Graded calculus is based on the Grassmann"s external algebra (ring), includes, in particularly, the graded tensor analysis over this ring, and unnecessarily uses a supersymmetry or supergravitation groups always acting in spaces of function values. To solve the problems of union of both the elementary particles and the fundamental interactions (B-model developed by the author), it is important that the Grassmannian grading is taken into account just of both the spaces of function values and the space of argument values. The latter serves as enlarged graded 9-dimensional space-time in bundle structures. To be exact, it includes one scalar dimension, four Minkowski"s vector dimensions and four spinor ones — all with respect to Lorentz group. However so far there was no natural criterion for the choice of graded (weight) signs in various sums of linear algebra over the Grassmann's ring. In the work reviewed, the natural, or canonical, principle of covariant sum construction with the weight signs is found. The canonical rules of weight signs never can be broken. In accordance with the canonical rules, almost all the basic notions of graded linear algebra are modified. The variety of important examples are given besides the theoretical material. New graded groups are found being incomplete affine extensions of the Lorentz group over the exterior Grassmann's algebra (ring). The extensions are specified as actions in the 9-dimensional base physical space playing the important role in the B-model union of elementary particles and fundamental interactions. The group properties of the extensions are consistent with the new rules in graded linear algebra. The proper explanation is given for the existence of three elementary particle generations (families). All the fermionic interactions (for leptons and quarks) — gravitational, electro-weak, and strong — are found to be caused by the covariant differentiation of the fermionic bundle sections and Riemannization procedure over the principal bundle with the structure group G=GL(B) x L x U(1) x SU(2) x SU(3).

• V. T. Berezin. Mathematical and Physical Principles of the B-model Union of Elementary Particles and Fundamental Interactions: Free Fields // Ukrainian J. Phys., 44(10), 1185–1197.

• Galkin Donetsk Institute for Physics and Engineeringne
• Donetsk Physical-Technical Institute of Ukraine Academy of Sciences