# A Guide to Arithmetic [Lecture notes] by Robin Chapman

By Robin Chapman

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Note that t acts on L as a semisimple transformation, and all eigenvalues are contained in GF(p). Then L is Z/(p)-graded by setting Li := {x ∈ L | t (x) = ix}. 9. Let H be a nilpotent subalgebra of L or Der L, and let k ∈ N. A subalgebra K of L is said to be a k-section with respect to H, if there are GF(p)independent H -roots α1 , . . ,αk Lα . We write K = L(α1 , . . , αk ) in this case. Clearly, L itself is a k-section with respect to H , where k is the GF(p)-dimension of the GF(p)-span of (L, H ).

Note that the mapping [p] is injective on T , hence on T ∩ C(I[p] ). Then T ∩ C(I[p] ) is spanned by (T ∩ C(I[p] ))[p] (cf. 1]). This implies that [T ∩ C(I[p] ), L[p] ] = [(T ∩ C(I[p] ))[p] , L[p] ] ⊂ [C(I[p] ), I ] = (0). Therefore T ∩ C(I[p] ) = T ∩ C(L[p] ), which proves (2). (3) For π : L → L/I and K = L, assertion (1) gives TR(L/I ) + TR(I, L) ≤ TR(L). As TR(I ) = TR(I, L), the inequality follows. 2 The absolute toral rank 29 Now suppose I ⊂ C(L). Due to (1) it remains to prove that TR(K, L) ≤ TR(K + I /I, L/I ).

Then L0 (H ) = {x ∈ L | for all h ∈ H there exists n ∈ N such that hn (x) = 0}, and L = L0 ⊕ L1 , [L0 , Li ] ⊂ Li , H (Li ) ⊂ Li (i = 1, 2). The proof of the following proposition is trivial. 5. Let H be a nilpotent subalgebra of Der L. Then L1 (H ) + [L1 (H ), L1 (H )] is an H -invariant ideal of L. For our purposes special cases are important. 6. (1) Let T ⊂ Der L be a torus. If 0 ⊂ (L, T ) is such that (L, T ) ⊂ Lμ + μ∈ then [Lλ , Lμ ] λ,μ∈ 0 0 0 is a non-zero T -invariant ideal of L. (2) If T is a maximal torus in a p-envelope L[p] of L and x ∈ Lα is a root vector which acts non-nilpotently on L, then Lμ + μ(x)=0 [Lλ , Lμ ] λ(x)=0, μ(x)=0 is a non-zero T -invariant ideal of L.