scholarly journals On the Structure of Graded Leibniz Algebras

2015 ◽  
Vol 22 (01) ◽  
pp. 83-96 ◽  
Author(s):  
Antonio J. Calderón Martín ◽  
José M. Sánchez Delgado
Keyword(s):  
Arbitrary Dimension ◽  
Unit Element ◽  
Homogeneous Component ◽  
Maximal Length ◽  
Base Field ◽  
Leibniz Algebras ◽  
Arbitrary Base

We study the structure of graded Leibniz algebras with arbitrary dimension and over an arbitrary base field 𝕂. We show that any of such algebras 𝔏 with a symmetric G-support is of the form [Formula: see text] with U a subspace of 𝔏1, the homogeneous component associated to the unit element 1 in G, and any Ij a well described graded ideal of 𝔏, satisfying [Ij, Ik]=0 if j ≠ k. In the case of 𝔏 being of maximal length, we characterize the gr-simplicity of the algebra in terms of connections in the support of the grading.

Regular Hom-algebras admitting a multiplicative basis

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio J. Calderón Martín
Keyword(s):  
Arbitrary Dimension ◽  
Base Field ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family

AbstractLet {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

n-Algebras admitting a multiplicative basis

2018 ◽  
Vol 17 (02) ◽  
pp. 1850025 ◽  
Author(s):  
Antonio J. Calderón Martín ◽  
Francisco J. Navarro Izquierdo ◽  
José M. Sánchez Delgado
Keyword(s):  
Mild Condition ◽  
Arbitrary Dimension ◽  
Base Field ◽  
Triple Systems ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family ◽  
Sum Formula

Let [Formula: see text] be an [Formula: see text]-algebra of arbitrary dimension and over an arbitrary base field [Formula: see text]. A basis [Formula: see text] of [Formula: see text] is said to be multiplicative if for any [Formula: see text], we have either [Formula: see text] or [Formula: see text] for some (unique) [Formula: see text]. If [Formula: see text], we are dealing with algebras admitting a multiplicative basis while if [Formula: see text] we are speaking about triple systems with multiplicative bases. We show that if [Formula: see text] admits a multiplicative basis then it decomposes as the orthogonal direct sum [Formula: see text] of well-described ideals admitting each one a multiplicative basis. Also, the minimality of [Formula: see text] is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals.

scholarly journals Determinantal ideals without minimal free resolutions

1990 ◽  
Vol 118 ◽  
pp. 203-216 ◽  
Author(s):  
Mitsuyasu Hashimoto
Keyword(s):  
Free Resolution ◽  
Unit Element ◽  
Free Resolutions ◽  
Minimal Free Resolution ◽  
Determinantal Ideals ◽  
Arbitrary Base ◽  
Generic Matrix ◽  
Base Ring ◽  
Minimal Free Resolutions ◽  
The Ideal

Let R be a Noetherian commutative ring with, unit element, and Xij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n. Let S = R[xij] be the polynomial ring over R, and It be the ideal in S, generated by the t × t minors of the generic matrix (xij) ∈ Mm, n(S). For many years there has been considerable interest in finding a minimal free resolution of S/It, over arbitrary base ring R. If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗z P. is a resolution of S/It over the base ring R′.

scholarly journals Genus-2 curves and Jacobians with a given number of points

2015 ◽  
Vol 18 (1) ◽  
pp. 170-197 ◽  
Author(s):  
Reinier Bröker ◽  
Everett W. Howe ◽  
Kristin E. Lauter ◽  
Peter Stevenhagen
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Polynomial Time ◽  
Rational Points ◽  
The Other ◽  
Base Field ◽  
Arbitrary Base ◽  
The Family ◽  
Genus 2 ◽  
Genus 2 Curves

AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.

scholarly journals BASIS OF MULTILINEAR PART OF LEIBNIZ ALGEBRAS MANIFOLDS \overbrace{V_1}

2017 ◽  
Vol 20 (3) ◽  
pp. 76-82
Author(s):  
S.P. Mishchenko ◽  
Y.R. Pestova
Keyword(s):  
Polynomial Growth ◽  
Base Field ◽  
Leibniz Algebras ◽  
Continue Research

In the case of trivial characteristic of base field, Leibniz algebras manifolds defined by the identity x_1(x_2x_3)(x_4x_5)\equiv 0. has almost polynomial growth. In the work we continue research of this manifold, in particular, we construct bases of multilinear parts.

scholarly journals Jacques Tits motivic measure

2021 ◽  
Author(s):  
Gonçalo Tabuada
Keyword(s):  
Quadratic Forms ◽  
Arbitrary Dimension ◽  
Base Field ◽  
Central Simple Algebras ◽  
Grothendieck Ring ◽  
Main Application ◽  
Simple Algebras ◽  
Central Simple

AbstractIn this article we construct a new motivic measure called the Jacques Tits motivic measure. As a first main application, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to 2-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period $$\{3, 4, 5, 6\}$$ { 3 , 4 , 5 , 6 } , have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension 6 or to quadratic forms of arbitrary dimension defined over a base field k with $$I^3(k)=0$$ I 3 ( k ) = 0 , have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.

scholarly journals On the Betti Numbers of Shifted Complexes of Stable Simplicial Complexes

2006 ◽  
Vol 13 (01) ◽  
pp. 47-56
Author(s):  
Zhongming Tang ◽  
Guifen Zhuang
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Simplicial Complexes ◽  
Base Field ◽  
Arbitrary Base

Let Δ be a stable simplicial complex on n vertexes. Over an arbitrary base field K, the symmetric algebraic shifted complex Δs of Δ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in the polynomial ring K[x1, x2, …, xn] of the symmetric algebraic shifted complex, exterior algebraic shifted complex and combinatorial shifted complex of Δ are equal.

scholarly journals PRYM VARIETIES AND THE INFINITE GRASSMANNIAN

1998 ◽  
Vol 09 (01) ◽  
pp. 75-93 ◽  
Author(s):  
FRANCISCO J. PLAZA MARTÍN
Keyword(s):  
Dynamical Systems ◽  
Moduli Space ◽  
Point Of View ◽  
Base Field ◽  
Prym Varieties ◽  
Arbitrary Base ◽  
Bkp Hierarchy ◽  
Definition Of

In this paper we study Prym varieties and their moduli space using the well-known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over [Formula: see text]); a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinte Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of view.

On Classification of Filiform Leibniz Algebras

2015 ◽  
Vol 22 (spec01) ◽  
pp. 757-774 ◽  
Author(s):  
J.R. Gómez ◽  
B.A. Omirov
Keyword(s):  
Lie Algebra ◽  
Arbitrary Dimension ◽  
Classification Problem ◽  
Graded Algebra ◽  
Leibniz Algebras ◽  
Special Basis

In this work we prove that in classifying of filiform Leibniz algebras whose naturally graded algebra is a non-Lie algebra, it suffices to consider some special basis transformations. Moreover, we derive a criterion for two such Leibniz algebras to be isomorphic in terms of such transformations. The classification problem of filiform Leibniz algebras whose naturally graded algebra is non-Lie in an arbitrary dimension, is reduced to the investigation of the conditions obtained.

scholarly journals On split Lie color triple systems

2019 ◽  
Vol 17 (1) ◽  
pp. 267-281
Author(s):  
Yan Cao ◽  
Jian Zhang ◽  
Yunan Cui
Keyword(s):  
Root System ◽  
Triple System ◽  
Natural Generalization ◽  
Maximal Length ◽  
Base Field ◽  
Triple Systems

Abstract In order to begin an approach to the structure of arbitrary Lie color triple systems, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Lie color triple systems as the natural generalization of split Lie triple systems. By developing techniques of connections of roots for this kind of triple systems, we show that any of such Lie color triple systems T with a symmetric root system is of the form T = U + ∑[α]∈Λ1/∼ I[α] with U a subspace of T0 and any I[α] a well described (graded) ideal of T, satisfying {I[α], T, I[β]} = 0 if [α] ≠ [β]. Under certain conditions, in the case of T being of maximal length, the simplicity of the triple system is characterized.

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