Volichenko-type metasymmetry of braided Majorana qubits

TL;DR

This paper explores the mathematical structures of braided Majorana qubits, introducing metasymmetries and mixed brackets that interpolate between fermionic and bosonic statistics, with implications for quantum algebra and differential equations.

Contribution

It introduces a framework connecting braided Majorana qubits with metasymmetries, mixed brackets, and quantum group interpretations, extending previous models to include interpolations and ternary algebra structures.

Findings

Revealed quantum group structures at roots of unity

Constructed generalized Heisenberg-Lie algebras with mixed brackets

Identified metasymmetries in differential equations

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.