Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras

• URL: http://arxiv.org/abs/2112.00840v3
• Date: Mon, 13 Feb 2023 14:03:22 GMT
• Title: Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• Authors: Zhanna Kuznetsova and Francesco Toppan
• Abstract summary: "$10-fold way" refers to the combined classification of the $3$ associative division algebras (of real, complex and quaternionic numbers) and of the $7$, $mathbb Z$-letter-graded, superdivision algebras. 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 $mathbb Z$timesmathbb Z$-graded physics, we classify the associative $mathbb Z$times
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The "$10$-fold way" refers to the combined classification of the $3$ associative division algebras (of real, complex and quaternionic numbers) and of the $7$, ${\mathbb Z}_2$-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 ${\mathbb Z}_2\times{\mathbb Z}_2$-graded physics (classical and quantum invariant models, parastatistics) we classify the associative ${\mathbb Z}_2\times {\mathbb Z}_2$-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 $2\times 2$ real matrices and in each word the symbol of tensor product is skipped). The $13$ inequivalent ${\mathbb Z}_2\times {\mathbb Z}_2$-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).

Related papers

• Two-body Coulomb problem and hidden $g^{(2)}$ algebra: superintegrability and cubic polynomial algebra [55.2480439325792]
  It is shown that the two-body Coulomb problem in the Sturm representation leads to a new two-dimensional, exactly-solvable, superintegrable quantum system in curved space. The two integrals are integral of orders two and four, they are made from two components of the angular momentum and from the modified LaplaceRunge-Lenz vector.
  arXiv  Detail & Related papers  (2023-09-28T22:47:18Z)
• Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom) [77.34726150561087]
  It is shown that if the symmetry of the three-dimensional system is $(r, rho, varphi)$, the variables $(r, rho, varphi)$ allow a separation of variable $varphi$ and eigenfunctions. Thoses occur in the study of the Zeeman effect on Hydrogen atom.
  arXiv  Detail & Related papers  (2022-12-02T20:11:17Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Monogamy of entanglement between cones [68.8204255655161]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones. Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• First quantization of braided Majorana fermions [0.0]
  A $mathbb Z$-graded qubit represents an even (bosonic) "vacuum state" Multiparticle sectors of $N$, braided, indistinguishable Majorana fermions are constructed via first quantization.
  arXiv  Detail & Related papers  (2022-03-03T15:43:38Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Inequivalent quantizations from gradings and ${\mathbb Z}_2\times {\mathbb Z}_2$ parabosons [0.0]
  It accommodates four kinds of particles: ordinary bosons and three types of parabosons which mutually anticommute when belonging to different type. It is shown how to detect $mathbb Ztimes mathbb Z$-graded parabosons in the multi-particle sector of a quantum model.
  arXiv  Detail & Related papers  (2021-04-19T23:56:33Z)
• Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes [0.5156484100374059]
  We present an explicit matrix algebra quantization of the algebra of volume-preserving diffeomorphisms of the $n$-torus. We approximate the corresponding classical Nambu brackets using $mathfraksl(Nlceilfracn2rceil,mathbbC)$-matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product.
  arXiv  Detail & Related papers  (2021-03-11T13:44:57Z)
• $C^*$-fermi systems and detailed balance [0.0]
  A systematic theory of product and diagonal states is developed for tensor products of $mathbb Z$-graded $*$-algebras. Twisted duals of positive linear maps between von Neumann algebras are then studied, and applied to solve a positivity problem on the infinite Fermi lattice.
  arXiv  Detail & Related papers  (2020-10-14T15:16:32Z)
• Polynomial algebras from $su(3)$ and the generic model on the two sphere [0.0]
  Construction of superintegrable systems based on Lie algebras have been introduced over the years. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. We develop a new approach reexamining the case of the generic superintegrable systems on the 2-sphere.
  arXiv  Detail & Related papers  (2020-07-22T02:20:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.