Homomorphism of Lie rings

From Groupprops
Jump to: navigation, search

Definition

Suppose L_1 and L_2 are Lie rings. A homomorphism of Lie rings is a map f:L_1 \to L_2 such that:

  1. f is a homomorphism of groups with respect to the additive group structures of L_1 and L_2.
  2. f([x,y]) = [f(x),f(y)], i.e., f commutes with the Lie bracket.

In other words, a homomorphism of Lie rings is a homomorphism in the variety of Lie rings.