Developments and Retrospectives in Lie Theory pp 99-109 | Cite as

# Three Results on Representations of Mackey Lie Algebras

## Abstract

I. Penkov and V. Serganova have recently introduced, for any nondegenerate pairing \(W \otimes V \rightarrow \mathbb{C}\) of vector spaces, the Lie algebra \(\mathfrak{g}\mathfrak{l}^{M} = \mathfrak{g}\mathfrak{l}^{M}(V,W)\) consisting of endomorphisms of *V* whose duals preserve \(W \subseteq V ^{{\ast}}\). In their work, the category \(\mathbb{T}_{\mathfrak{g}\mathfrak{l}^{M}}\) of \(\mathfrak{g}\mathfrak{l}^{M}\)-modules, which are finite length subquotients of the tensor algebra \(T(W \otimes V )\), is singled out and studied. Denoting by \(\mathbb{T}_{V \otimes W}\) the category with the same objects as \(\mathbb{T}_{\mathfrak{g}\mathfrak{l}^{M}}\) but regarded as *V* ⊗ *W*-modules, we first show that when *W* and *V* are paired by dual bases, the functor \(\mathbb{T}_{\mathfrak{g}\mathfrak{l}^{M}} \rightarrow \mathbb{T}_{V \otimes W}\) 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 \(\mathbb{T}_{\mathrm{End}(V )}\) have finite-length as \(\mathfrak{g}\mathfrak{l}^{M}\)-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in \(\mathbb{T}_{\mathrm{End}(V )}\) as a \(\mathfrak{g}\mathfrak{l}^{M}\)-module.

## Key words

Mackey Lie algebra Finite length module Large annihilator Weight module Socle filtration## Mathematics Subject Classification (2010):

17B10 17B20 17B65.## Notes

## References

- 1.E. Dan-Cohen, I. Penkov, and V. Serganova,
*A Koszul category of representations of finitary Lie algebras*. ArXiv e-prints, May 2011.Google Scholar - 2.M. Hazewinkel,
*Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions*. ArXiv Mathematics e-prints, October 2004.Google Scholar - 3.G. W. Mackey, On infinite-dimensional linear spaces.
*Trans. Amer. Math. Soc.*,**57**(1945), 155–207.MathSciNetCrossRefzbMATHGoogle Scholar - 4.I. Penkov and V. Serganova, Representation theory of Mackey Lie algebras and their dense subalgebras.
*This volume*.Google Scholar - 5.I. Penkov and K. Styrkas, Tensor representations of classical locally finite Lie algebras, in:
*Developments and trends in infinite-dimensional Lie theory*, Progress in Mathematics,**288**, Birkhäuser Boston, 2011, pp. 127–150.Google Scholar - 6.S. V. Sam and A. Snowden,
*Stability patterns in representation theory*. ArXiv e-prints, February 2013.Google Scholar