scholarly journals AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

2021 ◽  
pp. 1-25
Author(s):  
AYAKO ITABA ◽  
MASAKI MATSUNO
Keyword(s):  
Elliptic Curves ◽  
Three Dimensional ◽  
Preceding Paper ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Defining Relations ◽  
Morita Equivalent ◽  
Cubic Curve ◽  
Regular Algebras ◽  
Geometric Algebras

Abstract In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

scholarly journals GRADED MORITA EQUIVALENCES FOR GEOMETRIC AS-REGULAR ALGEBRAS

2012 ◽  
Vol 55 (2) ◽  
pp. 241-257 ◽  
Author(s):  
IZURU MORI ◽  
KENTA UEYAMA
Keyword(s):  
Algebraic Geometry ◽  
Three Dimensional ◽  
Koszul Algebras ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Regular Algebra ◽  
Morita Equivalent ◽  
Regular Algebras

AbstractClassification of AS-regular algebras is one of the major projects in non-commutative algebraic geometry. In this paper, we will study when given AS-regular algebras are graded Morita equivalent. In particular, for every geometric AS-regular algebra A, we define another graded algebra A, and show that if two geometric AS-regular algebras A and A' are graded Morita equivalent, then A and A' are isomorphic as graded algebras. We also show that the converse holds in many three-dimensional cases. As applications, we apply our results to Frobenius Koszul algebras and Beilinson algebras.

scholarly journals A Complete Classification of 3-dimensional Quadratic AS-regular Algebras of Type EC

2020 ◽  
pp. 1-19
Author(s):  
Masaki Matsuno
Keyword(s):  
Algebraic Geometry ◽  
Complete Classification ◽  
Complete List ◽  
Graded Algebra ◽  
Regular Algebra ◽  
3 Dimensional ◽  
Noncommutative Algebraic Geometry ◽  
Regular Algebras ◽  
Point Scheme

Abstract Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$ -dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb {P}^{2}$ . In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$ -dimensional quadratic AS-regular algebras that cannot be written as a twist of a Calabi–Yau AS-regular algebra by a graded algebra automorphism.

scholarly journals Algunos tipos especiales de determinantes en extensiones PBW torcidas graduadas

2021 ◽  
Vol 39 (1) ◽  
Author(s):  
Héctor Suárez ◽  
Duban Cáceres ◽  
Armando Reyes
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Regular Algebra ◽  
Noncommutative Algebraic Geometry ◽  
Noncommutative Differential Geometry ◽  
Regular Algebras ◽  
Skew Pbw Extensions ◽  
Finitely Presented

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over a finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = σ(R)<x1, x2> a graded skew PBW extension over a connected algebra R, we compute its P-determinant and the inverse of σ. In the particular case of quasi-commutative skew PBW extensions over Koszul Artin-Schelter regular algebras, we show explicitly the connection between the Nakayama automorphism of the ring of coefficients and the extension. Finally, we give conditions to guarantee that A is Calabi-Yau. We provide illustrative examples of the theory concerning algebras of interest in noncommutative algebraic geometry and noncommutative differential geometry.

Complete caps in AG(3, q) from elliptic curves

2014 ◽  
Vol 13 (08) ◽  
pp. 1450050 ◽  
Author(s):  
Irene Platoni
Keyword(s):  
Elliptic Curves ◽  
Lower Bounds ◽  
Three Dimensional ◽  
Complete Caps ◽  
Odd Order ◽  
Galois Space

In a three-dimensional Galois space of odd order q, the known infinite families of complete caps have size far from the theoretical lower bounds. In this paper, we investigate some caps defined from elliptic curves. In particular, we show that for each q between 100 and 350 they can be extended to complete caps, which turn out to be the smallest complete caps known in the literature.

scholarly journals The characterization of elliptic curves over finite fields

1988 ◽  
Vol 45 (2) ◽  
pp. 275-286 ◽  
Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Maximum Size ◽  
Desarguesian Plane ◽  
Cubic Curve ◽  
Odd Order ◽  
Even Order ◽  
Curves Over Finite Fields

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

scholarly journals Subaxolemmal cytoskeleton in squid giant axon. II. Morphological identification of microtubule- and microfilament-associated domains of axolemma.

1986 ◽  
Vol 102 (5) ◽  
pp. 1710-1725 ◽  
Author(s):  
S Tsukita ◽  
S Tsukita ◽  
T Kobayashi ◽  
G Matsumoto
Keyword(s):  
Electron Microscopy ◽  
Three Dimensional ◽  
Preceding Paper ◽  
Microscopic Analysis ◽  
Giant Axon ◽  
Squid Giant Axon ◽  
Molecular Organization ◽  
Fixation Method ◽  
Morphological Identification

In the preceding paper (Kobayashi, T., S. Tsukita, S. Tsukita, Y. Yamamoto, and G. Matsumoto, 1986, J. Cell Biol., 102:1710-1725), we demonstrated biochemically that the subaxolemmal cytoskeleton of the squid giant axon was highly specialized and mainly composed of tubulin, actin, axolinin, and a 255-kD protein. In this paper, we analyzed morphologically the molecular organization of the subaxolemmal cytoskeleton in situ. For thin section electron microscopy, the subaxolemmal cytoskeleton was chemically fixed by the intraaxonal perfusion of the fixative containing tannic acid. With this fixation method, the ultrastructural integrity was well preserved. For freeze-etch replica electron microscopy, the intraaxonally perfused axon was opened and rapidly frozen by touching its inner surface against a cooled copper block (4 degrees K), thus permitting the direct stereoscopic observation of the cytoplasmic surface of the axolemma. Using these techniques, it became clear that the major constituents of the subaxolemmal cytoskeleton were microfilaments and microtubules. The microfilaments were observed to be associated with the axolemma through a specialized meshwork of thin strands, forming spot-like clusters just beneath the axolemma. These filaments were decorated with heavy meromyosin showing a characteristic arrowhead appearance. The microtubules were seen to run parallel to the axolemma and embedded in the fine three-dimensional meshwork of thin strands. In vitro observations of the aggregates of axolinin and immunoelectron microscopic analysis showed that this fine meshwork around microtubules mainly consisted of axolinin. Some microtubules grazed along the axolemma and associated laterally with it through slender strands. Therefore, we were led to conclude that the axolemma of the squid giant axon was specialized into two domains (microtubule- and microfilament-associated domains) by its underlying cytoskeletons.

scholarly journals A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2936
Author(s):  
Zhidong Zhang ◽  
Osamu Suzuki
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Preceding Paper ◽  
Mathematical Structure ◽  
Topological Phases ◽  
Riemann Hilbert Problem ◽  
3D Ising Model

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.

scholarly journals Constructing cubic curves with involutions

2021 ◽  
Author(s):  
Lorenz Halbeisen ◽  
Norbert Hungerbühler
Keyword(s):  
Elliptic Curves ◽  
Simple Proof ◽  
Torsion Group ◽  
Cubic Curves ◽  
Cubic Curve ◽  
On Line

AbstractIn 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles’ Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter’s construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter’s construction we provide a new parametrisation of elliptic curves with torsion group $$\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/8\mathbb {Z}$$ Z / 2 Z × Z / 8 Z and give some configurations with all their points on a cubic curve.

Fixed-points and algebras with infinitely long expressions: Part II. μ-clones of regular algebras

1979 ◽  
Vol 2 (1) ◽  
pp. 317-335
Author(s):  
Jerzy Tiuryn
Keyword(s):  
Fixed Points ◽  
Regular Algebra ◽  
Arbitrary Signature ◽  
Regular Algebras ◽  
Main Notion

This paper is a continuation of Tiuryn [16]. The main notion presented in the latter paper is the notion of a regular algebra. As it was proved in Tiuryn [16] for an arbitrary signature Σ the algebra of regular Σ-trees is an initial regular algebra. This means that there are naturally defined “polynomials” for regular algebras, which are determined by infinitely long expressions. The aim of this paper is to investigate this phenomenon.

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