Delta methods in enveloping rings

Enveloping algebras that are principal ideal rings

“…Thus, Proposition 3.6 implies that L = L/T is infinite dimensional cyclic and the proof is complete. ✷ Following [4], for a restricted Lie algebra L we define…”

The adjoint representation of group algebras and enveloping algebras

“…For now we just make two quick observations . First, it is an easy exercise to show that AL = 0 in the restricted case and, in particular, we sea that AL is a characteristic restricted ideal of -L. Second, the argument of [BP2,Proposition 4.2] applies equally well to restricted enveloping algebas and yields the necessary sharpening of [BP1,Theorem 5 .1] . With this, the exact restricted arralog of [BP2; Theorern 4.5] holds and therefore a direct application of the preceding proof yields Theorem 2.2.…”

“…) Again, the latter formula is a linear identity in U(L) and this time we observe that S((arJ2) E U(InkL) and that S(p) = fp is plus or minus a straightened monomial in X . Thus, for any monomial T, [BP2,Theorem 4.5(ii)] yields 0 =~(a,), r S((a,)2) = aT -r for all r E U(L) .…”

“…Proof-We follow the notation of [BP2] . Let a E Z(L), choose a complementary basis X for AL in L and use it to write cx = El~pa, based on AL .…”

The inversion height of the free field is infinite

“…Note that S = ι rs (X 0 ) and T 0 = ι(X 1 )ι(X 0 ) −1 . Since Γ s is generated by S and T 0 , k[Γ s ] ⊆ E s ιrs (2). Therefore E s is generated, as a field, by the image of ι rs and h ιrs (k X 0 , .…”