The melikyan algebras as lie algebras of the type G₂

Tori in the Melikian Algebra

“…Both G 0 and G 0 are subalgebras of codimension 5 in G. As is shown by Kuznetsov [9,Theorem 4.7] any subalgebra of codimension 5 in G is either G 0 or of the form V + N G 0 V + G 1 + G 2 + · · · where V is a onedimensional subspace in G −1 and N G 0 V its normalizer in G 0 .…”

“…In [9] Kuznetsov described the automorphisms of G preserving its /3 -grading and pointed out without proof that the Lie algebra of the full automorphism group Aut G is G 0 . We will establish the results on the automorphism group using the technique of coinduced modules rather than explicitly constructing certain exponentials as was proposed in [9].…”

“…We use the realization of the Melikian algebra found by Kuznetsov [9]. Let A 2 1 = k x 1 x 2 , x p 1 = x p 2 = 0, be the commutative associative algebra of truncated polynomials in two variables x 1 x 2 , let W 2 1 be its derivation algebra, and W 2 1 = D D ∈ W 2 1 a vector space copy of W 2 1 .…”

Structures of G(2) Type and Nonintegrable Distributions in Characteristic p

“…Kuznetsov offered another description of me(N), see [Ku1]. From Yamaguchi's theorem cited above we know that the Cartan-Tanaka-Shchepochkina prolong of (g(2) − , g(2) 0 ) (in any Z-grading of g (2)) is isomorphic to g(2), at least, over C. There are two Z-gradings of g(2) with one "selected"generator: one of depth 2 and one of depth 3.…”

Фильтрованные деформации алгебр Ли серии $\mathscr{R}$

“…Под исклю-чительными алгебрами Ли понимаются алгебры, которые не изоморфны ни класси-ческим алгебрам Ли, ни алгебрам Ли картановского типа. При = 5 исключитель-ными простыми алгебрами Ли являются алгебры Меликяна, жесткость которых относительно фильтрованных деформаций доказана в [1], [2] (см. также [3]).…”