Lie algebra of a product

The Lie algebra of a product Lie group is the direct sum of the Lie algebras.

Lie algebra of a product

Let $G,H$ be Lie groups , and consider their product Lie group $G\times H$ .

Theorem

There is a natural Lie algebra isomorphism

\operatorname{Lie}(G\times H)\;\cong\; \operatorname{Lie}(G)\oplus \operatorname{Lie}(H),

where the right-hand side is the direct sum of Lie algebras with bracket

[(X_1,Y_1),(X_2,Y_2)] = \bigl([X_1,X_2],[Y_1,Y_2]\bigr).

Construction (canonical identification)

Using the definition $\operatorname{Lie}(G)=T_eG$ , we have

T_{(e_G,e_H)}(G\times H)\cong T_{e_G}G \oplus T_{e_H}H.

Under this identification, the Lie bracket on $\operatorname{Lie}(G\times H)$ (defined via brackets of left-invariant vector fields) becomes the componentwise bracket above.

Example

For matrix groups, this is visible directly: if $G\subset \mathrm{GL}(n,\Bbb R)$ and $H\subset \mathrm{GL}(m,\Bbb R)$ , then one model of $G\times H$ sits inside $\mathrm{GL}(n+m,\Bbb R)$ as block-diagonal matrices $\mathrm{diag}(A,B)$ . Differentiating at the identity shows

\operatorname{Lie}(G\times H) = \left\{\begin{pmatrix} X & 0\\ 0 & Y\end{pmatrix} : X\in \operatorname{Lie}(G),\ Y\in \operatorname{Lie}(H)\right\} \cong \operatorname{Lie}(G)\oplus \operatorname{Lie}(H),

and commutators are computed blockwise.