Branching laws for tensor modules over classical locally finite Lie algebras

Let g′g′ and gg be isomorphic to any two of the Lie algebras gl(∞),sl(∞),sp(∞)gl(∞),sl(∞),sp(∞), and so(∞)so(∞). Let M   be a simple tensor gg-module. We introduce the notion of an embedding g′⊂gg′⊂g of general tensor type and derive branching laws for triples g′,g,Mg′,g,M, where g′⊂gg′⊂g is an embedding of general tensor type. More precisely, since M   is in general not semisimple as a g′g′-module, we determine the socle filtration of M   over g′g′. Due to the description of embeddings of classical locally finite Lie algebras given by Dimitrov and Penkov in 2009, our results hold for all possible embeddings g′⊂gg′⊂g unless g′≅gl(∞)g′≅gl(∞).