Tensor product of representations

Let $G$ be a Lie group and let $\pi_V:G\to\operatorname{GL}(V)$ and $\pi_W:G\to\operatorname{GL}(W)$ be representations of $G$ . Their tensor product representation on $V\otimes W$ is

\pi_{V\otimes W}(g)=\pi_V(g)\otimes\pi_W(g).

Thus

\pi_{V\otimes W}(g)(v\otimes w) =\pi_V(g)v\otimes\pi_W(g)w,

extended linearly from pure tensors.

Lie-algebra version

Let $\mathfrak g$ be a Lie algebra, and let $\rho_V:\mathfrak g\to\mathfrak{gl}(V)$ and $\rho_W:\mathfrak g\to\mathfrak{gl}(W)$ be representations of $\mathfrak g$ . The tensor product representation $\rho_{V\otimes W}:\mathfrak g\to\mathfrak{gl}(V\otimes W)$ is

\rho_{V\otimes W}(X) =\rho_V(X)\otimes\operatorname{Id}_W +\operatorname{Id}_V\otimes\rho_W(X),

or, equivalently,

X\cdot (v\otimes w)=(X\cdot v)\otimes w + v\otimes (X\cdot w).

A direct computation using the commutator bracket on $\mathfrak{gl}(V\otimes W)$ shows that $\rho_{V\otimes W}$ is a Lie algebra homomorphism.

Weight behavior

If $\mathfrak g$ is semisimple and $\mathfrak h$ is a Cartan subalgebra, then tensor products interact cleanly with the weight-space decomposition: if $v\in V_\lambda$ and $w\in W_\mu$ , then

v\otimes w \in (V\otimes W)_{\lambda+\mu}.

Thus the set of weights of $V\otimes W$ is contained in the Minkowski sum of the weight sets of $V$ and $W$ . This underlies Clebsch–Gordan decompositions and highest-weight calculations.