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.

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.

(ANTI)COMMUTATIVE ALGEBRAS WITH A MULTIPLICATIVE BASIS

2014 ◽  
Vol 91 (2) ◽  
pp. 211-218 ◽  
Author(s):  
ANTONIO J. CALDERÓN MARTÍN
Keyword(s):  
Mild Condition ◽  
Base Field ◽  
Commutative Algebras ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family ◽  
Anticommutative Algebra

AbstractA basis ${\mathcal{B}}=\{u_{i}\}_{i\in I}$ of a commutative or anticommutative algebra $\mathfrak{C},$ over an arbitrary base field $\mathbb{F}$, is called multiplicative if for any $i,j\in I$ we have that $u_{i}u_{j}\in \mathbb{F}u_{k}$ for some $k\in I$. We show that if a commutative or anticommutative algebra $\mathfrak{C}$ admits a multiplicative basis then it decomposes as the direct sum $\mathfrak{C}=\bigoplus _{j}\mathfrak{i}_{j}$ of well-described ideals each one of which admits a multiplicative basis. Also the minimality of $\mathfrak{C}$ is characterised in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is indexed by the family of its minimal ideals admitting a multiplicative basis.

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.

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.

A note on the right-left symmetry of aR ⊕ bR = (a + b)R in rings

2021 ◽  
pp. 2250145
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Direct Sum ◽  
Sum Formula ◽  
Left And Right ◽  
The Right

We give an example to show that, for nonunital rings [Formula: see text], the direct sum [Formula: see text] with [Formula: see text] regular has no in general right-left symmetry. It is then proved that the right-left symmetry actually holds in a left and right faithful ring.

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.

Split Lie algebras of order 3

2019 ◽  
Vol 19 (04) ◽  
pp. 2050070
Author(s):  
Antonio J. Calderón ◽  
Rosa M. Navarro ◽  
José M. Sánchez
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Linear Subspace ◽  
Type Theorem ◽  
Natural Generalization ◽  
Lie Superalgebras ◽  
Direct Sum ◽  
The Family ◽  
Split Lie Algebra

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form [Formula: see text] with [Formula: see text] a linear subspace of [Formula: see text] and any [Formula: see text] a well-described (split) ideal of [Formula: see text] satisfying [Formula: see text], with [Formula: see text], if [Formula: see text]. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that [Formula: see text] is the direct sum of the family of its (split) simple ideals) is stated.

The Transformation of Commutative Phase Space to Noncommutative One, and its Lorentz Transformation-Like Forms

2021 ◽  
pp. 188-198
Author(s):  
Makoto Nakamura ◽  
Hiroshi Kakuhata ◽  
Kouichi Toda
Keyword(s):  
Phase Space ◽  
Lorentz Transformation ◽  
Linear Transformation ◽  
Arbitrary Dimension ◽  
Two Dimensional ◽  
Exact Form ◽  
Noncommutative Phase Space ◽  
Two Dimension ◽  
Direct Sum

Noncommutative phase space of arbitrary dimension is discussed. We introduce momentum-momentum noncommutativity in addition to co-ordinate-coordinate noncommutativity. We find an exact form for the linear transformation which relates a noncommutative phase space to the corresponding ordinary one. By using this form, we show that a noncommutative phase space of arbitrary dimension can be represented by the direct sum of two-dimensional noncommutative ones. In two-dimension, we obtain the transformation which relates a noncommutative phase space to commutative one. The transformation has the Lorentz transformation-like forms and can also describe the Bopp's shift.

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.

WEIGHT MODULES OVER SPLIT LIE ALGEBRAS

2013 ◽  
Vol 28 (05) ◽  
pp. 1350008 ◽  
Author(s):  
ANTONIO J. CALDERÓN MARTÍN ◽  
JOSÉ M. SÁNCHEZ-DELGADO
Keyword(s):  
Lie Algebras ◽  
Weight Module ◽  
Base Field ◽  
The Family ◽  
Weight Modules

We study the structure of weight modules V with restrictions neither on the dimension nor on the base field, over split Lie algebras L. We show that if L is perfect and V satisfies LV = V and [Formula: see text], then [Formula: see text] with any Ii an ideal of L satisfying [Ii, Ik] = 0 if i ≠k and any Vj a (weight) submodule of V in such a way that for any j ∈J there exists a unique i ∈I such that IiVj ≠0, being Vj a weight module over Ii. Under certain conditions, it is shown that the above decomposition of V is by means of the family of its minimal submodules, each one being a simple (weight) submodule.

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