By Helmut Strade

The matter of classifying the finite-dimensional easy Lie algebras over fields of attribute p > zero is a long-standing one. paintings in this query over the past 35 years has been directed by way of the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed box of attribute p > five a finite-dimensional constrained uncomplicated Lie algebra is classical or of Cartan sort. This conjecture was once proved for p > 7 by way of Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the final case of no longer unavoidably constrained Lie algebras and p > 7 used to be introduced in 1991 via Strade and Wilson and at last proved by means of Strade in 1998. the ultimate Block-Wilson-Strade-Premet class Theorem is a landmark results of sleek arithmetic and will be formulated as follows: each finite-dimensional uncomplicated Lie algebra over an algebraically closed box of attribute p > three is of classical, Cartan, or Melikian sort. within the two-volume booklet, the writer is assembling the evidence of the category Theorem with reasons and references. The aim is a cutting-edge account at the constitution and class thought of Lie algebras over fields of confident attribute resulting in the vanguard of present examine during this box. this primary quantity is dedicated to getting ready the floor for the category paintings to be played within the moment quantity.

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Note that i(rad L) ⊂ rad G. Thus i −1 (I ) = rad L and therefore L/ i −1 (I ) = L/ rad L. 6 (4) shows that (G/ rad G, [p], i) is a p-envelope of L/ rad L. Since C(G/ rad G) = (0), it is a minimal p-envelope. (2) By (1), there is a minimal p-envelope of L which is semisimple. 6 (3) shows that all minimal p-envelopes are semisimple. 7 then proves that all minimal p-envelopes are isomorphic as restricted Lie algebras. Note that, if L is semisimple, then there is an embedding L → Der L via the adrepresentation.

Then (a) μ∈ α,β ,μ=0 Kμ ⊂ n≥0 K (n) , (b) α(K β ) = 0, β(K α ) = 0, and all μ ∈ α, β , μ = 0, are T -roots of K. (3) If α([Kγ , Kβ−γ ]) = 0, γ (Kα ∪ [Kα , Kβ−α ]) = 0, then α, β, γ are T -roots of K, and α([K γ , K β−γ ]) = 0, γ (K α ∪ [K α , K β−α ]) = 0. Proof. (1) There is x ∈ Iα satisfying β(x) = 0. 2 (2) yields that adKβ,α x is invertible. Hence Kβ = (ad x)p (Kβ ) ⊂ I (1) . (2), (3) We only prove (3), as (2) can be treated similarly. Put In := (radT K)(n) . Suppose inductively that Kγ ⊂ In .

We obtain π(L[p] ) = V + π(J ). It has been computed above that the corresponding Lie algebra homomorphism σ : V → L[p] /J is a restricted homomorphism if one considers the p-mapping [p] on V and the p-mapping [p] on L[p] /J . 7 ensures that the ensuing deﬁnition does not depend on the choice of the p-envelope. 1. Let L be a ﬁnite dimensional Lie algebra and let (G, [p], i) be a p-envelope of L. Suppose that H is a subalgebra of L and H[p] is the restricted subalgebra of G generated by i(H ). (1) TR(H, L) := max{dim T | T is a torus of (H[p] + C(G))/C(G)} is called the toral rank of H in L.