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 , … , an interpolation between ordinary Majorana fermions (recovered at s = 2) and bosons (recovered in the s → ∞ 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 a (superselected) set of reps of the quantum superalgebra U q ( osp ( 1 | 2 ) ) ; - The reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products (in a minimal representation, the N -particle sector of the braided Majorana qubits is described by 2 N × 2 N matrices); - The introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg–Lie algebras; - The s → ∞ untruncated limit of the mixed-bracket Heisenberg–Lie algebras producing parafermionic oscillators; - ( meta )symmetries of ordinary differential equations given by matrix Schrödinger 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.