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Representations of Super Yang-Mills Algebras

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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Correspondence to Estanislao Herscovich.

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The author is an Alexander von Humboldt fellow.

Communicated by A. Connes

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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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Keywords

  • Hilbert Series
  • Homogeneous Element
  • Weyl Algebra
  • Primitive Ideal
  • Grade Vector Space