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Tensor Representations of Mackey Lie Algebras and Their Dense Subalgebras

  • Ivan PenkovEmail author
  • Vera Serganova
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 38)

Abstract

In this article we review the main results of the earlier papers [PStyr, PS] and [DPS], and establish related new results in considerably greater generality. We introduce a class of infinite-dimensional Lie algebras \(\mathfrak{g}^{M}\), which we call Mackey Lie algebras, and define monoidal categories \(\mathbb{T}_{\mathfrak{g}^{M}}\) of tensor \(\mathfrak{g}^{M}\)-modules. We also consider dense subalgebras \(\mathfrak{a} \subset \mathfrak{g}^{M}\) and corresponding categories \(\mathbb{T}_{\mathfrak{a}}\). The locally finite Lie algebras \(\mathfrak{s}\mathfrak{l}(V,W),\mathfrak{o}(V ),\mathfrak{s}\mathfrak{p}(V )\) are dense subalgebras of respective Mackey Lie algebras. Our main result is that if \(\mathfrak{g}^{M}\) is a Mackey Lie algebra and \(\mathfrak{a} \subset \mathfrak{g}^{M}\) is a dense subalgebra, then the monoidal category \(\mathbb{T}_{\mathfrak{a}}\) is equivalent to \(\mathbb{T}_{\mathfrak{s}\mathfrak{l}(\infty )}\) or \(\mathbb{T}_{\mathfrak{o}(\infty )}\); the latter monoidal categories have been studied in detail in [DPS]. A possible choice of \(\mathfrak{a}\) is the well-known Lie algebra of generalized Jacobi matrices.

Key words

Finitary Lie algebra Mackey Lie algebra Linear system Tensor representation Socle filtration 

Mathematics Subject Classification (2010):

Primary 17B10 17B65 Secondary 18D10. 

Notes

Acknowledgements

Both authors thank the Max Planck Institute for Mathematics in Bonn where a preliminary draft of this paper was written in 2012. We also thank Jacobs University for its hospitality.

We thank the referee for several thoughtful suggestions, and in particular for showing us the module \(\varLambda ^{[\frac{\infty } {2} ]}V\) from Sect. 3.1.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Jacobs University BremenBremenGermany
  2. 2.Department of MathematicsBerkeleyUSA

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