Lie algebras MOC

Tensor product of Lie algebra representations

Let 𝜋𝑖 :𝔤 →𝔤𝔩(𝑉𝑖) be representations of a Lie algebra 𝔤 for 𝑖 =1,2. The tensor product 𝜋1 ⊗𝜋2 :𝔤 →𝔤𝔩(𝑉1 ⊗𝑉2) is a representation defined by

(𝜋1⊗𝜋2)(𝑥)=𝜋1(𝑥)⊗1+1⊗𝜋2(𝑥)

The corresponding module over a Lie algebra is denoted 𝑉1 ⊗𝕂𝑉2.

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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