NONCOMMUTATIVE DIFFERENTIAL GEOMETRY OF MATRIX ALGEBRAS

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) ⊗ ∧sl(n,ℂ)*,sl(n,ℂ) acting by inner derivations on M n (ℂ). A canonical symplectic structure is exhibited for M n (ℂ) for which the Poisson bracket is, to within a factor i, the commutator. Also, a canonical Riemannian structure is described for M n (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. © 1990 American Institute of Physics.