Shapiro's lemma and its consequences in the cohomology theory of modular Lie algebras

Infinitesimal deformations of restricted simple Lie algebras II

“…This follows from the general results of [16]. Indeed, it is easily seen that K (n) is the restricted K (n)-module induced from the restricted K (n) ≥0 -submodule x τ F ⊂ K (n).…”

“…Moreover, using Lemma 2.5, it is straightforward to check that, in the notation of [16], we have the equality…”

“…where F λ is the one-dimensional K (n) ≥0 -module corresponding to the Lie algebra homomorphism λ : K (n) ≥0 → F whose only non-zero value is λ(x n ) = −2m − 4 ≡ deg x τ = (2m + 2)(p − 1) − 2 mod p. Moreover, consider the Lie algebra homomorphism σ : K (n) ≥0 → F given by σ (x) := tr(ad K (n)/K (n) ≥0 x) for x ∈ K (n) ≥0 (see [16,Pag. 155]).…”

“…Indeed, it is easily seen that K (n) is the restricted K (n)-module induced from the restricted K (n) ≥0 -submodule x τ F ⊂ K (n). In the notation of [16], the K (n) ≥0 -module x τ…”

Almost split sequences of Verma modules

“…By (13), (14), (15), and (19) we see that Ext 1 u(g) (Y i , X i ) = (0) for each simple submodule X i of Soc(Ω 2 (Z(γ n ))) and Y i ⊂ Z(γ n ) as above. Consequently, our result follows from Proposition 3.4(ii) and Lemma 4.3(a).…”