Born geometry on ρ-commutative algebra

2020 ◽  
Vol 17 (14) ◽  
pp. 2050210
Author(s):  
Zahra Bagheri ◽  
Esmaeil Peyghan
Keyword(s):  
Commutative Algebra ◽  
Riemannian Geometry ◽  
Fundamental Theorem ◽  
Existence And Uniqueness ◽  
Commutative Algebras

The aim of this paper is to establish a generalization of the Born geometry to [Formula: see text]-commutative algebras. We introduce the notion of Born [Formula: see text]-commutative algebras and study the existence and uniqueness of a torsion connection which preserves the Born structure. Also, an analogue of the fundamental theorem of Riemannian geometry will be proved for these algebras.

scholarly journals Riemannian Structures on Z 2 n -Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1469
Author(s):  
Andrew James Bruce ◽  
Janusz Grabowski
Keyword(s):  
Riemannian Geometry ◽  
Fundamental Theorem ◽  
Scalar Product ◽  
Riemannian Metric ◽  
Theoretical Framework ◽  
Similarities And Differences

Very loosely, Z2n-manifolds are ‘manifolds’ with Z2n-graded coordinates and their sign rule is determined by the scalar product of their Z2n-degrees. A little more carefully, such objects can be understood within a sheaf-theoretical framework, just as supermanifolds can, but with subtle differences. In this paper, we examine the notion of a Riemannian Z2n-manifold, i.e., a Z2n-manifold equipped with a Riemannian metric that may carry non-zero Z2n-degree. We show that the basic notions and tenets of Riemannian geometry directly generalize to the setting of Z2n-geometry. For example, the Fundamental Theorem holds in this higher graded setting. We point out the similarities and differences with Riemannian supergeometry.

FUZZY MEALY MACHINES: HOMOMORPHISMS, ADMISSIBLE RELATIONS AND MINIMAL MACHINES

1996 ◽  
Vol 04 (01) ◽  
pp. 27-43 ◽  
Author(s):  
J.N. MORDESON ◽  
P.S. NAIR
Keyword(s):  
Group Theory ◽  
Fundamental Theorem ◽  
Existence And Uniqueness ◽  
Normal Subgroups

Homomorphisms and admissible relations of fuzzy Mealy machines are studied. Admissible relations play a role similar to normal subgroups in group theory. The kernel of a homomorphism is shown to be an admissible relation. Conversely, corresponding to an admissible relation, there exists a homomorphism. The fundamental theorem on homomorphisms; and the existence and uniqueness of minimal machines are also presented.

scholarly journals TROPICAL ALGEBRAIC SETS, IDEALS AND AN ALGEBRAIC NULLSTELLENSATZ

2008 ◽  
Vol 18 (06) ◽  
pp. 1067-1098 ◽  
Author(s):  
ZUR IZHAKIAN
Keyword(s):  
Algebraic Geometry ◽  
Classical Theory ◽  
Commutative Algebra ◽  
Fundamental Theorem ◽  
Polynomial Algebra ◽  
Algebraic Sets ◽  
Tropical Semiring ◽  
Fundamental Theorem Of Algebra

This paper introduces the foundations of the polynomial algebra and basic structures for algebraic geometry over the extended tropical semiring. Our development, which includes the tropical version for the fundamental theorem of algebra, leads to the reduced polynomial semiring — a structure that provides a basis for developing a tropical analogue to the classical theory of commutative algebra. The use of the new notion of tropical algebraic com-sets, built upon the complements of tropical algebraic sets, eventually yields the tropical algebraic Nullstellensatz.

Rotation fields and the fundamental theorem of Riemannian geometry in

2006 ◽  
Vol 343 (6) ◽  
pp. 415-421 ◽  
Author(s):  
Philippe G. Ciarlet ◽  
Liliana Gratie ◽  
Oana Iosifescu ◽  
Cristinel Mardare ◽  
Claude Vallée
Keyword(s):  
Riemannian Geometry ◽  
Fundamental Theorem

scholarly journals GL-EQUIVARIANT MODULES OVER POLYNOMIAL RINGS IN INFINITELY MANY VARIABLES. II

2019 ◽  
Vol 7 ◽  
Author(s):  
STEVEN V SAM ◽  
ANDREW SNOWDEN
Keyword(s):  
Commutative Algebra ◽  
Polynomial Rings ◽  
Commutative Algebras ◽  
Representation Stability ◽  
Veronese Embeddings ◽  
Twisted Commutative Algebras

Twisted commutative algebras (tca’s) have played an important role in the nascent field of representation stability. Let $A_{d}$ be the tca freely generated by $d$ indeterminates of degree 1. In a previous paper, we determined the structure of the category of $A_{1}$-modules (which is equivalent to the category of $\mathbf{FI}$-modules). In this paper, we establish analogous results for the category of $A_{d}$-modules, for any $d$. Modules over $A_{d}$ are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra.

scholarly journals Nonnoetherian geometry

2016 ◽  
Vol 15 (09) ◽  
pp. 1650176 ◽  
Author(s):  
Charlie Beil
Keyword(s):  
Commutative Algebra ◽  
Projective Dimension ◽  
Krull Dimension ◽  
Maximal Dimension ◽  
Algebraic Varieties ◽  
Finitely Generated ◽  
One Dimensional ◽  
Simple Modules ◽  
Commutative Algebras

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of [Formula: see text] are all isomorphic if and only if [Formula: see text] is noetherian, if and only if the center [Formula: see text] of [Formula: see text] is noetherian, if and only if [Formula: see text] is a finitely generated [Formula: see text]-module. Furthermore, we show that [Formula: see text] is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.

Embedding Theorems for L-Algebras and Anti-commutative Algebras

2019 ◽  
Vol 26 (01) ◽  
pp. 51-64
Author(s):  
Qiuhui Mo
Keyword(s):  
Associative Algebra ◽  
Commutative Algebra ◽  
Embedding Theorems ◽  
Commutative Algebras

Bokut, Chen and Huang proved that every countably generated L-algebra over a countable field can be embedded into a simple two-generated L-algebra. In this paper, we prove that every countably generated L-algebra can be embedded into a simple two-generated L-algebra. We also prove that every anti-commutative algebra can be embedded into a simple anti-commutative algebra, and that every countably generated anti-commutative algebra can be embedded into a simple two-generated anti-commutative algebra. Finally, we prove that every anti-commutative algebra can be embedded into its universal enveloping non-associative algebra.

Geodesics and Jacobi fields in singular semi-riemannian geometry

1994 ◽  
Vol 446 (1928) ◽  
pp. 441-452 ◽  
Keyword(s):  
Singular Point ◽  
Uniqueness Theorem ◽  
Riemannian Geometry ◽  
Existence And Uniqueness ◽  
Smooth Manifold ◽  
Tensor Field ◽  
Final Section ◽  
Existence And Uniqueness Theorem ◽  
Jacobi Fields ◽  
Neutral Vector

This paper proves an existence and uniqueness theorem for geodesics tangent to a neutral vector at a stable singular point of a smooth symmetric two tensor field g on a smooth manifold M . The final section is devoted to a proof of existence and uniqueness of Jacobi fields along the above mentioned geodesics.

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF THE DISCRETIZED CONTACT ELASTO-PLASTIC PROBLEM WITH ISOTROPIC HARDENING

2000 ◽  
Vol 10 (08) ◽  
pp. 1151-1179 ◽  
Author(s):  
EDUARD ROHAN
Keyword(s):  
Variational Inequalities ◽  
Fundamental Theorem ◽  
Existence And Uniqueness ◽  
Contact Problems ◽  
Isotropic Hardening ◽  
Plastic Strains ◽  
Uniqueness Of Solutions ◽  
Operator Equations

A class of quasistatic contact problems for elasto-plastic bodies with isotropic hardening is considered. The problems involve displacements, plastic strains and plastic multipliers. In the framework of multi-valued operator equations, the existence and uniqueness assertions for discretized reduced subproblems are proved using a fundamental theorem on variational inequalities.

ANTI-COMMUTATIVE ALGEBRAS WITH SKEW-SYMMETRIC IDENTITIES

2009 ◽  
Vol 08 (02) ◽  
pp. 157-180 ◽  
Author(s):  
A. S. DZHUMADIL'DAEV
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Algebra ◽  
Symmetric Polynomial ◽  
Commutative Algebras ◽  
Associative Commutative Algebra ◽  
Symmetric Identities ◽  
Generalized Lie Algebras

Generalizing Lie algebras, we consider anti-commutative algebras with skew-symmetric identities of degree > 3. Given a skew-symmetric polynomial f, we call an anti-commutative algebra f-Lie if it satisfies the identity f = 0. If sn is a standard skew-symmetric polynomial of degree n, then any s4-Lie algebra is f-Lie if deg f ≥ 4. We describe a free anti-commutative super-algebra with one odd generator. We exhibit various constructions of generalized Lie algebras, for example: given any derivations D, F of an associative commutative algebra U, the algebras (U, D ∧ F) and (U, id ∧ D2) are s4-Lie. An algebra (U, id ∧ D3 - 2D ∧ D2) is s'5-Lie, where s'5 is a non-standard skew-symmetric polynomial of degree 5.

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