Lie algebras of linear systems and their automorphisms

2014/06/18 by Mengyuan Zhang, Zhang, Mengyuan
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1406.4753

openalex publication_date 2014/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The objective of this thesis is to study the automorphism groups of the Lie algebras attached to linear systems. A linear system is a pair of vector spaces (U,W) with a nondegenerate pairing ⟨⋅,⋅⟩\colon U⊗ W→ ℂ, to which we attach three Lie algebras \mathfraksl_U,W⊂ \mathfrakgl_U,W⊂\mathfrakgl^M_U,W. If both U and W are countable dimensional, then, up to isomorphism, there is a unique linear system (V,V_*). In this case \mathfraksl_{V,V_*} and \mathfrakgl_{V,V_*} are the well-known Lie algebras \mathfraksl_∞ and \mathfrakgl_∞, while the Lie algebra \mathfrakgl^M_{V,V_*} is the Mackey Lie algebra introduced in \citePSer. We review results about the monoidal categories \mathbbT_\mathfraksl_U,W and \mathbbT_\mathfrakgl^M_U,W of tensor modules, both of which turn out to be equivalent as monoidal categories to the category \mathbbT_\mathfraksl_∞ introduced earlier in \citeDPS. Using the relations between the categories \mathbbT_\mathfraksl_∞ and \mathbbT_\mathfrakgl^M_∞, we compute the automorphism group of \mathfrakgl^M_∞.

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