{
  "id": "0704.3278",
  "version": "v2",
  "published": "2007-04-24T22:05:29.000Z",
  "updated": "2016-02-16T11:37:10.000Z",
  "title": "Zeroth Hochschild homology of preprojective algebras over the integers",
  "authors": [
    "Travis Schedler"
  ],
  "comment": "69 pages, 2 figures; final pre-publication version; many corrections and improvements throughout. Note though the previous version has additional results",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new $p$-torsion classes in degrees 2p^l, l >= 1, We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the appendix. In the previous version, additional results are included, such as: the Poisson center of $\\Sym HH_0(\\Pi)$ for all quivers, the BV algebra structure on Hochschild cohomology, including how the Lie algebra structure $HH_0(\\Pi_Q)$ naturally arises from it, and the cyclic homology groups of $\\Pi_Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-04-24T22:05:29.000Z",
      "title": "Hochschild homology of preprojective algebras over the integers",
      "abstract": "We determine the $\\Z$-module structure and explicit bases for the preprojective algebra $\\Pi$ and all of its Hochschild (co)homology, for any non-Dynkin quiver. This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new $p$-torsion elements in degrees $2 p^\\ell, \\ell \\geq 1$. We relate these elements by $p$-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. We also define a Lie bialgebra structure on $HH_0(\\Pi)$ (from the necklace Lie bialgebra), relate it to Goldman/Turaev's Lie bialgebra of loops, compute it for extended Dynkin quivers, and compute the Poisson center of $\\Sym HH_0(\\Pi)$ for all quivers. We then compute the BV algebra structure on Hochschild cohomology, show that the Lie algebra structure $HH_0(\\Pi_Q)$ naturally arises from it, and compute all cyclic homology groups of $\\Pi_Q$. In the process, we define and study related algebraic structures: a ``noncommutative BV structure'' generalizing the necklace Lie bialgebra, and ``free-product'' deformations of $\\Pi_Q$, which yield all ordinary deformations as quotients.",
      "comment": "103 pages, 2 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-02-16T11:37:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "preprojective algebra",
      "hochschild homology",
      "necklace lie bialgebra",
      "goldman/turaevs lie bialgebra",
      "study related algebraic structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0704.3278S"
    }
  }
}