Tensor product of representations

The diagonal action on : for Lie algebras acts by Leibniz, for Lie groups by tensoring operators.

Tensor product of representations

Definition (Lie groups)

Let $G$ be a Lie group and let $\pi_V:G\to GL(V)$ and $\pi_W:G\to GL(W)$ be representations of $G$ . The tensor product representation is

\pi_{V\otimes W}(g)=\pi_V(g)\otimes \pi_W(g)\in GL(V\otimes W),

defined by $(\pi_V(g)\otimes \pi_W(g))(v\otimes w)=\pi_V(g)v\otimes \pi_W(g)w$ and extended linearly.

Definition (Lie algebras)

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 given by

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

i.e.

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 $\rho_{V\otimes W}$ is a Lie algebra homomorphism .

Weight behavior (motivation)

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 is one of the basic mechanisms behind Clebsch–Gordan type decompositions and highest-weight calculations (compare highest-weight representations ).