Tensor product of Lie algebra representations

Let $𝜋 𝑖 : 𝔤 \to 𝔤 𝔩 (𝑉 𝑖)$ be representations of a Lie algebra $𝔤$ for $𝑖 = 1, 2$ . The tensor product $𝜋 1 \otimes 𝜋 2 : 𝔤 \to 𝔤 𝔩 (𝑉 1 \otimes 𝑉 2)$ is a representation defined by

(𝜋 1 \otimes 𝜋 2) (𝑥) = 𝜋 1 (𝑥) \otimes 1 + 1 \otimes 𝜋 2 (𝑥)

The corresponding module over a Lie algebra is denoted $𝑉 1 \otimes 𝕂 𝑉 2$ .

Proof of representation

Clearly each summand defines a representation, furthermore these representations commute. Thus by Sum of commuting Lie algebra representations the above defines a representation.

Properties

This is the infinitesimal representation of the corresponding Tensor product of group representations.

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