scholarly journals UNIVERSAL CALABI–YAU ALGEBRA: TOWARDS AN UNIFICATION OF COMPLEX GEOMETRY

2003 ◽  
Vol 18 (30) ◽  
pp. 5541-5612 ◽  
Author(s):  
F. ANSELMO ◽  
J. ELLIS ◽  
D. V. NANOPOULOS ◽  
G. VOLKOV
Keyword(s):  
Lie Algebras ◽  
Complex Geometry ◽  
Algebraic Approach ◽  
Higher Dimensions ◽  
Normal Algebra ◽  
Natural Extensions ◽  
Recurrence Formulae ◽  
Different Dimensions ◽  
Arbitrary Dimensions

We present a universal normal algebra suitable for constructing and classifying Calabi–Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a "dual" construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi–Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi–Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan–Lie algebras. This Universal Calabi–Yau algebra is a powerful tool for deciphering the Calabi–Yau genome in all dimensions.

scholarly journals UNIVERSAL CALABI–YAU ALGEBRA: CLASSIFICATION AND ENUMERATION OF FIBRATIONS

2003 ◽  
Vol 18 (10) ◽  
pp. 699-710 ◽  
Author(s):  
F. ANSELMO ◽  
J. ELLIS ◽  
G. VOLKOV ◽  
D. V. NANOPOULOS
Keyword(s):  
Algebraic Approach ◽  
Higher Dimensions ◽  
Recurrence Formulas ◽  
Natural Extensions ◽  
Arbitrary Dimensions

We apply a universal normal Calabi–Yau algebra to the construction and classification of compact complex n-dimensional spaces with SU (n) holonomy and their fibrations. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions and a "dual" construction based on the Diophantine decomposition of invariant monomials. The latter provides recurrence formulas for the numbers of fibrations of Calabi–Yau spaces in arbitrary dimensions, which we exhibit explicitly for some Weierstrass and K3 examples.

Extending the Search for Symmetries in the Genetic Code

2003 ◽  
Vol 17 (17) ◽  
pp. 3135-3204 ◽  
Author(s):  
Fernando Antoneli ◽  
Lígia Braggion ◽  
Michael Forger ◽  
José Eduardo M. Hornos
Keyword(s):  
Genetic Code ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
Simple Lie Algebras ◽  
Symmetry Breakings

We report on the search for symmetries in the genetic code involving the medium rank simple Lie algebras [Formula: see text] and [Formula: see text], in the context of the algebraic approach originally proposed by one of the present authors, which aims at explaining the degeneracies encountered in the genetic code as the result of a sequence of symmetry breakings that have occurred during its evolution.

Quantum algebra embeddings: deforming functionals and algebraic approach

1994 ◽  
Vol 72 (7-8) ◽  
pp. 519-526 ◽  
Author(s):  
J. Van der Jeugt
Keyword(s):  
Hypergeometric Function ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
Quantum Algebra ◽  
Physical Models ◽  
Quantum Algebras ◽  
Basic Hypergeometric Function

The study of subalgebras of Lie algebras arising in physical models has been important for many applications. In the present paper we examine the q-deformation of such embeddings; the Lie algebras are then replaced by quantum algebras. Two methods are presented: one based upon deforming functionals, and a direct algebraic approach. A number of examples are given, e.g., [Formula: see text] and [Formula: see text]. For the last example, we give the q-boson construction, and the relevant overlap coefficients are related to a generalized basic hypergeometric function [Formula: see text].

LIE-ALGEBRAIC APPROACH TO THE PROBLEM OF QUASI-EXACT SOLUBILITY IN QUANTUM MECHANICS

1990 ◽  
Vol 05 (23) ◽  
pp. 1891-1899 ◽  
Author(s):  
A. G. USHVERIDZE
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
New Method ◽  
Exactly Solvable Models ◽  
Solvable Models ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Exactly Solvable

A new method of constructing quasi-exactly solvable models of quantum mechanics is proposed. This method is based on the use of infinite-dimensional representations of simple and semi-simple Lie algebras.

Near Integrability in Low Dimensional Gross–Neveu Models

2011 ◽  
Vol 66 (8-9) ◽  
pp. 468-480 ◽  
Author(s):  
Ognyan Christov
Keyword(s):  
Lie Algebras ◽  
Degrees Of Freedom ◽  
Normal Forms ◽  
Kam Theory ◽  
Higher Dimensions ◽  
Low Dimensional ◽  
Three Degrees Of Freedom

Abstract The low-dimensional Gross-Neveu models are studied. For the systems, related to the Lie algebras so(4), so(5), sp(4), sl(3), we prove that they have Birkhoff-Gustavson normal forms which are integrable and non-degenerate in Kolmogorov-Arnold-Moser (KAM) theory sense. Unfortunately, this is not the case for systems with three degrees of freedom, related to the Lie algebras so(6) ~ sl(4), so(7), sp(6); their Birkhoff-Gustavson normal forms are proven to be non-integrable in the Liouville sense. The last result can easily be extended to higher dimensions.

scholarly journals Algebraic approach to Rump’s results on relations between braces and pre-lie algebras

2020 ◽  
pp. 2250054
Author(s):  
Agata Smoktunowicz
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Algebraic Approach ◽  
Geometric Approach ◽  
Prime Characteristic ◽  
Affine Manifolds ◽  
Algebraic Interpretation

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right [Formula: see text]-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump’s correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.

Hypervolume Subset Selection with Small Subsets

2019 ◽  
Vol 27 (4) ◽  
pp. 611-637
Author(s):  
Benoît Groz ◽  
Silviu Maniu
Keyword(s):  
Evolutionary Algorithms ◽  
Subset Selection ◽  
Higher Dimensions ◽  
Efficient Algorithms ◽  
Selection Problem ◽  
Multiobjective Evolutionary Algorithms ◽  
Optimal Subset ◽  
Arbitrary Dimensions ◽  
Hypervolume Indicator

The hypervolume subset selection problem (HSSP) aims at approximating a set of [Formula: see text] multidimensional points in [Formula: see text] with an optimal subset of a given size. The size [Formula: see text] of the subset is a parameter of the problem, and an approximation is considered best when it maximizes the hypervolume indicator. This problem has proved popular in recent years as a procedure for multiobjective evolutionary algorithms. Efficient algorithms are known for planar points ([Formula: see text]), but there are hardly any results on HSSP in larger dimensions ([Formula: see text]). So far, most algorithms in higher dimensions essentially enumerate all possible subsets to determine the optimal one, and most of the effort has been directed toward improving the efficiency of hypervolume computation. We propose efficient algorithms for the selection problem in dimension 3 when either [Formula: see text] or [Formula: see text] is small, and extend our techniques to arbitrary dimensions for [Formula: see text].

An Algebraic Approach to Weakly Symmetric Finsler Spaces

2010 ◽  
Vol 62 (1) ◽  
pp. 52-73 ◽  
Author(s):  
Shaoqiang Deng
Keyword(s):  
Large Class ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
Comparison Theorems ◽  
Finsler Spaces ◽  
Volume Comparison ◽  
Algebraic Description ◽  
Interesting Class ◽  
Weakly Symmetric

AbstractIn this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. Thismeans that reversible non-Berwaldian Finsler spaces with vanishing S-curvaturemay exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.

ON AMINO ACID AND CODON ASSIGNMENT IN ALGEBRAIC MODELS FOR THE GENETIC CODE

2010 ◽  
Vol 24 (04) ◽  
pp. 435-463 ◽  
Author(s):  
FERNANDO ANTONELI ◽  
MICHAEL FORGER ◽  
PAOLA A. GAVIRIA ◽  
JOSÉ EDUARDO M. HORNOS
Keyword(s):  
Amino Acid ◽  
Genetic Code ◽  
Lie Groups ◽  
Lie Algebras ◽  
Finite Groups ◽  
Algebraic Approach ◽  
Lie Superalgebras ◽  
Algebraic Models ◽  
Symmetry Principles

We give a list of all possible schemes for performing amino acid and codon assignments in algebraic models for the genetic code, which are consistent with a few simple symmetry principles, in accordance with the spirit of the algebraic approach to the evolution of the genetic code proposed by Hornos and Hornos. Our results are complete in the sense of covering all the algebraic models that arise within this approach, whether based on Lie groups/Lie algebras, on Lie superalgebras or on finite groups.

scholarly journals Some Topological and Algebraic Features of Symmetric Spaces

2020 ◽  
pp. 144-155
Author(s):  
Um Salama ◽  
Ahmed Abd Alla ◽  
A. Elemam
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Lie Groups ◽  
Lie Algebras ◽  
Symmetric Spaces ◽  
Algebraic Approach ◽  
Root Systems ◽  
Algebraic Properties ◽  
Important Field

In this study, we introduce some approaches, geometrical and algebraic, which help to give further understanding of symmetric spaces. Symmetric space is a very important field for understanding abstract and applied features of spaces. We have introduced Riemannian Manifold, Lie groups and Lie algebras, and some of their topological and algebraic properties, with some concentration on Lie algebras and root systems , which help classification and many applications of symmetric spaces. The paper is an attempt to explain some algebraic features of symmetric spaces and how to get some of their properties using algebraic approach, concluded with some results.

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