Noncommutative differential geometry of matrix algebras
- 1 February 1990
- journal article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 31 (2), 316-322
- https://doi.org/10.1063/1.528916
Abstract
The noncommutative differential geometry of the algebra Mn (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebra C(sl(n,C),Mn (C))=Mn (C)⊗Λsl(n,C)*,sl(n,C) acting by inner derivations on Mn (C). A canonical symplectic structure is exhibited for Mn (C) for which the Poisson bracket is, to within a factor i, the commutator. Also, a canonical Riemannian structure is described for Mn (C). Finally, the analog of the Maxwell potential is constructed and it is pointed out that there is a potential with a vanishing curvature that is not a pure gauge.Keywords
This publication has 2 references indexed in Scilit:
- Noncommutative differential geometry and new models of gauge theoryJournal of Mathematical Physics, 1990
- Cohomology theory of Lie groups and Lie algebrasTransactions of the American Mathematical Society, 1948