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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Freed, Daniel S. "Higher algebraic structures and quantization." Communications in Mathematical Physics 159, no. 2 (January 1994): 343–98. http://dx.doi.org/10.1007/bf02102643.

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2

Gangopadhyay, Sunandan. "Lie algebraic noncommuting structures from reparametrization symmetry." Journal of Mathematical Physics 48, no. 5 (May 2007): 052302. http://dx.doi.org/10.1063/1.2723551.

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3

Duplij, Steven. "Arity Shape of Polyadic Algebraic Structures." Zurnal matematiceskoj fiziki, analiza, geometrii 15, no. 1 (March 25, 2019): 3–56. http://dx.doi.org/10.15407/mag15.01.003.

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4

Kulish, P. P., and E. K. Sklyanin. "Algebraic structures related to reflection equations." Journal of Physics A: Mathematical and General 25, no. 22 (November 21, 1992): 5963–75. http://dx.doi.org/10.1088/0305-4470/25/22/022.

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5

Kramer, P. "Algebraic structures for 1D quasiperiodic systems." Journal of Physics A: Mathematical and General 26, no. 2 (January 21, 1993): 213–28. http://dx.doi.org/10.1088/0305-4470/26/2/010.

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6

Hammad, M. M., H. E. Shetawy, Ayman A. Aly, and S. B. Doma. "Nuclear supersymmetry and dual algebraic structures." Physica Scripta 94, no. 10 (August 6, 2019): 105207. http://dx.doi.org/10.1088/1402-4896/ab2442.

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7

Rajchel-Mieldzioć, Grzegorz, Kamil Korzekwa, Zbigniew Puchała, and Karol Życzkowski. "Algebraic and geometric structures inside the Birkhoff polytope." Journal of Mathematical Physics 63, no. 1 (January 1, 2022): 012202. http://dx.doi.org/10.1063/5.0046581.

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8

Cardoso, Pedro G. S., Ernesto P. Borges, Thierry C. P. Lobão, and Suani T. R. Pinho. "Nondistributive algebraic structures derived from nonextensive statistical mechanics." Journal of Mathematical Physics 49, no. 9 (September 2008): 093509. http://dx.doi.org/10.1063/1.2982233.

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9

Harrelson, Eric, Alexander A. Voronov, and J. Javier Zúñiga. "Open-Closed Moduli Spaces and Related Algebraic Structures." Letters in Mathematical Physics 94, no. 1 (September 1, 2010): 1–26. http://dx.doi.org/10.1007/s11005-010-0418-0.

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10

KERNER, RICHARD. "TERNARY AND NON-ASSOCIATIVE STRUCTURES." International Journal of Geometric Methods in Modern Physics 05, no. 08 (December 2008): 1265–94. http://dx.doi.org/10.1142/s0219887808003326.

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Анотація:

We discuss ternary algebraic structures appearing in various domains of theoretical and mathematical physics. Some of them are associative, and some are not. Their interesting and curious properties can be exploited in future applications to enlarged and generalized field theoretical models in the years to come. Many ideas presented here have been developed and clarified in countless discussions with Michel Dubois-Violette.

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11

Figueroa-O’Farrill, José. "Lie algebraic Carroll/Galilei duality." Journal of Mathematical Physics 64, no. 1 (January 1, 2023): 013503. http://dx.doi.org/10.1063/5.0132661.

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We characterize Lie groups with bi-invariant bargmannian, galilean, or carrollian structures. Localizing at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian, or galilean structures are actually determined by the same data: a metric Lie algebra with a skew-symmetric derivation. This is the same data defining a one-dimensional double extension of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with such double extensions, containing carrollian Lie algebras as an ideal and projecting to galilean Lie algebras. This sets up a canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras. This reformulation allows us to use the structure theory of metric Lie algebras to give a list of bargmannian, carrollian, and galilean Lie algebras in the positive-semidefinite case. We also characterize Lie groups admitting a bi-invariant (ambient) leibnizian structure. Leibnizian Lie algebras extend the class of bargmannian Lie algebras and also set up a non-canonical correspondence between carrollian and galilean Lie algebras.

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12

Cioroianu, Eugen-Mihăiţă, and Cornelia Vizman. "A linear algebraic setting for Jacobi structures." Journal of Geometry and Physics 159 (January 2021): 103904. http://dx.doi.org/10.1016/j.geomphys.2020.103904.

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13

Santini, P. M. "The algebraic structures underlying integrability." Inverse Problems 6, no. 1 (February 1, 1990): 99–114. http://dx.doi.org/10.1088/0266-5611/6/1/010.

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14

Santini, P. M. "The algebraic structures underlying integrability." Inverse Problems 6, no. 3 (June 1, 1990): 479. http://dx.doi.org/10.1088/0266-5611/6/3/516.

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15

Kaptsov, O. V. "Algebraic and geometric structures of analytic partial differential equations." Theoretical and Mathematical Physics 189, no. 2 (November 2016): 1592–608. http://dx.doi.org/10.1134/s0040577916110052.

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16

Zeitlin, Anton M. "String field theory-inspired algebraic structures in gauge theories." Journal of Mathematical Physics 50, no. 6 (June 2009): 063501. http://dx.doi.org/10.1063/1.3142964.

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17

Avan, Jean, and Antal Jevicki. "Algebraic structures and eigenstates for integrable collective field theories." Communications in Mathematical Physics 150, no. 1 (November 1992): 149–66. http://dx.doi.org/10.1007/bf02096570.

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18

Sen, Ashoke, and Barton Zwiebach. "Background independent algebraic structures in closed string field theory." Communications in Mathematical Physics 177, no. 2 (April 1996): 305–26. http://dx.doi.org/10.1007/bf02101895.

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19

Chen, Deng‐yuan, and Da‐jun Zhang. "Lie algebraic structures of (1+1)‐dimensional Lax integrable systems." Journal of Mathematical Physics 37, no. 11 (November 1996): 5524–38. http://dx.doi.org/10.1063/1.531742.

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20

Cariñena, José F., Janusz Grabowski, and Giuseppe Marmo. "Contractions: Nijenhuis and Saletan tensors for general algebraic structures." Journal of Physics A: Mathematical and General 34, no. 18 (April 27, 2001): 3769–89. http://dx.doi.org/10.1088/0305-4470/34/18/306.

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21

BAI, CHENGMING. "A FURTHER STUDY ON NON-ABELIAN PHASE SPACES: LEFT-SYMMETRIC ALGEBRAIC APPROACH AND RELATED GEOMETRY." Reviews in Mathematical Physics 18, no. 05 (June 2006): 545–64. http://dx.doi.org/10.1142/s0129055x06002711.

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Анотація:

The notion of non-abelian phase space of a Lie algebra was first formulated and then discussed by Kuperschmidt. In this paper, we further study the non-abelian phase spaces in terms of left-symmetric algebras. We interpret the natural appearance of left-symmetric algebras from the intrinsic algebraic properties and the close relations with the classical Yang–Baxter equation. Furthermore, using the theory of left-symmetric algebras, we study some interesting geometric structures related to phase spaces. Moreover, we also discuss the generalized phase spaces with certain non-trivial algebraic structures on the dual spaces.

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22

Sokolowski, Andrzej. "Constructing Wave Functions Using Mathematical Reasoning." Physics Teacher 61, no. 2 (February 2023): 124–27. http://dx.doi.org/10.1119/5.0062821.

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Анотація:

Transfer of structural math knowledge to physics is difficult for students. While research suggests various improvement techniques, enhancing parallelism of algebraic structures used in physics to those studied in mathematics courses seems underrepresented. This paper proposes an alternative way of introducing wave functions as a set of parametric equations. Research shows that the mathematical underpinnings of the mechanical wave function are problematic for students. As seen from the math perspective, there can be several drawbacks that likely impede scientific analysis and interpretation of the wave function: [Formula: see text]

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23

Arik, Metin, Natig M. Atakishiyev, and Kurt Bernardo Wolf. "Quantum algebraic structures compatible with the harmonic oscillator Newton equation." Journal of Physics A: Mathematical and General 32, no. 33 (August 11, 1999): L371—L376. http://dx.doi.org/10.1088/0305-4470/32/33/101.

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24

Jing-Fa, Lu, and Ge Mo-Lin. "On algebraic structures in the non-linear principal chiral model." Journal of Physics A: Mathematical and General 21, no. 8 (April 21, 1988): L435—L438. http://dx.doi.org/10.1088/0305-4470/21/8/001.

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25

Bychkov, S. A., and D. V. Yur'ev. "Three algebraic structures of quantum projective [sl(2, C)-invariant] field theory." Theoretical and Mathematical Physics 97, no. 3 (December 1993): 1333–39. http://dx.doi.org/10.1007/bf01015762.

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26

LÉANDRE, RÉMI. "STOCHASTIC PROCESSES ON CLASSIFYING SPACES AND STRING STRUCTURES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 07, no. 03 (September 2004): 361–81. http://dx.doi.org/10.1142/s0219025704001700.

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27

Caprio, M. A., J. H. Skrabacz, and F. Iachello. "Dual algebraic structures for the two-level pairing model." Journal of Physics A: Mathematical and Theoretical 44, no. 7 (January 28, 2011): 075303. http://dx.doi.org/10.1088/1751-8113/44/7/075303.

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28

Ma, Wen-Xiu. "The algebraic structures of isospectral Lax operators and applications to integrable equations." Journal of Physics A: Mathematical and General 25, no. 20 (October 21, 1992): 5329–43. http://dx.doi.org/10.1088/0305-4470/25/20/014.

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29

Bonezzi, Roberto, and Olaf Hohm. "Duality Hierarchies and Differential Graded Lie Algebras." Communications in Mathematical Physics 382, no. 1 (February 2021): 277–315. http://dx.doi.org/10.1007/s00220-021-03973-8.

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Анотація:

AbstractThe gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.

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30

Lannes, A. "Phase and amplitude calibration in aperture synthesis. Algebraic structures." Inverse Problems 7, no. 2 (April 1, 1991): 261–98. http://dx.doi.org/10.1088/0266-5611/7/2/009.

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31

Gutman, I., Y. N. Yeh, S. L. Lee, H. Hosoya, and S. J. Cyvin. "Calculating the Determinant of the Adjacency Matrix and Counting Kekulé Structures in Circulenes." Zeitschrift für Naturforschung A 49, no. 11 (November 1, 1994): 1053–58. http://dx.doi.org/10.1515/zna-1994-1110.

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32

Urazmukhamedova, Z., D. Juraev, and M. Mirsaidov. "Assessment of stress state and dynamic characteristics of plane and spatial structure." Journal of Physics: Conference Series 2070, no. 1 (November 1, 2021): 012156. http://dx.doi.org/10.1088/1742-6596/2070/1/012156.

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Анотація:

Abstract This study is devoted to the assessment of the stress state and dynamic characteristics of various structures. The actual task at the design stage is to determine the parameters of a structure. In this article, a mathematical model was developed for assessing the stress state and dynamic characteristics of plane and spatial structures based on the Lagrange variational equation using the d’Alembert principle. The variational problem for the structures considered by the finite element method leads to the solution of nonhomogeneous algebraic equations or to the solution of algebraic eigenvalue problems. To assess the adequacy of the model and the accuracy of the numerical results obtained, a plane and spatial test problem with an exact solution was solved. Using the proposed model, the eigenfrequencies and modes of oscillations of the gravitational and earth dams (296 m high) of the Nurek reservoir were investigated. At that, it was revealed that in the natural modes of vibration of earth dams, the greatest displacements under low frequencies are observed at the crest part or at the middle of the slopes.

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33

Balinsky, A. A., and A. I. Balinsky. "On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type." Journal of Physics A: Mathematical and General 26, no. 7 (April 7, 1993): L361—L364. http://dx.doi.org/10.1088/0305-4470/26/7/002.

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34

Kaushal Rana. "Homological Algebra and Its Application: A Descriptive Study." Integrated Journal for Research in Arts and Humanities 2, no. 1 (January 31, 2022): 29–35. http://dx.doi.org/10.55544/ijrah.2.1.47.

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Анотація:

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

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35

Ghikas, Demetris P. K. "From Complexity to Information Geometry and Beyond." Nonlinear Phenomena in Complex Systems 23, no. 2 (July 9, 2020): 212–20. http://dx.doi.org/10.33581/1561-4085-2020-23-2-212-220.

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Анотація:

Complex Systems are ubiquitous in nature and man-made systems. In natural sciences, in social and economic models and in mathematical constructions are studied and analyzed, are applied in practical problems but without a clear and universal definition of "complexity", let alone classification and quantification. Following the "three-level scheme" of physical theories, observations/experiments, phenomenology, microscopic interactions, we need, starting from the experience of observation to establish appropriate phenomenological parameters and concepts, and in conjunction with a possible knowledge of the nature of microscopic structures to deepen our understanding of a particular system which we "understand as complex". Information Geometry seems to be a useful phenomenological framework, which using generalized entropies, provides some classification and quantification tools. But we need the next level, microscopic structure and interactions of the parts of complex systems. A useful direction is the conceptual niche of hyper-networks and super graphs, where a strong involvement of algebra offers concrete techniques. We believe that appropriate algebraic structures may systematize our approach to microscopic structures of complex systems, and help associate the information geometric phenomenology with concrete properties. In this paper after a short discussion of the problem of "definition of complexity", we introduce our information geometric quantities derived from generalized entropies. Then we present our results of application of information geometry for classification of complex systems. Finally we present our ideas for an abstract algebraic approach which may offer a framework for the microscopic study of complex systems.

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36

Prykarpatsky, Yarema A., Orest D. Artemovych, Maxim V. Pavlov, and Anatolij K. Prykarpatski. "The Differential-Algebraic Analysis of Symplectic and Lax Structures Related with New Riemann-Type Hydrodynamic Systems." Reports on Mathematical Physics 71, no. 3 (June 2013): 305–51. http://dx.doi.org/10.1016/s0034-4877(13)60035-x.

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37

Majewski, Wladyslaw Adam. "On Quantum Statistical Mechanics: A Study Guide." Advances in Mathematical Physics 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/9343717.

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Анотація:

We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed.

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38

Gutman, I., A. V. Teodorović, and N. Kolaković. "Algebraic Studies of Kekulé Structures. A Semilattice Based on the Sextet Rotation Concept." Zeitschrift für Naturforschung A 44, no. 11 (November 1, 1989): 1097–101. http://dx.doi.org/10.1515/zna-1989-1109.

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39

Goktas, Sertac. "A New Type of Sturm-Liouville Equation in the Non-Newtonian Calculus." Journal of Function Spaces 2021 (October 31, 2021): 1–8. http://dx.doi.org/10.1155/2021/5203939.

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Анотація:

In mathematical physics (such as the one-dimensional time-independent Schrödinger equation), Sturm-Liouville problems occur very frequently. We construct, with a different perspective, a Sturm-Liouville problem in multiplicative calculus by some algebraic structures. Then, some asymptotic estimates for eigenfunctions of the multiplicative Sturm-Liouville problem are obtained by some techniques. Finally, some basic spectral properties of this multiplicative problem are examined in detail.

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40

Chruściński, Dariusz, and Giuseppe Marmo. "Remarks on the GNS Representation and the Geometry of Quantum States." Open Systems & Information Dynamics 16, no. 02n03 (September 2009): 157–77. http://dx.doi.org/10.1142/s1230161209000128.

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Анотація:

It is shown how to introduce a geometric formulation of the algebraic approach to the standard non-relativistic quantum mechanics. It turns out that the GNS representation provides not only symplectic but even Hermitian realization of a 'quantum Poisson algebra'. We discuss alternative Hamiltonian structures emerging out of different GNS representations which provide a natural setting for quantum bi-Hamiltonian systems.

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41

Trawiński, T., A. Kochan, P. Kielan, and D. Kurzyk. "Inversion of selected structures of block matrices of chosen mechatronic systems." Bulletin of the Polish Academy of Sciences Technical Sciences 64, no. 4 (December 1, 2016): 853–63. http://dx.doi.org/10.1515/bpasts-2016-0093.

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AbstractThis paper describes how to calculate the number of algebraic operations necessary to implement block matrix inversion that occurs, among others, in mathematical models of modern positioning systems of mass storage devices. The inversion method of block matrices is presented as well. The presented form of general formulas describing the calculation complexity of inverted form of block matrix were prepared for three different cases of division into internal blocks. The obtained results are compared with a standard Gaussian method and the “inv” method used in Matlab. The proposed method for matrix inversion is much more effective in comparison in standard Matlab matrix inversion “inv” function (almost two times faster) and is much less numerically complex than standard Gauss method.

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42

Cheng, Ji-Wen, Qin-Sheng Zhu, Xiao-Yu Kuang, Shi-Xun Zhang, and Cai-Xia Zhang. "Homotrinuclear Spin Cluster with Orbital Degeneracy in a Magnetic Field: Algebraic Dynamic Studies of the Geometric Phase January 25, 2008." Zeitschrift für Naturforschung A 63, no. 7-8 (August 1, 2008): 405–11. http://dx.doi.org/10.1515/zna-2008-7-804.

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Анотація:

Based on the homotrinuclear spin cluster having SU(2)⊗SU(2) symmetry with twofold orbital degeneracy τ = 1/2) and the SU(2) algebraic structures of both ŝ and τˆ subspaces in the external magnetic field, we calculate exactly the non-adiabatic energy levels and the cyclic and non-cyclic non-adiabatic geometric phase of the homotrinuclear spin cluster by making use of the method of algebraic dynamics. The solution will show that the Berry phase is much influenced by the parameters N =γs/γτ (γs and γτ are the magnetic momentums of ŝ and τ̂ subspaces, respectively) in addition to ω/Ω in a rotating magnetic field. The change of the Berry phase in the basis state of the system is demonstrated from the changing diagram.

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43

Arshad, Muhammad, Aly R. Seadawy, Dianchen Lu, and Farman Ullah Khan. "Optical solitons of the paraxial wave dynamical model in kerr media and its applications in nonlinear optics." International Journal of Modern Physics B 34, no. 09 (April 10, 2020): 2050078. http://dx.doi.org/10.1142/s0217979220500782.

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Анотація:

The solitons and other solutions illustrate nondiffractive and nondispersive spatio-temporal localized packets of wave propagating in the media of optical Kerr. In this paper, solitons, elliptic function and other solutions of dimensionless time-dependent paraxial wave model are constructed via employing three mathematical techniques, namely, the improved simple equation technique, [Formula: see text]-expansion technique and modified extended direct algebraic technique. These wave solutions have key applications and help to understand the physical phenomena of this wave model. By giving appropriate parameter values, different types of solitons structures can be depicted graphically. Several precise solutions and computations have proved the straightforwardness, consistency and power of the these techniques.

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44

Ma, Tianshui, Jie Li, Liangyun Chen, and Shuanhong Wang. "Rota-Baxter operators on Turaev's Hopf group (co)algebras I: Basic definitions and related algebraic structures." Journal of Geometry and Physics 175 (May 2022): 104469. http://dx.doi.org/10.1016/j.geomphys.2022.104469.

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45

Muhiuddin, Ghulam, Mohamed E. Elnair та Deena Al-Kadi. "𝒩-Structures Applied to Commutative Ideals of BCI-Algebras". Symmetry 14, № 10 (26 вересня 2022): 2015. http://dx.doi.org/10.3390/sym14102015.

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Анотація:

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object’s symmetries. In order to provide a mathematical tool for dealing with negative information, a negative-valued function came into existence along with N-structures. In the present analysis, the notion of N-structures is applied to the ideals, especially the commutative ideals of BCI-algebras. Firstly, several properties of N-subalgebras and N-ideals in BCI-algebras are investigated. Furthermore, the notion of a commutative N-ideal is defined, and related properties are investigated. In addition, useful characterizations of commutative N-ideals are established. A condition for a closed N-ideal to be a commutative N-ideal is provided. Finally, it is proved that in a commutative BCI-algebra, every closed N-ideal is a commutative N-ideal.

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46

He, Zeyuan, and Simon D. Guest. "On rigid origami I: piecewise-planar paper with straight-line creases." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2232 (December 2019): 20190215. http://dx.doi.org/10.1098/rspa.2019.0215.

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Анотація:

Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.

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47

Richter, Wolf-Dieter. "On Complex Numbers in Higher Dimensions." Axioms 11, no. 1 (January 7, 2022): 22. http://dx.doi.org/10.3390/axioms11010022.

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The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.

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48

Cederwall, Martin, and Jakob Palmkvist. "Tensor Hierarchy Algebra Extensions of Over-Extended Kac–Moody Algebras." Communications in Mathematical Physics 389, no. 1 (December 3, 2021): 571–620. http://dx.doi.org/10.1007/s00220-021-04243-3.

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AbstractTensor hierarchy algebras are infinite-dimensional generalisations of Cartan-type Lie superalgebras. They are not contragredient, exhibiting an asymmetry between positive and negative levels. These superalgebras have been a focus of attention due to the fundamental rôle they play for extended geometry. In the present paper, we examine tensor hierarchy algebras which are super-extensions of over-extended (often, hyperbolic) Kac–Moody algebras. They contain novel algebraic structures. Of particular interest is the extension of a over-extended algebra by its fundamental module, an extension that contains and generalises the extension of an affine Kac–Moody algebra by a Virasoro derivation $$L_1$$ L 1 . A conjecture about the complete superalgebra is formulated, relating it to the corresponding Borcherds superalgebra.

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49

Marquette, Ian, Luke Yates, and Peter D. Jarvis. "Generalized quadratic commutator algebras of PBW-type." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 121703. http://dx.doi.org/10.1063/5.0096769.

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In recent years, various nonlinear algebraic structures have been obtained in the context of quantum systems as symmetry algebras, Painlevé transcendent models, and missing label problems. In this paper, we treat all these algebras as instances of the class of quadratic (and higher degree) commutator bracket algebras of Poincaré–Birkhoff–Witt type. We provide a general approach for simplifying the constraints arising from the diamond lemma and apply this in particular to give a comprehensive analysis of the quadratic case. We present new examples of quadratic algebras, which admit a cubic Casimir invariant. The connection with other approaches, such as Gröbner bases, is developed, and we suggest how our explicit and computational techniques can be relevant in other contexts.

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50

Albuquerque, C. D., R. Palazzo Jr., and E. B. Silva. "New classes of TQC associated with self-dual, quasi self-dual and denser tessellations." Quantum Information and Computation 10, no. 11&12 (November 2010): 956–70. http://dx.doi.org/10.26421/qic10.11-12-6.

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In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\frac{k}{n} \rightarrow 1$ as $n \rightarrow \infty$, except for the one case where $\frac{k}{n} \rightarrow \frac{1}{3}$ as $n \rightarrow \infty$.

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