[LiePRing] 2 Lie p-rings

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# 2 Lie p-rings

In this preliminary chapter we recall some of theoretic background of Lie rings and Lie p-rings. We refer to Chapter 5 in Khu for some further details. Throughout we assume that p stands for a rational prime.

A Lie ring L is an additive abelian group with a multiplication that is alternating, bilinear and satisfies the Jacobi identity. We denote the product of two elements g and h of L with g h.

A subset I ⊆ L is an idealin the Lie ring L if it is a subgroup of the additive group of L and it satisfies a l ∈ I for all a ∈ I and l ∈ L. As the multiplication in L is alternating, it follows that l a ∈ I for all l ∈ L and a ∈ I. Note that if I and J are ideals in L, then I + J = { a + b | a ∈ I, b ∈ J} and I J = 〈a b | a ∈ I, b ∈ J 〉+ are ideals in L.

A subset U ⊆ L is a subringof the Lie ring L if U is a Lie ring with respect to the addition and the multiplication of L. Every ideal in L is also a subring of L. As usual, for an ideal I in L the quotient L/I has the structure of a Lie ring, but this does not hold for subrings.

The lower central seriesof the Lie ring L is the series of ideals L = γ1(L) ≥ γ2(L) ≥ … defined by γi(L) = γi−1(L) L. We say that L is nilpotentif there exists a natural number c with γc+1(L) = {0}. The smallest natural number with this property is the classof L.

The notion of nilpotence now allows to state the central definition of this package. A

Lie p-ring is a Lie ring that is nilpotent and has pn elements for some natural number n.

Every finite dimensional Lie algebra over a field with p elements is an example for a Lie ring with pn elements. Note that there exist non-nilpotent Lie algebras of this type: the Lie algebra consisting of all n ×n matrices with trace 0 and n ≥ 3 is an example. Thus not every Lie ring with pn elements is nilpotent. (In contrast to the group case, where every group with pn elements is nilpotent!)

For a Lie p-ring L we define the series L = λ1(L) ≥ λ2(L) ≥ … via λi+1(L) = λi(L) L + p λi(L). This series is the lower exponent-p central seriesof L. Its length is the p-classof L. If |L/λ2(L)| = pd, then d is the minimal generator numberof L. Similar to the p-group case, one can observe that this is indeed the cardinality of a generating set of smallest possible size.

Each Lie p-ring L has a central series L = L1 ≥ … ≥ Ln ≥ {0} with quotients of order p. Choose li ∈ Li \Li+1 for 1 ≤ i ≤ n. Then (l1, …, ln) is a generating set of L satisfying that p li ∈ Li+1 and li lj ∈ Li+1 for 1 ≤ j < i ≤ n. We call such a generating sequence a basisfor L and we say that L has dimensionn.

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