{
  "id": "1406.4753",
  "version": "v1",
  "published": "2014-06-18T15:02:52.000Z",
  "updated": "2014-06-18T15:02:52.000Z",
  "title": "Lie algebras of linear systems and their automorphisms",
  "authors": [
    "Mengyuan Zhang"
  ],
  "categories": [
    "math.RT"
  ],
  "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 $\\langle\\cdot,\\cdot\\rangle\\colon U\\otimes W\\to \\mathbb{C}$, to which we attach three Lie algebras $\\mathfrak{sl}_{U,W}\\subset \\mathfrak{gl}_{U,W}\\subset\\mathfrak{gl}^M_{U,W}$. If both $U$ and $W$ are countable dimensional, then, up to isomorphism, there is a unique linear system $(V,V_*)$. In this case $\\mathfrak{sl}_{V,V_*}$ and $\\mathfrak{gl}_{V,V_*}$ are the well-known Lie algebras $\\mathfrak{sl}_\\infty$ and $\\mathfrak{gl}_\\infty$, while the Lie algebra $\\mathfrak{gl}^M_{V,V_*}$ is the Mackey Lie algebra introduced in \\cite{PSer}. We review results about the monoidal categories $\\mathbb{T}_{\\mathfrak{sl}_{U,W}}$ and $\\mathbb{T}_{\\mathfrak{gl}^M_{U,W}}$ of tensor modules, both of which turn out to be equivalent as monoidal categories to the category $\\mathbb{T}_{\\mathfrak{sl}_\\infty}$ introduced earlier in \\cite{DPS}. Using the relations between the categories $\\mathbb{T}_{\\mathfrak{sl}_\\infty}$ and $\\mathbb{T}_{\\mathfrak{gl}^M_\\infty}$, we compute the automorphism group of $\\mathfrak{gl}^M_\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-18T15:02:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "automorphism group",
      "monoidal categories",
      "mackey lie algebra",
      "unique linear system",
      "well-known lie algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.4753Z"
    }
  }
}