Noncommutative differential geometry of matrix algebras.

*(English)*Zbl 0704.53081This 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.

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 |

##### Keywords:

noncommutative differential geometry; gauge potential; Hermitian connection; graded differential algebra; symplectic 2-form; Poisson bracket
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\textit{M. Dubois-Violette} et al., J. Math. Phys. 31, No. 2, 316--322 (1990; Zbl 0704.53081)

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##### 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 · 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 · doi:10.1063/1.528917 |

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