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We investigate the observational consequences of the light-like deformations of the Poincaré algebra induced by the jordanian and the extended jordanian classes of Drinfel’d twists. Twist-deformed generators belonging to a Universal Enveloping Algebra close nonlinear algebras. In some cases the nonlinear algebra is responsible for the existence of bounded domains of the deformed generators. The Hopf algebra coproduct implies associative nonlinear additivity of the multi-particle states. A subalgebra of twist-deformed observables is recovered whenever the twist-deformed generators are either hermitian or pseudo-hermitian with respect to a common invertible hermitian operator.

I report the recent advances in applying (graded) Hopf algebras with braided tensor product in two scenarios: i ) paraparticles beyond bosons and fermions living in any space dimensions and transforming under the permutation group; ii ) physical models of anyons living in two space-dimensions and transforming under the braid group. In the first scenario simple toy models based on the so-called 2-bit parastatistics show that, in the multiparticle sector, certain observables can discriminate paraparticles from ordinary bosons/fermions (thus, providing a counterexample to the widespread belief of the “conventionality of parastatistics” argument). In the second scenario the notion of (braided) Majorana qubit is introduced as the simplest building block to implement the Kitaev’s proposal of a topological quantum computer which protects from decoherence.

In a recent paper (Balbino-de Freitas-Rana-FT, arXiv:2309.00965) we proved that the supercharges of the supersymmetric quantum mechanics can be statistically transmuted and accommodated into a Z 2 n -graded parastatistics. In this talk I derive the 6 = 1 + 2 + 3 transmuted spectrum-generating algebras (whose respective Z 2 n gradings are n = 0, 1, 2) of the = 2 Superconformal Quantum Mechanics. These spectrum-generating algebras allow to compute, in the corresponding multiparticle sectors of the de Alfaro-Fubini-Furlan deformed oscillator, the degeneracies of each energy level. The levels induced by the Z 2 × Z 2 -graded paraparticles cannot be reproduced by the ordinary bosons/fermions statistics. This implies the theoretical detectability of the Z 2 × Z 2 -graded parastatistics.

The dynamical symmetries of 1+1-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial de Alfaro-Fubini-Furlan, DFF, term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the sl(2)⊕u(1) invariance of the Calogero Conformal Mechanics, an osp2∣2 invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the universal enveloping algebra of the deformed Heisenberg algebra, which is shown to be equivalent to a deformed version of the Schrödinger algebra. This vector space also gives rise to a higher-spin (gravity) superalgebra. We furthermore prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical symmetries, being related by a similarity transformation and a redefinition of the time variable.

The “10-fold way” refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, Z 2 -graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in Z 2 × Z 2 -graded physics (classical and quantum invariant models, parastatistics) we classify the associative Z 2 × Z 2 -graded superdivision algebras and show that 13 inequivalent cases have to be added to the 10-fold way. Our scheme is based on the “alphabetic presentation of Clifford algebras”, here extended to graded superdivision algebras. The generators are expressed as equal-length words in a 4-letter alphabet (the letters encode a basis of invertible 2 × 2 real matrices and in each word the symbol of tensor product is skipped). The 13 inequivalent Z 2 × Z 2 -graded superdivision algebras are split into real series (4 subcases with 4 generators each), complex series (5 subcases with 8 generators) and quaternionic series (4 subcases with 16 generators). As an application, the connection of Z 2 × Z 2 -graded superdivision algebras with a parafermionic Hamiltonian possessing time-reversal and particle-hole symmetries is presented.

Some key features of the symmetries of the Schrodinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra superalgebra duality involving first and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation- dependent notion of on-shell symmetry is introduced. The difference in associating the time-derivative symmetry operator with either a root or a Cartan generator of the sl(2) subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric.

The non-commutativity induced by a Drinfel’d twist produces Bopp-shift-like transformations for deformed operators. In a single-particle setting the Drinfel’d twist allows to recover the non-commutativity obtained from various methods which are not based on Hopf algebras. In multi-particle sector, on the other hand, the Drinfel’d twist implies novel features. In conventional approaches to non-commutativity, deformed primitive operators are postulated to act additively. A Drinfel’d twist implies non-additive effects which are controlled by the coproduct. We stress that in our framework, the central element denoted as ħ is associated to an additive operator whose physical interpretation is that of the Particle Number operator. We illustrate all these features for a class of (abelian twist-deformed) 2D Hamiltonians. Suitable choices of the parameters lead to the Hamiltonian of the non-commutative Quantum Hall Effect, the harmonic oscillator, the quantization of the configuration space. The non-additive effects in the multi-particle sector, leading to results departing from the existing literature, are pointed out.

The first-order Lévy-Leblond differential equations (LLEs) are the non-relativistic analogous of the Dirac equation: they are the “square roots” of the Schrödinger equation in (1 + d ) dimensions and admit spinor solutions. In this paper we show how to extend to the Lévy-Leblond spinors the real/complex/quaternionic classification of the relativistic spinors (which leads to the notions of Dirac, Weyl, Majorana, Majorana-Weyl, Quaternionic spinors). Besides the free equations, we also consider the presence of potential terms. Applied to a conformal potential, the simplest (1 + 1)-dimensional LLE induces a new differential realization of the osp (1 | 2) superalgebra in terms of first-order differential operators depending on the time and space coordinates.

In this talk I discuss a recently developed \"Unfolded Quantization Framework\". It allows to introduce a Hamiltonian Second Quantization based on a Hopf algebra endowed with a coproduct satisfying, for the Hamiltonian, the physical requirement of being a primitive element. The scheme can be applied to theories deformed via a Drinfel'd twist. I discuss in particular two cases: the abelian twist deformation of a rotationally invariant nonrelativistic Quantum Mechanics (the twist induces a standard noncommutativity) and the Jordanian twist of the harmonic oscillator. In the latter case the twist induces a Snyder non-commutativity for the space-coordinates, with a pseudo-Hermitian deformed Hamiltonian. The \"Unfolded Quantization Framework\" unambiguously fixes the non-additive effective interactions in the multi-particle sector of the deformed quantum theory. The statistics of the particles is preserved even in the presence of a deformation.