Higher algebraic geometrization emerging from noncommutativity. (Q2754470)

The author presents a quick review of the geometrization process occurred in physics during the past century. After describing in a chronological order some fundamental steps in this process, the internal supersymmetry is discussed in a more detailed way. Superalgebras, supermatrices and related superobjects play a relevant rôle and are reviewed in Section 3 (see [\textit{L. Corwin, Y. Ne'eman} and \textit{S. Sternberg}, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry). Rev. Mod. Phys. 47, 573--603 (1975; Zbl 0557.17004)], for a review). The theory of superconnections as introduced by \textit{D. Quillen} [Topology 24, 89--95 (1985; Zbl 0569.58030)] is shortly explained in Sec. 4. The use of alternative \({\mathbb Z}_2\)-gradings and the electroweak geometrization are analysed from the perspective of Noncommutative Geometry (see e.g. the recent book by \textit{J. M. Gracia-Bondía, J. C. Várilly} and \textit{H. Figueroa}, [Elements of noncommutative geometry. (Birkhäuser Advanced Texts. Boston, MA: Birkhäuser) (2001; Zbl 0958.46039)]. The higher algebraic geometrization developed by the author in a recent paper [Phys. Lett. B 427, 19--25 (1988; Zbl 1355.81134)] is also discussedNEWLINENEWLINEFor the entire collection see [Zbl 0967.00114].