Cyclic structures of Cliffordian supergroups and particle representations of Spin_+(1,3)

Supergroups are defined in the framework of ${\text{dZ}}_2$-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 ${\text{spin}}_{+}(1,3)$) of the proper orthochronous Lorentz group are constructed via the Clifford algebra formalism. Complex and real supergroups are defined on the representation system of ${\text{spin}}_{+}(1,3)$. 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 ${\text{spin}}_{+}(1,3)$.