Noncommutative Supergeometry of Graded Matrix Algebras

  • H. Grosse
  • G. Reiter
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 543)


Generalizing the ungraded case one can build up \( \mathbb{Z}_2 \)-graded differential calculi over arbitrary \( \mathbb{Z}_2 \)-graded \( \mathbb{Z}_2 \)-algebras based on their respective Lie superalgebra of graded derivations. Especially for the \( \mathbb{Z}_2 \)-graded \( \mathbb{Z}_2 \)-algebra (M n|m) of (n+m) × (n+m)-matrices with block-matrix grading (n, m ∈ N0, nm) the resulting differential algebra (Ω g (M n|m), d) coincides— as far as we are interested only in its linear structure - with the cochain complex of the Lie superalgebra sl( n|m) with coefficients in (M n|m). Here we want to point out two remarkable facts about this differential algebra (for proofs see Grosse and Reiter 1999)


Quantum Field Theory Linear Structure Differential Calculus Remarkable Fact Differential Algebra 
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  1. Grosse H., Reiter G. (1999): Graded Differential Geometry of Graded Matrix Algebras. Preprint UWThPh-6-1999.Google Scholar
  2. Kostant B. (1977): Graded manifolds, graded Lie theory, and prequantization. Lecture Notes in Mathematics 570.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • H. Grosse
    • 1
  • G. Reiter
    • 1
    • 2
  1. 1.Institut für Theoretische PhysikUniversität WienWienAustria
  2. 2.Institut für Theoretische PhysikTechnische Universität GrazGrazAustria

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