Three results on representations of Mackey Lie algebras

2014/03/11 by Alexandru Chirvăsitu, Chirvasitu, Alexandru
Mathematics · #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1403.2481

Abstract

I. Penkov and V. Serganova have recently introduced, for any non-degenerate pairing W⊗ V→\mathbb C of vector spaces, the Lie algebra \mathfrakgl^M=\mathfrakgl^M(V,W) consisting of endomorphisms of V whose duals preserve W⊆ V^*. In their work, the category \mathbbT_\mathfrakgl^M of \mathfrakgl^M-modules which are finite length subquotients of the tensor algebra T(W⊗ V) is singled out and studied. In this note we solve three problems posed by these authors concerning the categories \mathbbT_\mathfrakgl^M. Denoting by \mathbbT_{V⊗ W} the category with the same objects as \mathbbT_\mathfrakgl^M but regarded as V⊗ W-modules, we first show that when W and V are paired by dual bases, the functor \mathbbT_\mathfrakgl^M→ \mathbbT_{V⊗ W} taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of V⊗ W is a tensor equivalence. Secondly, we prove that when W and V are countable-dimensional, the objects of \mathbbT_End(V) have finite length as \mathfrakgl^M-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in \mathbbT_End(V) as a \mathfrakgl^M-module.

Related

• Representations of large Mackey Lie algebras and universal tensor categories
• Universal tensor categories generated by dual pairs
• Ordered tensor categories and representations of the Mackey Lie algebra of infinite matrices
• Representation theory of Mackey Lie algebras and their dense subalgebras
• Topological tensor representations of gl(V) for a space V of countable dimension