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Journal articles on the topic 'Noncommutative derived algebraic geometry'

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Published: 4 April 2023

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1

Chen, Yiping, and Wei Hu. "Approximations, ghosts and derived equivalences." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 2 (January 26, 2019): 813–40. http://dx.doi.org/10.1017/prm.2018.120.

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AbstractApproximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly n-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.

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2

Cirio, Lucio S., Giovanni Landi, and Richard J. Szabo. "Instantons and vortices on noncommutative toric varieties." Reviews in Mathematical Physics 26, no. 09 (October 2014): 1430008. http://dx.doi.org/10.1142/s0129055x14300088.

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We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

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3

Dolbeault, Pierre. "On a noncommutative algebraic geometry." Banach Center Publications 107 (2015): 119–31. http://dx.doi.org/10.4064/bc107-0-8.

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4

Nikolaev, Igor V. "Noncommutative geometry of algebraic curves." Proceedings of the American Mathematical Society 137, no. 10 (October 1, 2009): 3283. http://dx.doi.org/10.1090/s0002-9939-09-09917-1.

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5

Sharygin, G. I. "Geometry of Noncommutative Algebraic Principal Bundles." Journal of Mathematical Sciences 134, no. 2 (April 2006): 1911–82. http://dx.doi.org/10.1007/s10958-006-0092-z.

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6

Škoda, Zoran. "Some Equivariant Constructions in Noncommutative Algebraic Geometry." gmj 16, no. 1 (March 2009): 183–202. http://dx.doi.org/10.1515/gmj.2009.183.

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Abstract We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.

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7

Beil, Charlie. "The Bell states in noncommutative algebraic geometry." International Journal of Quantum Information 12, no. 05 (August 2014): 1450033. http://dx.doi.org/10.1142/s0219749914500336.

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We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices ϕ1, ϕ2 such that ϕ1ϕ2 = ϕ2ϕ1 = p id. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on non-noetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime.

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8

Reineke, Markus, J. Toby Stafford, Catharina Stroppel, and Michel Van den Bergh. "Interactions between Algebraic Geometry and Noncommutative Algebra." Oberwolfach Reports 11, no. 2 (2014): 1365–402. http://dx.doi.org/10.4171/owr/2014/25.

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9

Reineke, Markus, J. Toby Stafford, Catharina Stroppel, and Michel Van den Bergh. "Interactions between Algebraic Geometry and Noncommutative Algebra." Oberwolfach Reports 15, no. 2 (April 11, 2019): 1465–515. http://dx.doi.org/10.4171/owr/2018/24.

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10

Toën, Bertrand. "Derived algebraic geometry." EMS Surveys in Mathematical Sciences 1, no. 2 (2014): 153–245. http://dx.doi.org/10.4171/emss/4.

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11

Ozawa, Narutaka. "NONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY OF KAZHDAN’S PROPERTY (T)." Journal of the Institute of Mathematics of Jussieu 15, no. 1 (July 30, 2014): 85–90. http://dx.doi.org/10.1017/s1474748014000309.

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It is well known that a finitely generated group ${\rm\Gamma}$ has Kazhdan’s property (T) if and only if the Laplacian element ${\rm\Delta}$ in $\mathbb{R}[{\rm\Gamma}]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in $\mathbb{R}[{\rm\Gamma}]$. Namely, ${\rm\Gamma}$ has property (T) if and only if there exist a constant ${\it\kappa}>0$ and a finite sequence ${\it\xi}_{1},\ldots ,{\it\xi}_{n}$ in $\mathbb{R}[{\rm\Gamma}]$ such that ${\rm\Delta}^{2}-{\it\kappa}{\rm\Delta}=\sum _{i}{\it\xi}_{i}^{\ast }{\it\xi}_{i}$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in $\mathbb{R}[{\rm\Gamma}]$, possibly with the assistance of computers.

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12

Franco, Nicolas, and Michał Eckstein. "An algebraic formulation of causality for noncommutative geometry." Classical and Quantum Gravity 30, no. 13 (June 7, 2013): 135007. http://dx.doi.org/10.1088/0264-9381/30/13/135007.

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13

Cimprič, Jakob. "A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings." Canadian Mathematical Bulletin 52, no. 1 (March 1, 2009): 39–52. http://dx.doi.org/10.4153/cmb-2009-005-4.

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AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

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14

Khoroshkov, Yu V. "Mirror Symmetry as an Algebra of Operators in Noncommutative Geometry of Space-Time." Ukrainian Journal of Physics 67, no. 2 (April 1, 2022): 117. http://dx.doi.org/10.15407/ujpe67.2.117.

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The analysis of the geometric and algebraic properties of mirror mappings allowed the latter to be used as the operator algebra of a noncommutative geometry. The coordinates of the noncommutative geometry are auto- or cross-correlation coordinates in the mirror-mapped spaces. A particular case of the six-dimensional Kahler manifold which is mapped on the noncommutative geometry with the vector Clifford algebra Cl4 has been considered. This mapping corresponds to a tetraquark composed from two quark–anti-quark pairs with the charges ±2/3q taken from different generations.

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15

MARTINS, RACHEL A. D. "NONCOMMUTATIVE FERMION MASS MATRIX AND GRAVITY." International Journal of Modern Physics A 28, no. 25 (October 8, 2013): 1350120. http://dx.doi.org/10.1142/s0217751x13501200.

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The first part is an introductory description of a small cross-section of the literature on algebraic methods in nonperturbative quantum gravity with a specific focus on viewing algebra as a laboratory in which to deepen understanding of the nature of geometry. This helps to set the context for the second part, in which we describe a new algebraic characterization of the Dirac operator in noncommutative geometry and then use it in a calculation on the form of the fermion mass matrix. Assimilating and building on the various ideas described in the first part, the final part consists of an outline of a speculative perspective on (noncommutative) quantum spectral gravity. This is the second of a pair of papers so far on this project.

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16

Verschoren, A., and L. Willaert. "Noncommutative algebraic geometry: from pi-algebras to quantum groups." Bulletin of the Belgian Mathematical Society - Simon Stevin 4, no. 5 (1997): 557–88. http://dx.doi.org/10.36045/bbms/1105737761.

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17

ZHANG, R. B., and XIAO ZHANG. "PROJECTIVE MODULE DESCRIPTION OF EMBEDDED NONCOMMUTATIVE SPACES." Reviews in Mathematical Physics 22, no. 05 (June 2010): 507–31. http://dx.doi.org/10.1142/s0129055x10004028.

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An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in [8]. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi–Civita connection and related curvatures, which were introduced geometrically in [8]. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.

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18

Lorenz, Martin. "Algebraic group actions on noncommutative spectra." Transformation Groups 14, no. 3 (June 20, 2009): 649–75. http://dx.doi.org/10.1007/s00031-009-9059-8.

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19

Iwanari, Isamu. "Tannakization in derived algebraic geometry." Journal of K-theory 14, no. 3 (December 2014): 642–700. http://dx.doi.org/10.1017/is014008019jkt278.

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AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

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20

WULKENHAAR, RAIMAR. "SO(10) UNIFICATION IN NONCOMMUTATIVE GEOMETRY REVISITED." International Journal of Modern Physics A 14, no. 04 (February 10, 1999): 559–88. http://dx.doi.org/10.1142/s0217751x99000282.

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We investigate the SO(10) unification model in a Lie-algebraic formulation of noncommutative geometry. The SO(10) symmetry is broken by a 45-Higgs and the Majorana mass term for the right neutrinos (126-Higgs) to the standard model structure group. We study the case where the fermion masses are as general as possible, which leads to two 10-multiplets, four 120-multiplets and two additional 126-multiplets of Higgs fields. This Higgs structure differs considerably from the two Higgs multiplets 16 ⊗ 16* and 16c ⊗ 16* used by Chamseddine and Fröhlich. We find the usual tree level predictions of noncommutative geometry: [Formula: see text], [Formula: see text] and g2=g3 as well as mH≤ mt.

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21

Besnard, F., and S. Farnsworth. "Particle models from special Jordan backgrounds and spectral triples." Journal of Mathematical Physics 63, no. 10 (October 1, 2022): 103505. http://dx.doi.org/10.1063/5.0107136.

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We put forward a definition for spectral triples and algebraic backgrounds based on Jordan coordinate algebras. We also propose natural and gauge-invariant bosonic configuration spaces of fluctuated Dirac operators and compute them for general, almost-associative, Jordan, coordinate algebras. We emphasize that the theory so obtained is not equivalent with usual associative noncommutative geometry, even when the coordinate algebra is the self-adjoint part of a C*-algebra. In particular, in the Jordan case, the gauge fields are always unimodular, thus curing a long-standing problem in noncommutative geometry.

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22

Jiang, Yue, Mai-Lin Liang, and Ya-Bin Zhang. "A general derivation of the Hall conductivity on the noncommutative plane." Canadian Journal of Physics 89, no. 7 (July 2011): 769–72. http://dx.doi.org/10.1139/p11-054.

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By algebraic approach, Hall conductivity on the noncommutative plane is derived exactly. The calculations are carried out in noncommutative phase spaces without any representations or perturbation expansions of the coordinate and momentum operators.

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23

COQUEREAUX, R., R. HÄUβLING, and F. SCHECK. "ALGEBRAIC CONNECTIONS ON PARALLEL UNIVERSES." International Journal of Modern Physics A 10, no. 01 (January 10, 1995): 89–98. http://dx.doi.org/10.1142/s0217751x95000048.

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For any manifold M we introduce a ℤ-graded differential algebra Ξ, which, in particular, is a bimodule over the associative algebra C(M⋃M). We then introduce the corresponding covariant differentials and show how this construction can be interpreted in terms of Yang-Mills and Higgs fields. This is a particular example of noncommutative geometry. It differs from the prescription of Connes in the following way: the definition of Ξ does not rely on a given Dirac-Yukawa operator acting on a space of spinors.

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24

Antieau, Benjamin. "Étale twists in noncommutative algebraic geometry and the twisted Brauer space." Journal of Noncommutative Geometry 11, no. 1 (2017): 161–92. http://dx.doi.org/10.4171/jncg/11-1-5.

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25

Reyes, Armando, and Jason Hernández-Mogollón. "A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type." Ingeniería y Ciencia 16, no. 31 (June 19, 2020): 27–52. http://dx.doi.org/10.17230/ingciencia.16.31.2.

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In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.

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26

VAN DEN DUNGEN, KOEN, and WALTER D. VAN SUIJLEKOM. "PARTICLE PHYSICS FROM ALMOST-COMMUTATIVE SPACETIMES." Reviews in Mathematical Physics 24, no. 09 (October 2012): 1230004. http://dx.doi.org/10.1142/s0129055x1230004x.

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Our aim in this review paper is to present the applications of Connes' noncommutative geometry to elementary particle physics. Whereas the existing literature is mostly focused on a mathematical audience, in this paper we introduce the ideas and concepts from noncommutative geometry using physicists' terminology, gearing towards the predictions that can be derived from the noncommutative description. Focusing on a light package of noncommutative geometry (so-called "almost-commutative manifolds"), we shall introduce in steps: electrodynamics, the electroweak model, culminating in the full Standard Model. We hope that our approach helps in understanding the role noncommutative geometry could play in describing particle physics models, eventually unifying them with Einstein's (geometrical) theory of gravity.

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27

Reyes, Armando, and Héctor Suárez. "Skew Poincaré–Birkhoff–Witt extensions over weak compatible rings." Journal of Algebra and Its Applications 19, no. 12 (November 18, 2019): 2050225. http://dx.doi.org/10.1142/s0219498820502254.

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In this paper, we introduce weak [Formula: see text]-compatible rings and study skew Poincaré–Birkhoff–Witt extensions over these rings. We characterize the weak notion of compatibility for several noncommutative rings appearing in noncommutative algebraic geometry and some quantum algebras of theoretical physics. As a consequence of our treatment, we unify and extend results in the literature about Ore extensions and skew PBW extensions over compatible rings.

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28

Hacon, Christopher, Daniel Huybrechts, Richard P. W. Thomas, and Chenyang Xu. "Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects." Oberwolfach Reports 17, no. 2 (July 1, 2021): 977–1021. http://dx.doi.org/10.4171/owr/2020/19.

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29

Yu, Hefu, and Bo-Qiang Ma. "Origin of fermion generations from extended noncommutative geometry." International Journal of Modern Physics A 33, no. 29 (October 20, 2018): 1850168. http://dx.doi.org/10.1142/s0217751x18501683.

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We propose a way to understand the three fermion generations by the algebraic structures of noncommutative geometry, which is a promising framework to unify the standard model and general relativity. We make the tensor product extension and the quaternion extension on the framework. Each of the two extensions alone keeps the action invariant, and we consider them as the almost trivial structures of the geometry. We combine the two extensions, and show the corresponding physical effects, i.e. the emergence of three fermion generations and the mass relationships among those generations. We define the coordinate fiber space of the bundle of the manifold as the space in which the classical noncommutative geometry is expressed, then the tensor product extension explicitly shows the contribution of structures in the non-coordinate base space of the bundle to the action. The quaternion extension plays an essential role to reveal the physical effect of the structure in the non-coordinate base space.

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30

Shojaei-Fard, Ali. "Application of Deformed Lie Algebras to Non-Perturbative Quantum Field Theory." Journal of the Indian Mathematical Society 84, no. 1-2 (January 2, 2017): 109. http://dx.doi.org/10.18311/jims/2017/5839.

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The manuscript implements Connes-Kreimer Hopf algebraic renormalization of Feynman diagrams and Dubois-Violette type noncommutative differential geometry to discover a new class of differential calculi with respect to infinite formal expansions of Feynman diagrams which are generated by Dyson-Schwinger equations.

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31

Lee, Hyun Ho. "A note on nonlinear σ-models in noncommutative geometry." Infinite Dimensional Analysis, Quantum Probability and Related Topics 19, no. 01 (March 2016): 1650006. http://dx.doi.org/10.1142/s0219025716500065.

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We study nonlinear [Formula: see text]-models defined on a noncommutative torus as a two-dimensional string worldsheet. We consider (i) a two-point space, (ii) a circle, (iii) a noncommutative torus, (iv) a classical group [Formula: see text] as examples of space-time. Based on established results, the trivial harmonic unitaries of the noncommutative chiral model known as local minima are shown not to be global minima by comparing them to the symmetric unitaries derived from instanton solutions of the noncommutative Ising model corresponding to a two-point space. In addition, a [Formula: see text]-action on field maps is introduced to a noncommutative torus, and its action on solutions of various Euler–Lagrange equations is described.

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32

Berest, Yuri, Ajay Ramadoss, and Xiang Tang. "The Picard group of a noncommutative algebraic torus." Journal of Noncommutative Geometry 7, no. 2 (2013): 335–56. http://dx.doi.org/10.4171/jncg/119.

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33

Annala, Toni. "On line bundles in derived algebraic geometry." Annals of K-Theory 5, no. 2 (June 20, 2020): 317–25. http://dx.doi.org/10.2140/akt.2020.5.317.

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34

Masuda, Tetsuya, Yoshiomi Nakagami, and Junsei Watanabe. "Noncommutative differential geometry on the quantum SU(2), I: An algebraic viewpoint." K-Theory 4, no. 2 (March 1990): 157–80. http://dx.doi.org/10.1007/bf00533155.

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35

LIU, LIYU, and WEN MA. "NAKAYAMA AUTOMORPHISMS OF ORE EXTENSIONS OVER POLYNOMIAL ALGEBRAS." Glasgow Mathematical Journal 62, no. 3 (June 17, 2019): 518–30. http://dx.doi.org/10.1017/s0017089519000259.

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AbstractNakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.

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36

DASZKIEWICZ, MARCIN. "TWISTED RINDLER SPACETIMES." Modern Physics Letters A 25, no. 13 (April 30, 2010): 1059–70. http://dx.doi.org/10.1142/s0217732310032858.

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The (linearized) noncommutative Rindler spacetimes associated with canonical, Lie-algebraic and quadratic twist-deformed Minkowski spaces are provided. The corresponding deformed Hawking spectra detected by Rindler observers are derived as well.

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37

MORETTI, VALTER. "ASPECTS OF NONCOMMUTATIVE LORENTZIAN GEOMETRY FOR GLOBALLY HYPERBOLIC SPACETIMES." Reviews in Mathematical Physics 15, no. 10 (December 2003): 1171–217. http://dx.doi.org/10.1142/s0129055x03001886.

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Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally-hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a C*-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of C*-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called loci, are realized as the elements of the inductive limit of the spaces of the algebraic states on the C*-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the role of a Lorentzian metric. Specializing back the formalism to the usual globally-hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.

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Brav, Christopher, and Tobias Dyckerhoff. "Relative Calabi–Yau structures." Compositio Mathematica 155, no. 2 (February 2019): 372–412. http://dx.doi.org/10.1112/s0010437x19007024.

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We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

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39

Yazdani, Aref. "Field Equations and Radial Solutions in a Noncommutative Spherically Symmetric Geometry." Advances in High Energy Physics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/349659.

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We study a noncommutative theory of gravity in the framework of torsional spacetime. This theory is based on a Lagrangian obtained by applying the technique of dimensional reduction of noncommutative gauge theory and that the yielded diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity. Field equations are derived in the framework of teleparallel gravity through Weitzenbock geometry. We solve these field equations by considering a mass that is distributed spherically symmetrically in a stationary static spacetime in order to obtain a noncommutative line element. This new line element interestingly reaffirms the coherent state theory for a noncommutative Schwarzschild black hole. For the first time, we derive the Newtonian gravitational force equation in the commutative relativity framework, and this result could provide the possibility to investigate examples in various topics in quantum and ordinary theories of gravity.

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40

Masuda, Tetsuya, Yoshiomi Nakagami, and Junsei Watanabe. "Noncommutative differential geometry on the quantum two sphere of podleś. I: An algebraic viewpoint." K-Theory 5, no. 2 (March 1991): 151–75. http://dx.doi.org/10.1007/bf01254546.

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41

LERNER, BORIS, and STEFFEN OPPERMANN. "A RECOLLEMENT APPROACH TO GEIGLE–LENZING WEIGHTED PROJECTIVE VARIETIES." Nagoya Mathematical Journal 226 (October 13, 2016): 71–105. http://dx.doi.org/10.1017/nmj.2016.39.

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We introduce a new method for expanding an abelian category and study it using recollements. In particular, we give a criterion for the existence of cotilting objects. We show, using techniques from noncommutative algebraic geometry, that our construction encompasses the category of coherent sheaves on Geigle–Lenzing weighted projective lines. We apply our construction to some concrete examples and obtain new weighted projective varieties, and analyze the endomorphism algebras of their tilting bundles.

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42

Hacon, Christopher, Daniel Huybrechts, Bernd Siebert, and Chenyang Xu. "Algebraic Geometry: Birational Classification, Derived Categories, and Moduli Spaces." Oberwolfach Reports 14, no. 3 (July 4, 2018): 2703–67. http://dx.doi.org/10.4171/owr/2017/45.

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43

Ben-Zvi, David, John Francis, and David Nadler. "Integral transforms and Drinfeld centers in derived algebraic geometry." Journal of the American Mathematical Society 23, no. 4 (April 1, 2010): 909–66. http://dx.doi.org/10.1090/s0894-0347-10-00669-7.

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44

Antieau, Benjamin, and David Gepner. "Brauer groups and étale cohomology in derived algebraic geometry." Geometry & Topology 18, no. 2 (April 7, 2014): 1149–244. http://dx.doi.org/10.2140/gt.2014.18.1149.

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45

VIET, NGUYEN AI, and KAMESHWAR C. WALI. "NONCOMMUTATIVE GEOMETRY AND A DISCRETIZED VERSION OF KALUZA-KLEIN THEORY WITH A FINITE FIELD CONTENT." International Journal of Modern Physics A 11, no. 03 (January 30, 1996): 533–51. http://dx.doi.org/10.1142/s0217751x96000249.

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We consider a four-dimensional space-time supplemented by two discrete points assigned to a Z2-algebraic structure and develop the formalism of noncommutative geometry. By setting up a generalized vielbein, we study the metric structure. Metric-compatible torsion-free connection defines a unique finite field content in the model and leads to a discretized version of Kaluza-Klein theory. We study some special cases of this model that illustrate the rich and complex structure with massive modes and the possible presence of a cosmological constant.

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46

Cornelissen, Gunther, and Giovanni Landi. "Editors’ preface for the topical issue on “Noncommutative algebraic geometry and its applications to physics”." Journal of Geometry and Physics 72 (October 2013): 1–2. http://dx.doi.org/10.1016/j.geomphys.2013.03.022.

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Tabuada, Gonçalo. "VANISHING OF THE NEGATIVE HOMOTOPY -THEORY OF QUOTIENT SINGULARITIES." Journal of the Institute of Mathematics of Jussieu 18, no. 3 (May 8, 2017): 619–27. http://dx.doi.org/10.1017/s1474748017000172.

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Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy$K$-theory groups of a Noetherian scheme$X$of Krull dimension$d$vanish below$-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy$K$-theory groups vanish below$-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy$K$-theory group.

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Fernandes Pereira, Francisco Revson, Giuliano Gadioli La Guardia, and Francisco Marcos de Assis. "Classical and Quantum Convolutional Codes Derived From Algebraic Geometry Codes." IEEE Transactions on Communications 67, no. 1 (January 2019): 73–82. http://dx.doi.org/10.1109/tcomm.2018.2875754.

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Cao, Wensheng. "Quadratic Equation in Split Quaternions." Axioms 11, no. 5 (April 20, 2022): 188. http://dx.doi.org/10.3390/axioms11050188.

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Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x2+bx+c=0 in split quaternion algebra.

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GOSWAMI, DEBASHISH. "TWISTED ENTIRE CYCLIC COHOMOLOGY, J-L-O COCYCLES AND EQUIVARIANT SPECTRAL TRIPLES." Reviews in Mathematical Physics 16, no. 05 (June 2004): 583–602. http://dx.doi.org/10.1142/s0129055x04002114.

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We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.

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