Properties
Representations of a Lie algebra are in one-to-one correspondence with algebra representations of the associated universal enveloping algebra. This follows from the universal property of that construction.
If the Lie algebra is semisimple, then all reducible representations are decomposable. Otherwise, that's not true in general.
If we have two representations, with V1 and V2 as their underlying vector spaces and ·1 and ·2 as the representations, then the product of both representations would have V1 ⊗ V2 as the underlying vector space and
If L is a real Lie algebra and ρ: L × V→ V is a complex representation of it, we can construct another representation of L called its dual representation as follows.
Let V∗ be the dual vector space of V. In other words, V∗ is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that for any z in C, ω in V∗ and X in V. This is usually rewritten as a contraction with a sesquilinear form 〈·,·〉. i.e. 〈ω,X〉 is defined to be ω.
We define as follows:
- 〈(A),X〉 + 〈ω, ρA〉 = 0,
for any A in L, ω in V∗ and X in V. This defines uniquely.
Read more about this topic: Lie Algebra Representation
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