Filtered Multiplicative Bases of Restricted Enveloping Algebras

Abstract

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p.

On Filtered Multiplicative Bases of Some Associative Algebras

“…Moreover, by Theorem 2.2 of [17] one has u(gr(L)) ∼ = gr(u(L)), hence gr(u(L)) has an f.m.b. On the other hand, for what was showed in [7] (see the example on page 607), in this case u(L) cannot have any filtered multiplicative basis.…”

“…Consider the abelian restricted Lie algebra It is easy to see that gr(L) is isomorphic to the direct sum of three cyclic restricted Lie algebras and so u(gr(L)) has an f.m.b. (see [7], Theorem 1). Moreover, by Theorem 2.2 of [17] one has u(gr(L)) ∼ = gr(u(L)), hence gr(u(L)) has an f.m.b.…”

“…by virtue of (F3) and Theorem 1 of [7] it enough to show that u(h (s) ) has an f.m.b. Let L be the relatively free nilpotent restricted Lie algebra of class 2 on the set {x, y} and I the restricted ideal of L generated by x [2] s , y [2] s and [x, y] [2] s .…”

“…Let L be a finite-dimensional p-nilpotent restricted Lie algebra over a field of characteristic p > 0. In [7] we proved that if L is abelian then u(L) has a filtered multiplicative basis if and only if it is a direct sum of cyclic restricted subalgebras. Moreover, we showed that if L has nilpotent class 2 and p > 2 then u(L) does not have any filtered multiplicative basis.…”

“…bs(u(L)) such that 1 ∈ bs(u(L)). By Theorem 3 of[7] the ground field must have characteristic 2. Let Γ = bs(u(L))\ ω(L) 2 ∪ {1} = {g 1 , .…”