is isomorphic to \(L\). The goal of this paper is to show that for a Frank algebra, the only filtered deformation is itself. The Frank algebras are an infinite family (parameterized by the set of positive integers) of simple Lie algebras over a field of characteristic 3. Each Frank algebra \(T(m)\) (for an integer \(m\)) can be realized as a subalgebra of the contact algebra \({\mathcal K}(3:1,1,m)\). Each \(T(m)\) admits a \({\mathbb Z}_2\)-grading as well as a \({\mathbb Z}\)-grading with depth 2. For a given \(m\), set \(T := T(m)\) and \({\mathcal K} := {\mathcal K}(3:1,1,m)\). Based on work of \textit{M. Kuznetsov} [Commun. Algebra 18, No. 9, 2943--3013 (1990; Zbl 0716.17016)], the filtered deformations of the Frank algebras are described by the positive (relative to the grading) portion of the first cohomology group of \(T\) with coefficients in \({\mathcal K}/T\). In particular, the claim follows if that cohomology is zero. To show this latter fact, the author embeds the desired cohomology in a sum of Spencer cohomology groups which are shown to all vanish. The vanishing is shown in part by explicit computations with cocycles.