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]
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]
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},
} -
@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]
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]
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},
} -
@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]
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},
} -
@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]
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},
} -
@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},
}