2013/11/20 by Penkov, Ivan, Serganova, Vera
#17B10 #17B65 #18D10 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1311.5217
In this article we review the main results of the earlier papers [I. Penkov, K. Styrkas, Tensor representations of infinite-dimensional root-reductive Lie algebras, in Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, Birkhäuser, 2011, pp. 127-150], [I. Penkov, V. Serganova, Categories of integrable \mathfraksl(∞)-, \mathfrako(∞)-, \mathfraksp(∞)-modules, in "Representation Theory and Mathematical Physics", Contemporary Mathematics 557 (2011), pp. 335-357] and [E. Dan-Cohen, I. Penkov, V. Serganova, A Koszul category of representations of finitary Lie algebras, preprint 2011, arXiv:1105.3407], and establish related new results in considerably greater generality. We introduce a class of infinite-dimensional Lie algebras \mathfrakgM, which we call Mackey Lie algebras, and define monoidal categories \mathbbT_\mathfrakgM of tensor \mathfrakgM-modules. We also consider dense subalgebras \mathfraka ⊂ \mathfrakgM and corresponding categories \mathbbT_\mathfraka. The locally finite Lie algebras \mathfraksl(V,W), \mathfrako(V), \mathfraksp(V) are dense subalgebras of respective Mackey Lie algebras. Our main result is that if \mathfrakgM is a Mackey Lie algebra and \mathfraka ⊂ \mathfrakgM is a dense subalgebra, then the monoidal category \mathbbT_\mathfraka is equivalent to \mathbbT_\mathfraksl(∞) or \mathbbT_\mathfrako(∞); the latter monoidal categories have been studied in detail in [E. Dan-Cohen, I. Penkov, V. Serganova, A Koszul category of representations of finitary Lie algebras, preprint 2011, arXiv:1105.3407]. A possible choice of \mathfraka is the well-known Lie algebra of generalized Jacobi matrices.