Tensor Representations of Classical Locally Finite Lie Algebras

Summary

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.

2000 Mathematics Subject Classifications: Primary 17B65, 17B10.