Centralizer of a subset of a Lie ring

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ANALOGY: This is an analogue in Lie rings of a term encountered in group. That term is: centralizer.
View other analogues of centralizer

Definition

Let L be a Lie ring and A be a subset of L. The centralizer of A in L, denoted C_L(A), is defined as:

C_L(A) := \{ x \in L \mid [a,x] = 0 \ \forall \ a \in A \}.

The centralizer of any subset of a Lie ring is a Lie subring. For full proof, refer: Centralizer of subset of Lie ring is Lie subring

As a Galois correspondence

Brief description

The centralizer operator can be viewed as a Galois correspondence from the collection of subsets of the Lie ring to itself, corresponding to the symmetric relation of commuting in the sense of the Lie bracket being zero. In particular:

  • S_1 \subseteq S_2 \implies C_L(S_2) \subseteq C_L(S_1).
  • S \subseteq C_L(C_L(S)).

Implications

The implications of the above correspondence are as follows. Define the bicentralizer of a subset as the centralizer of its centralizer. Thus, a subset equals its own bicentralizer if and only if it occurs as the centralizer of some subset. A subset that occurs as a bicentralizer is termed a c-closed Lie subring. In particular, it is a subring. Also, the centralizer of any subset equals the centralizer of the Lie subring generated by that subset.