Tensor Representations of Classical Locally Finite Lie Algebras

Abstract

The structure of tensor representations of the classical finite-dimensional Lie algebras was described by H. Weyl. In this paper we extend Weyl's results to the classical infinite-dimensional locally finite Lie algebras gl ∞ , sl ∞ , sp ∞ and so ∞ , and study important new features specific to the infinite-dimensional setting.Let g be one of the above locally finite Lie algebras and let V be the natural representation of g. The tensor representations of g have the form V ⊗d for the cases g = sp ∞ , so ∞ , and the form V ⊗p ⊗ V ⊗q * for the cases g = gl ∞ , sl ∞ , where V * is the restricted dual of V. In contrast with the finite-dimensional case, these tensor representations are not semisimple. We explicitly describe their Jordan-Hölder constituents, socle filtrations, and indecomposable direct summands.

A Koszul category of representations of finitary Lie algebras

“…The fact that V + λ = soc(T |λ 1 |+1,|λ 2 | ) ∩ (V ⊗Ṽ λ ) follows directly from [PStyr,Theorem 2.3].…”

“…Both statements follow from the obvious infinite-dimensional version of Schur-Weyl duality for the tensor algebra T considered as an sl(∞)-module (see for instance [PStyr]). Moreover, according to [PStyr,Theorems 3.2,4.2],…”

“…It is easy to check that the length of M n as a g n -module stabilizes for n ≥ k, and moreover it coincides with the length of M ; a formula for the length of M is implied by [PStyr,Theorem 2.3]…”

Stability Patterns in Representation Theory

“…The most important results of the section are Proposition 3.1.8 and its consequences. These results can be deduced from those in [PSt,Section 2]; however, our proofs are different and shorter (though we prove less than [PSt, Section 2]).…”

“…Again, a more intrinsic definition is given in [PSt,Section 4]. We denote by Rep(O) the category of algebraic representations.…”