RESEARCH PAPER Graded Lie Algebroids: A Framework for Geometrization of Matter and Forces Unification Ghodratallah Fasihi Ramandi 1 • Naser Boroojerdian 1 Received: 16 November 2015 / Accepted: 1 March 2016 Ó Shiraz University 2018 Abstract In this paper, we introduce a geometric structure that is capable of describing matter and forces simultaneously. This structure can be established by using the notion of Z 2 -graded Lie algebroid structures and graded semi-Riemannian metrics on them. Using calculus of variations, we derive field equations from the extended Hilbert–Einstein action. The derived equations contain Yang–Mills and Einstein field equations simultaneously. The even part of the graded Lie algebroid describes forces and its odd part is related to matter and particles. Keywords Z 2 -graded Lie algebroid  Graded metric  Hilbert–Einstein action  Unified field equation 1 Introduction Attempts at the unification of gravitation and electro- magnetism have been made ever since the advent of gen- eral relativity. Most of these attempts share the idea that Einstein’s original theory must in some way be generalized that some part of geometry describes electromagnetism. These theories only discus on forces and, particles will be added to the theory in some way that is not directly related to the geometric structure of that theory. In our world, particles and forces exist and are closely related to each other and we must not separate them. Each fundamental force in nature is related to a group of symmetries, and this group is locally determined by its Lie algebra. So, in general a Lie algebra can be interpreted as a force in nature. Particles are modeled by representations of the Lie algebras. In Z 2 -graded Lie algebras, the odd part is a representation of the even part, therefore a Z 2 -graded Lie algebra can show a force and its related particles simulta- neously. The basic idea of this paper is just to use notion of Z 2 -graded transitive Lie algebroids over the graded tangent bundle, which is defined in (Boroojerdian 2013). Fortu- nately suitable semi-Riemannian metrics on Z 2 -graded transitive Lie algebroid is a good apparatus to describe forces and matter in a same manner. 2 Preliminaries In this section, we summarize definitions and essential facts about Lie algebroids that are needed in this paper. For details on most of the ideas touched on here, one can consult (Mackenzie 2005) and (Boroojerdian 2013). Definition 2.1 A vector bundle ðE; p; MÞ is said Z 2 -graded if fibers of E are Z 2 -graded vector spaces that are com- patible to the vector bundle structure. A Z 2 -graded vector bundle ðE; p; MÞ has a decomposi- tion E ¼ E 0  E 1 ; where E 0 and E 1 are vector sub-bundles of E In a Z 2 -graded vector bundle ðE; p; MÞ the sections of ^ T that take their values in even(resp. odd) part are called even(resp. odd) sections. Also, elements of E 0 (resp. E 1 ) are called even (resp. odd) vectors. Even and odd elements of a Z 2 -graded vector bundle are called homogeneous and their parity are defined as follows: a jj¼ 0 a is even 1 a is odd  & Naser Boroojerdian broojerd@aut.ac.ir Ghodratallah Fasihi Ramandi gh_fasihi@aut.ac.ir 1 Department of Pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran 123 Iran J Sci Technol Trans Sci https://doi.org/10.1007/s40995-018-0516-x