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 $${\mathfrak{g}{{\mathfrak{l}}}_{\infty}},\,\,{\mathfrak{s}{{\mathfrak{{l}}}_{\infty}}},\,\,{\mathfrak{s}{{\mathfrak{{p}}}_{\infty}}}\,\,\,{\rm and}\,\,\, {\mathfrak{s}{{\mathfrak{{o}}}_{\infty}}},$$ and study important new features specific to the infinite-dimensional setting. Let $$\mathfrak{g}$$ be one of the above locally finite Lie algebras and let v be the natural representation of $$\mathfrak{g}.$$ The tensor representations of $$\mathfrak{g}.$$ have the form V ⊗p ⊗ V ⊗q * for the cases $${\mathfrak{g}}=\,\,{\mathfrak{g}}{{\mathfrak{{l}}}}_{\infty},\,\,{\mathfrak{s}{{\mathfrak{{l}}}_{\infty}}},$$ 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’Höllder constituents, socle filtrations, and indecomposable direct summands.