{
  "id": "1308.3190",
  "version": "v2",
  "published": "2013-08-14T17:45:28.000Z",
  "updated": "2014-03-18T15:57:43.000Z",
  "title": "The number of independent Traces and Supertraces on Symplectic Reflection Algebras",
  "authors": [
    "S. E. Konstein",
    "I. V. Tyutin"
  ],
  "comment": "29 pages, LaTeX. arXiv admin note: substantial text overlap with arXiv:1211.6600",
  "categories": [
    "math.RT"
  ],
  "abstract": "It is shown that $A:=H_{1,\\eta}(G)$, the Sympectic Reflection Algebra, has $T_G$ independent traces, where $T_G$ is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group $G$ generated by the system of symplectic reflections. Simultaneously, we show that the algebra $A$, considered as a superalgebra with a natural parity, has $S_G$ independent supertraces, where $S_G$ is the number of conjugacy classes of elements without eigenvalue -1 belonging to $G$. We consider also $A$ as a Lie algebra $A^L$ and as a Lie superalgebra $A^S$. It is shown that if $A$ is a simple associative algebra, then the supercommutant $[A^{S},A^{S}]$ is a simple Lie superalgebra having at least $S_G$ independent supersymmetric invariant non-degenerate bilinear forms, and the quotient $[A^L,A^L]/([A^L,A^L]\\cap\\mathbb C)$ is a simple Lie algebra having at least $T_G$ independent symmetric invariant non-degenerate bilinear forms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-18T15:57:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symplectic reflection algebras",
      "independent traces",
      "independent symmetric invariant non-degenerate bilinear",
      "supersymmetric invariant non-degenerate bilinear forms"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.3190K"
    }
  }
}