Tensor product of Lie algebra representations

Let be representations of a Lie algebra for . The tensor product is a representation defined by

The corresponding module over a Lie algebra is denoted .

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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