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Cyclic Structures of Cliffordian Supergroups and Particle Representations of Spin+(1, 3)

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Abstract

Supergroups are defined in the framework of \({\mathbb{Z}_2}\) 2-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups related with the modulo 2 and modulo 8 periodicities of the complex and real Clifford algebras. Particle (fermionic and bosonic) representations of a universal covering (spinor group Spin +(1, 3)) of the proper orthochronous Lorentz group are constructed via the Clifford algebra formalism. Complex and real supergroups are defined on the representation system of Spin +(1, 3). It is shown that a cyclic (modulo 2) structure of the complex supergroup is equivalent to a supersymmetric action, that is, it converts fermionic representations into bosonic representations and vice versa. The cyclic action of the real supergroup leads to a much more high-graded symmetry related with the modulo 8 periodicity of the real Clifford algebras. This symmetry acts on the system of real representations of Spin +(1, 3).

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  1. Department of Mathematics, Siberian State Industrial University, Kirova 42, Novokuznetsk, 654007, Russia

    V. V. Varlamov

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Varlamov, V.V. Cyclic Structures of Cliffordian Supergroups and Particle Representations of Spin+(1, 3). Adv. Appl. Clifford Algebras 24, 849–874 (2014). https://doi.org/10.1007/s00006-014-0446-4

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