## Noncommutative differential geometry of matrix algebras.(English)Zbl 0704.53081

This paper is concerned with a noncommutative generalization of the exterior calculus in the following sense. Let Der(A) denote the Lie algebra of derivations of an associative algebra A with unit. Then the complex C(Der(A),A) of A-valued cochains of Der(A) is a graded differential algebra, and the smallest differential subalgebra $$\Omega_ D(A)$$ of this complex which contains A is a natural generalization of the algebra of differential forms, the role of the Lie algebra of vector fields being played by Der(A).
In the paper under review the algebra A is identified with the algebra $$M_ n({\mathbb{C}})$$ of complex $$n\times n$$ matrices (n$$\geq 2)$$. This gives rise to the so-called differential geometry of $$M_ n({\mathbb{C}})$$ in which $$\Omega_ D(M_ n({\mathbb{C}})$$ assumes the role of the algebra of differential forms. The existence of an invariant canonical symplectic 2- form is demonstrated for which the corresponding Poisson bracket is given by $$\{A,B\}=i[A,B]$$. An invariant canonical Riemannian structure for $$M_ n({\mathbb{C}})$$ is introduced and the resulting Hodge theory on $$\Omega_ D(M_ n({\mathbb{C}}))$$ is developed. A Hermitian connection on the free Hermitian $$M_ n({\mathbb{C}})$$ module of rank one is regarded as an analog of an electromagnetic potential. It is found that there is a unique potential with vanishing curvature that is not a pure gauge. This potential is gauge invariant and is related to the symplectic form.
Reviewer: H.Rund

### MSC:

 53C80 Applications of global differential geometry to the sciences 15A75 Exterior algebra, Grassmann algebras 81T13 Yang-Mills and other gauge theories in quantum field theory
Full Text:

### References:

 [1] Dubois-Violette M., C. R. Acad. Sci. Paris 307 pp 403– (1988) [2] Connes A., Publi. I.H.E.S. 62 pp 257– (1986) [3] Karoubi M., C. R. Acad. Sci. Paris 297 pp 381– (1983) [4] DOI: 10.1090/S0002-9947-1948-0024908-8 [5] Connes A., C. R. Acad. Sci. Paris 290 pp 599– (1980) [6] DOI: 10.1063/1.528917 · Zbl 0704.53082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.