Right-Invariant Vector Field

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

where $R_g$ is right translation .

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

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

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

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 .