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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>s</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:mrow></mml:math> , an interpolation between ordinary Majorana fermions (recovered at s = 2) and bosons (recovered in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>s</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi class="MJX-tex-calligraphic">U</mml:mi></mml:mrow><mml:mi>q</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi mathvariant="fraktur">osp</mml:mi></mml:mrow></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> ; - 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mi>N</mml:mi></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math> matrices); - The introduction of mixed brackets for the braided creation/annihilation operators which define generalized Heisenberg–Lie algebras; - The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>s</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:math> untruncated limit of the mixed-bracket Heisenberg–Lie algebras producing parafermionic oscillators; - ( meta )symmetries of ordinary differential equations given by matrix Schrödinger equations in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mn>0</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> 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.