Let $g_{1}$, $g_{2}$ be Lie algebras , $g_{1} \oplus g_{2}$ their direct sum. As a vector space it is a direct sum of $g_{1}$ and $g_{2}$ and Lie algebra structure is given by $[(x,y),(x',y')]=([x,x'],[y,y'])$. Is it true that every irreducible representation of $g_{1}\oplus g_{2}$ is isomorphic to $V_{1}\otimes V_{2}$ where $V_{1}$, $V_{2}$ are irreducible represenrations of $g_{1}$, $g_{2}$ respectively?
Irr. representations of semi-direct product of Lie algebras
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representation-theory
lie-algebras
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0$V_1 \otimes V_2$ only naturally inherits the structure of a representation of the direct sum; there's no natural action of any other semidirect product. I expect the classification to resemble the corresponding classification for finite groups, which is known; look up Clifford theory. – 2017-02-23
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0Why is this a semidirect product/sum? Isn't this just the direct product/sum? – 2017-02-24
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0I suppose it would be right to call it direct sum. – 2017-02-24