Volichenko-type metasymmetry of braided Majorana qubits

Abstract

This paper presents different mathematical structures connected with the parastatistics of braided Majorana qubits and clarifies their role; in particular, mixed-bracket Heisenberg-Lie algebras are introduced. These algebras belong to a more general framework than the Volichenko algebras defined in 1990 by Leites-Serganova as metasymmetries which do not respect even/odd gradings and lead to mixed brackets interpolating ordinary commutators and anticommutators. In a previous paper braided $Z_2$-graded Majorana qubits were first-quantized within a graded Hopf algebra framework endowed with a braided tensor product. The resulting system admits truncations at roots of unity and realizes, for a given integer $s=2,3,4,\ldots$, an interpolation between ordinary Majorana fermions (recovered at $s=2$) and bosons (recovered in the $s\rightarrow \infty$ limit); it implements a parastatistics where at most $s-1$ indistinguishable particles are accommodated in a multi-particle sector. The structures discussed in this work are: - the quantum group interpretation of the roots of unity truncations recovered from reps of the quantum superalgebra ${\cal U}_q({{osp}}(1|2))$; - the reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products; - the introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg-Lie algebras; - the $s\rightarrow \infty$ untruncated limit of the mixed-bracket Heisenberg-Lie algebras producing parafermionic oscillators; - (meta)symmetries of ordinary differential equations given by matrix Schr\"{o}dinger equations in $0+1$ dimension induced by the braided creation/annihilation operators; - in the special case of a third root of unity truncation, a nonminimal realization of the intertwining operators defines the system as a ternary algebra.