Abstract
We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are an extension of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in (Lett Math Phys 61(2):149–158, 2002), and in fact they appear as a “background independent” formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras \({{\rm {Cliff}}_{q}(k) \otimes A_{p}(k)}\), for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.
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Aubry M., Lemaire J.-M.: Zero divisors in enveloping algebras of graded Lie algebras. J. Pure Appl. Alg. 38(2–3), 159–166 (1985)
Artin M., Zhang J.J.: Noncommutative projective schemes. Adv. Math. 109(2), 228–287 (1994)
Bavula V., Bekkert V.: Indecomposable representations of generalized Weyl algebras. Comm. Alg. 28(11), 5067–5100 (2000)
Bell A.D., Musson I.M.: Primitive factors of enveloping algebras of nilpotent Lie superalgebras. J. London Math. Soc. (2) 42(3), 401–408 (1990)
Berger R., Ginzburg V.: Higher symplectic re ection algebras and non-homogeneous N-Koszul property. J. Alg. 304(1), 577–601 (2006)
Berger R., Marconnet N.: Koszul and Gorenstein properties for homogeneous algebras. Alg. Rep. Th. 9(1), 67–97 (2006)
Berger R., Taillefer R.: Poincaré-Birkhoff-Witt deformations of Calabi-Yau algebras. J. Noncom. Geom. 1(2), 241–270 (2007)
Bergman G.M.: The diamond lemma for ring theory. Adv. in Math. 29(2), 178–218 (1978)
Cohen M., Montgomery S.: Group-graded rings, smash products, and group actions. Trans. Amer. Math. Soc. 282(1), 237–258 (1984)
Connes A., Dubois-Violette M.: Yang-Mills algebra. Lett. Math. Phys. 61(2), 149–158 (2002)
Connes, A., Dubois-Violette, M.: Yang-Mills and some related algebras. In: Rigorous quantum field theory, Progr. Math., Vol. 251, Basel: Birkhäuser, 2007, pp. 65–78
Deligne, P., Freed, D.S.: Classical field theory. (Princeton, NJ, 1996/1997), Providence, RI: Amer. Math. Soc., 1999, pp. 137–225
Deligne, P., Freed, D.S.: Supersolutions. (Princeton, NJ, 1996/1997), Providence, RI: Amer. Math. Soc., 1999, pp. 227–355
Douglas M.R., Nekrasov N.A.: Noncommutative field theory. Rev. Mod. Phys. 73(4), 977–1029 (2001)
Ferrero M., Kishimoto K., Motose K.: On radicals of skew polynomial rings of derivation type. J. London Math. Soc. (2) 28(1), 8–16 (1983)
Fröberg R.: Determination of a class of Poincaré series. Math. Scand. 37(1), 29–39 (1975)
Gordon R., Green E.L.: Graded Artin algebras. J. Algebra 76(1), 111–137 (1982)
Hô Hai P., Kriegk B., Lorenz M.: N-homogeneous superalgebras. J. Noncom. Geom. 2(1), 1–51 (2008)
Herscovich E.: The Dixmier map for nilpotent super Lie algebras. Commun. Math. Phy. 313(2), 295–328 (2012)
Herscovich E., Solotar A.: Representation theory of Yang-Mills algebras. Ann. of Math. (2) 173(2), 1043–1080 (2011)
Herscovich E., Solotar A.: Hochschild and cyclic homology of Yang-Mills algebras. J. Reine Ange. Math. 665, 73–156 (2012)
Lang, S.: Algebra. 3rd ed., Graduate Texts in Mathematics, Vol. 211, New York: Springer-Verlag, 2002
Letzter E.: Primitive ideals in finite extensions of Noetherian rings. J. London Math. Soc. (2) 39(3), 427–435 (1989)
Mori I., Minamoto H.: The structure of AS-Gorenstein algebras. Adv. Math. 226(5), 4061–4095 (2011)
Movshev, M., Schwarz, A.: Algebraic structure of Yang-Mills theory. In: The unity of mathematics, Progr. Math., Vol. 244, Boston, MA: Birkhäuser Boston, 2006, pp. 473–523; Movshev, M.: Yang-Mills theories in dimesiions 3, 4, 6, 10 and Bar-duality. http://arxiv.org/abs/hep-th/0503165v2, 2005
Musson, I.M., Pinczon, G., Ushirobira, R.: Hochschild cohomology and deformations of Clifford-Weyl algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 5, Paper 028, 27 (2009)
Piontkovski D.: Coherent algebras and noncommutative projective lines. J. Alg. 319(8), 3280–3290 (2008)
Rotman, J.J.: An introduction to homological algebra. 2nd ed., Universitext, New York: Springer, 2009
Tanaka J.: On homology and cohomology of Lie superalgebras with coefficients in their finite-dimensional representations. Proc. Japan Acad. Ser. A Math. Sci. 71(3), 51–53 (1995)
Weibel, C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge: Cambridge University Press, 1994
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Communicated by A. Connes
The author is an Alexander von Humboldt fellow.
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Herscovich, E. Representations of Super Yang-Mills Algebras. Commun. Math. Phys. 320, 783–820 (2013). https://doi.org/10.1007/s00220-012-1648-z
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DOI: https://doi.org/10.1007/s00220-012-1648-z