Representations of Yang-Mills algebras

Abstract

The aim of this article is to describe families of representations of the Yang-Mills algebras $\mathrm{YM}(n)$ ($n \in \mathbb{N}_{\geq 2}$) defined by A. Connes and M. Dubois-Violette. We first describe some irreducible finite dimensional representations. Next, we provide families of infinite dimensional representations of $\mathrm{YM}$, big enough to separate points of the algebra. In order to prove this result, we prove and use that all Weyl algebras $A_{r}(k)$ are epimorphic images of $\mathrm{YM}(n)$.

  • [BB1] Go to document V. Bavula and V. Bekkert, "Indecomposable representations of generalized Weyl algebras," Comm. Algebra, vol. 28, iss. 11, pp. 5067-5100, 2000.
    @article {BB1, MRKEY = {1785490},
      AUTHOR = {Bavula, Vladimir and Bekkert, Viktor},
      TITLE = {Indecomposable representations of generalized {W}eyl algebras},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {28},
      YEAR = {2000},
      NUMBER = {11},
      PAGES = {5067--5100},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {16S32 (16G60 16S35)},
      MRNUMBER = {1785490},
      MRREVIEWER = {Jacques Alev},
      DOI = {10.1080/00927870008827145},
      ZBLNUMBER = {0963.17004},
      }
  • [Ber1] Go to document R. Berger, "Koszulity for nonquadratic algebras," J. Algebra, vol. 239, iss. 2, pp. 705-734, 2001.
    @article {Ber1, MRKEY = {1832913},
      AUTHOR = {Berger, Roland},
      TITLE = {Koszulity for nonquadratic algebras},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {239},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {705--734},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {16S37},
      MRNUMBER = {1832913},
      MRREVIEWER = {Dmitriy A. Rumynin},
      DOI = {10.1006/jabr.2000.8703},
      ZBLNUMBER = {1035.16023},
      }
  • [Ber2] Go to document R. Berger, La catégorie des modules gradués sur une algèbre graduée.
    @misc{Ber2,
      author = {Berger, Roland},
      TITLE = {La catégorie des modules gradués sur une algèbre graduée},
      URL={http://webperso.univ-st-etienne.fr/~rberger/mes-textes.html},
      }
  • [CD1] Go to document A. Connes and M. Dubois-Violette, "Yang-Mills algebra," Lett. Math. Phys., vol. 61, iss. 2, pp. 149-158, 2002.
    @article {CD1, MRKEY = {1936574},
      AUTHOR = {Connes, Alain and Dubois-Violette, Michel},
      TITLE = {Yang-{M}ills algebra},
      JOURNAL = {Lett. Math. Phys.},
      FJOURNAL = {Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics},
      VOLUME = {61},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {149--158},
      ISSN = {0377-9017},
      CODEN = {LMPHDY},
      MRCLASS = {58B34 (16S37 81R60)},
      MRNUMBER = {1936574},
      MRREVIEWER = {Jacek Brodzki},
      DOI = {10.1023/A:1020733628744},
      ZBLNUMBER = {1028.53025},
      }
  • [Cou1] Go to document S. C. Coutinho, A Primer of Algebraic $D$-Modules, Cambridge: Cambridge Univ. Press, 1995, vol. 33.
    @book {Cou1, MRKEY = {1356713},
      AUTHOR = {Coutinho, S. C.},
      TITLE = {A Primer of Algebraic {$D$}-Modules},
      SERIES = {London Math. Soc. Student Texts},
      VOLUME = {33},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1995},
      PAGES = {xii+207},
      ISBN = {0-521-55119-6; 0-521-55908-1},
      MRCLASS = {32C38 (16S32)},
      MRNUMBER = {1356713},
      MRREVIEWER = {Arno van den Essen},
      DOI = {10.1017/CBO9780511623653},
      ZBLNUMBER = {0848.16019},
      }
  • [Dix1] J. Dixmier, Enveloping Algebras, Providence, RI: Amer. Math. Soc., 1996, vol. 11.
    @book {Dix1, MRKEY = {1393197},
      AUTHOR = {Dixmier, Jacques},
      TITLE = {Enveloping Algebras},
      SERIES = {Grad. Studies in Math.},
      VOLUME = {11},
      NOTE = {revised reprint of the 1977 translation},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1996},
      PAGES = {xx+379},
      ISBN = {0-8218-0560-6},
      MRCLASS = {17B35},
      MRNUMBER = {1393197},
      ZBLNUMBER = {0867.17001},
      }
  • [Doug1] Go to document M. Douglas, Report on the status of the Yang-Mills Millenium Prize Problem.
    @misc{Doug1,
      author={Douglas, M.},
      TITLE={Report on the status of the {Y}ang-{M}ills {M}illenium {P}rize {P}roblem},
      URL={http://www.claymath.org/millennium/Yang-Mills\_Theory/ym2.pdf},
      }
  • [HKL1] Go to document P. Hô. Hai, B. Kriegk, and M. Lorenz, "$N$-homogeneous superalgebras," J. Noncommut. Geom., vol. 2, iss. 1, pp. 1-51, 2008.
    @article {HKL1, MRKEY = {2366123},
      AUTHOR = {Hai, Ph{ù}ng H{ô} and Kriegk, Benoit and Lorenz, Martin},
      TITLE = {{$N$}-homogeneous superalgebras},
      JOURNAL = {J. Noncommut. Geom.},
      FJOURNAL = {Journal of Noncommutative Geometry},
      VOLUME = {2},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {1--51},
      ISSN = {1661-6952},
      MRCLASS = {16W50 (05A19 16W55)},
      MRNUMBER = {2366123},
      MRREVIEWER = {Ian M. Musson},
      DOI = {10.4171/JNCG/15},
      ZBLNUMBER = {1151.16028},
      }
  • [Hum1] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, New York: Springer-Verlag, 1978, vol. 9.
    @book {Hum1, MRKEY = {0499562},
      AUTHOR = {Humphreys, James E.},
      TITLE = {Introduction to {L}ie Algebras and Representation Theory},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {9},
      NOTE = {Second printing, revised},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1978},
      PAGES = {xii+171},
      ISBN = {0-387-90053-5},
      MRCLASS = {17Bxx},
      MRNUMBER = {0499562},
      MRREVIEWER = {I. P. Shestakov},
      ZBLNUMBER = {0447.17001},
      }
  • [Kel2] Go to document B. Keller, Corrections to `Sur les ${A}_{\infty}$-catégories’, preprint.
    @misc{Kel2,
      author={Keller, B.},
      TITLE={Corrections to `{S}ur les ${A}_{\infty}$-catégories', preprint},
      URL={http://www.math.jussieu.fr/~keller/lefevre/TheseFinale/corrainf.pdf},
      }
  • [La2] S. Lang, Algebra, third ed., New York: Springer-Verlag, 2002, vol. 211.
    @book {La2, MRKEY = {1878556},
      AUTHOR = {Lang, Serge},
      TITLE = {Algebra},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {211},
      EDITION = {third},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2002},
      PAGES = {xvi+914},
      ISBN = {0-387-95385-X},
      MRCLASS = {00A05 (15-02)},
      MRNUMBER = {1878556},
      ZBLNUMBER = {0984.00001},
      }
  • [Lef1] K. Lefèvre-Hasegawa, Sur les ${A}_{\infty}$-catégories, thèse de doctorat, spécialité: mathématiques, Université Paris 7, Novembre 2003.
    @misc{Lef1,
      author={Lefèvre-Hasegawa, K.},
      TITLE={Sur les ${A}_{\infty}$-catégories, thèse de doctorat, spécialité: mathématiques, {U}niversité {P}aris 7, {N}ovembre 2003},
      }
  • [McR1] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Providence, RI: Amer. Math. Soc., 2001, vol. 30.
    @book {McR1, MRKEY = {1811901},
      AUTHOR = {McConnell, J. C. and Robson, J. C.},
      TITLE = {Noncommutative {N}oetherian Rings},
      SERIES = {Grad. Studies in Math.},
      VOLUME = {30},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2001},
      PAGES = {xx+636},
      ISBN = {0-8218-2169-5},
      MRCLASS = {16P40 (16-02)},
      MRNUMBER = {1811901},
      ZBLNUMBER = {0980.16019},
      }
  • [Mov1] M. Movshev, On deformations of Yang-Mills algebras.
    @misc{Mov1,
      author = {Movshev, M.},
      TITLE={On deformations of {Y}ang-{M}ills algebras},
      ARXIV={hep-th/0509119},
      }
  • [MS1] M. Movshev and A. Schwarz, "Algebraic structure of Yang-Mills theory," in The Unity of Mathematics, Boston, MA: Birkhäuser, 2006, vol. 244, pp. 473-523.
    @incollection {MS1, MRKEY = {2181815},
      AUTHOR = {Movshev, M. and Schwarz, A.},
      TITLE = {Algebraic structure of {Y}ang-{M}ills theory},
      BOOKTITLE = {The Unity of Mathematics},
      SERIES = {Progr. Math.},
      VOLUME = {244},
      PAGES = {473--523},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2006},
      MRCLASS = {81T70 (16E40 81T13)},
      MRNUMBER = {2181815},
      MRREVIEWER = {Beatrice Bucker},
      ZBLNUMBER = {05050086},
      }
  • [Ne1] Go to document N. A. Nekrasov, "Lectures on open strings, and noncommutative gauge theories," in Unity from Duality: Gravity, Gauge Theory and Strings, EDP Sci., Les Ulis, 2003, pp. 477-495.
    @incollection {Ne1, MRKEY = {2010977},
      AUTHOR = {Nekrasov, N. A.},
      TITLE = {Lectures on open strings, and noncommutative gauge theories},
      BOOKTITLE = {Unity from Duality: Gravity, Gauge Theory and Strings},
      VENUE={{L}es {H}ouches, 2001},
      SERIES = {{\rm NATO} Adv. Study Inst.},
      PAGES = {477--495},
      PUBLISHER = {EDP Sci., Les Ulis},
      YEAR = {2003},
      MRCLASS = {81T75 (81-02 81T30)},
      MRNUMBER = {2010977},
      MRREVIEWER = {Alessandro Torrielli},
      DOI = {10.1007/3-540-36245-2_8},
      }
  • [PP1] A. Polishchuk and L. Positselski, Quadratic Algebras, Providence, RI: Amer. Math. Soc., 2005, vol. 37.
    @book {PP1, MRKEY = {2177131},
      AUTHOR = {Polishchuk, Alexander and Positselski, Leonid},
      TITLE = {Quadratic Algebras},
      SERIES = {Univ. Lect. Series},
      VOLUME = {37},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      PAGES = {xii+159},
      ISBN = {0-8218-3834-2},
      MRCLASS = {16S37 (16E30)},
      MRNUMBER = {2177131},
      MRREVIEWER = {Ralf Fr{ö}berg},
      ZBLNUMBER = {1145.16009},
      }
  • [Sl1] Go to document N. J. A. Sloane, The on-line encyclopedia of integer sequences, 2010.
    @misc{Sl1,
      author = {Sloane, N. J. A.},
      TITLE = {The on-line encyclopedia of integer sequences},
      NOTE={published electronically},
      URL={http://oeis.org/},
      YEAR = {2010},
      PAGES = {912--915},
      }
  • [Wei1] C. A. Weibel, An Introduction to Homological Algebra, Cambridge: Cambridge Univ. Press, 1994.
    @book {Wei1, MRKEY = {1269324},
      AUTHOR = {Weibel, Charles A.},
      TITLE = {An Introduction to Homological Algebra},
      SERIES = {Cambridge Stud. Adv. Math.},
      NUMBER = {38},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1994},
      PAGES = {xiv+450},
      ISBN = {0-521-43500-5; 0-521-55987-1},
      MRCLASS = {18-01 (16-01 17-01 20-01 55Uxx)},
      MRNUMBER = {1269324},
      MRREVIEWER = {Kenneth A. Brown},
      ZBLNUMBER = {0797.18001},
      }

Authors

Estanislao Herscovich

Universidad de Buenos Aires
Buenos Aires
Argentina

Andrea Solotar

Universidad de Buenos Aires
Buenos Aires
Argentina