Lie ring of derivations

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Definition

Let L be a Lie ring. The Lie ring of derivations of L, denoted \operatorname{Der}(L), is defined as a Lie ring whose elements are the derivations of L, where:

  • The additive structure is given by pointwise addition. Thus, the zero of this ring is the zero derivation.
  • The Lie bracket is given by the commutator. Thus, if d_1,d_2:L \to L are derivations, their Lie bracket is defined as:

[d_1,d_2] := d_1 \circ d_2 - d_2 \circ d_1.

In other words, [d_1,d_2](x) = d_1(d_2(x)) - d_2(d_1(x)).