{
  "id": "0809.2833",
  "version": "v2",
  "published": "2008-09-17T00:47:42.000Z",
  "updated": "2010-10-23T21:56:16.000Z",
  "title": "Second cohomology groups for algebraic groups and their Frobenius kernels",
  "authors": [
    "Caroline B. Wright"
  ],
  "comment": "49 pages, 4 appendices, 6 tables",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic $p > 0$. Let $T$ be a maximal split torus in $G$, $B \\supset T$ be a Borel subgroup of $G$ and $U$ its unipotent radical. Let $F: G \\rightarrow G$ be the Frobenius morphism. For $r \\geq 1$ define the Frobenius kernel, $G_r$, to be the kernel of $F$ iterated with itself $r$ times. Define $U_r$ (respectively $B_r$) to be the kernel of the Frobenius map restricted to $U$ (respectively $B$). Let $X(T)$ be the integral weight lattice and $X(T)_+$ be the dominant integral weights. The computations of particular importance are $\\h^2(U_1,k)$, $\\h^2(B_r,\\la)$ for $\\la \\in X(T)$, $\\h^2(G_r,H^0(\\la))$ for $\\la \\in X(T)_+$, and $\\h^2(B,\\la)$ for $\\la \\in X(T)$. The above cohomology groups for the case when the field has characteristic 2 one computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen for $p \\geq 3$ \\cite{BNP2}.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-10-23T21:56:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06B15"
    ],
    "keywords": [
      "second cohomology groups",
      "frobenius kernel",
      "algebraic group scheme",
      "integral weight",
      "maximal split torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.2833W"
    }
  }
}