Tensor Representations of Classical Locally Finite Lie Algebras

By Ivan Penkov, Konstantin Styrkas. 2007

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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’Hollder constituents, socle filtrations, and indecomposable direct summands.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/306ace51ecc5dc8fdbafe07153217b9a8a85a6fb