Representation theory of Mackey Lie algebras and their dense subalgebras

Abstract

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 $\mathfrak{sl}(\infty )$-, $\mathfrak{o}(\infty )$-, $\mathfrak{sp}(\infty )$-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 ${\mathfrak{g}}^M$, 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 $\mathfrak{sl}(V,W),\mathfrak{o}(V),\mathfrak{sp}(V)$ 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 ${\mathbb{T}}_{\mathfrak{sl}(\infty )}$ or ${\mathbb{T}}_{\mathfrak{o}(\infty )}$; 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 $\mathfrak{a}$ is the well-known Lie algebra of generalized Jacobi matrices.

24 pages, 1 figure, key words: finitary Lie algebra, Mackey Lie algebra, linear system, tensor representation, socle filtration

References in corpus (2)

• GL-equivariant modules over polynomial rings in infinitely many variables
• Cartan subalgebras of root-reductive Lie algebras