{
  "id": "1511.00024",
  "version": "v1",
  "published": "2015-10-30T20:37:29.000Z",
  "updated": "2015-10-30T20:37:29.000Z",
  "title": "Extensions for Generalized Current Algebras",
  "authors": [
    "Brian D. Boe",
    "Christopher M. Drupieski",
    "Tiago R. Macedo",
    "Daniel K. Nakano"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Given a complex semisimple Lie algebra ${\\mathfrak g}$ and a commutative ${\\mathbb C}$-algebra $A$, let ${\\mathfrak g}[A] = {\\mathfrak g} \\otimes A$ be the corresponding generalized current algebra. In this paper we explore questions involving the computation and finite-dimensionality of extension groups for finite-dimensional ${\\mathfrak g}[A]$-modules. Formulas for computing $\\operatorname{Ext}^{1}$ and $\\operatorname{Ext}^{2}$ between simple ${\\mathfrak g}[A]$-modules are presented. As an application of these methods and of the use of the first cyclic homology, we completely describe $\\operatorname{Ext}^{2}_{{\\mathfrak g}[t]}(L_{1},L_{2})$ for ${\\mathfrak g}=\\mathfrak{sl}_{2}$ when $L_{1}$ and $L_{2}$ are simple ${\\mathfrak g}[t]$-modules that are each given by the tensor product of two evaluation modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-30T20:37:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B55",
      "17B65"
    ],
    "keywords": [
      "complex semisimple lie algebra",
      "first cyclic homology",
      "extension groups",
      "evaluation modules",
      "corresponding generalized current algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}