Right-Invariant Vector Field

Let $G$ be a Lie group. A vector field $X$ on $G$ is right-invariant if for every $g\in G$ ,

(R_g)_*X = X,

where $R_g$ is right translation.

Determined by the value at the identity

As with left-invariance, a right-invariant vector field is determined by $X_e\in T_eG$ , and any $v\in T_eG$ defines a unique right-invariant vector field

X_g := (dR_g)_e(v).

So right-invariant fields also identify (as a vector space) with $\mathfrak{g}=T_eG$ ; see Lie algebra of a Lie group.

Bracket sign convention

Under the identification $v\mapsto X^R$ , the commutator of right-invariant vector fields satisfies

[X^R,Y^R] = -([v,w])^R,

so the bracket corresponds to the negative of the usual Lie bracket on $\mathfrak{g}$ . (Left-invariant fields match the bracket without the minus sign; see left-invariant vector fields.)

Right-invariant fields are often convenient when studying conjugation and the adjoint action.

Equivalent characterizations

Equivalently, for all $g,h\in G$ ,

X_{hg} = (dR_g)_h(X_h).