Year: 2024
Type: article
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 theVolichenko algebrasdefined in 1990 by Leites–Serganova asmetasymmetrieswhich do not respect even/odd gradings and lead to mixed brackets interpolating ordinary commutators and anticommutators. In a previous paper braided Z2 -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 ats = 2) and bosons (recovered in the s→∞ limit); it implements a parastatistics where at mosts − 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 Uq(osp(1|2)) ; - The reconstruction, via suitable intertwining operators, of the braided tensor products as ordinary tensor products (in a minimal representation, theN-particle sector of the braided Majorana qubits is described by 2N×2N 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.
Author: Francesco Toppan
Institution: Centro Brasileiro de Pesquisas Físicas
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Awards: G2433182178, G7327006035
OpenAlex ID: W4403054859