Classification of compatible left-symmetric conformal algebraic structures on the Lie conformal algebra W(a,b)

2018 ◽  
Vol 46 (12) ◽  
pp. 5381-5398
Author(s):  
Deng Liu ◽  
Yanyong Hong ◽  
Hao Zhou ◽  
Nuan Zhang
Keyword(s):  
Conformal Algebra ◽  
Algebraic Structures

scholarly journals CLASSIFICATION OF GRADED LEFT-SYMMETRIC ALGEBRAIC STRUCTURES ON WITT AND VIRASORO ALGEBRAS

2011 ◽  
Vol 22 (02) ◽  
pp. 201-222 ◽  
Author(s):  
XIAOLI KONG ◽  
HONGJIA CHEN ◽  
CHENGMING BAI
Keyword(s):  
Virasoro Algebra ◽  
Indecomposable Module ◽  
Multiplication Operators ◽  
Algebraic Structures ◽  
Witt Algebra ◽  
Left Multiplication ◽  
Symmetric Algebras ◽  
Nonlinear Math ◽  
Math Phys

We find that a compatible graded left-symmetric algebraic structure on the Witt algebra induces an indecomposable module V of the Witt algebra with one-dimensional weight spaces by its left-multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebraic structures on the Witt algebra are classified. All of them are simple and they include the examples given by [Comm. Algebra32 (2004) 243–251; J. Nonlinear Math. Phys.6 (1999) 222–245]. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebraic structures on the Virasoro algebra. They coincide with the examples given by [J. Nonlinear Math. Phys.6 (1999) 222–245].

scholarly journals THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT

2019 ◽  
Vol 108 (2) ◽  
pp. 262-277 ◽  
Author(s):  
ANDREW D. BROOKE-TAYLOR ◽  
SHEILA K. MILLER
Keyword(s):  
Isomorphism Problem ◽  
Algebraic Structures ◽  
Knot Invariant ◽  
Borel Reducibility ◽  
Special Case

We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.

scholarly journals Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimes

2015 ◽  
Vol 12 (03) ◽  
pp. 1550033 ◽  
Author(s):  
A. Paliathanasis ◽  
M. Tsamparlis ◽  
M. T. Mustafa
Keyword(s):  
Wave Equation ◽  
Gordon Equation ◽  
Invariant Solution ◽  
Lie Symmetries ◽  
Conformal Algebra ◽  
Noether Symmetries ◽  
Bianchi I ◽  
Klein Gordon ◽  
Klein Gordon Equation

In this work we perform the symmetry classification of the Klein–Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein–Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also Noether symmetries for the Klein–Gordon equation. We use these results in order to determine all the potentials in which the Klein–Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein–Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.

scholarly journals On the Classification of Bol-Moufang Type of Some Varieties of Quasi Neutrosophic Triplet Loop (Fenyves BCI-Algebras)

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 427 ◽  
Author(s):  
Tèmítọ́pẹ́ Jaíyéọlá ◽  
Emmanuel Ilojide ◽  
Memudu Olatinwo ◽  
Florentin Smarandache
Keyword(s):  
Research Finding ◽  
Research Work ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Structures ◽  
Associative Algebras ◽  
Necessary And Sufficient

In this paper, Bol-Moufang types of a particular quasi neutrosophic triplet loop (BCI-algebra), chritened Fenyves BCI-algebras are introduced and studied. 60 Fenyves BCI-algebras are introduced and classified. Amongst these 60 classes of algebras, 46 are found to be associative and 14 are found to be non-associative. The 46 associative algebras are shown to be Boolean groups. Moreover, necessary and sufficient conditions for 13 non-associative algebras to be associative are also obtained: p-semisimplicity is found to be necessary and sufficient for a F 3 , F 5 , F 42 and F 55 algebras to be associative while quasi-associativity is found to be necessary and sufficient for F 19 , F 52 , F 56 and F 59 algebras to be associative. Two pairs of the 14 non-associative algebras are found to be equivalent to associativity ( F 52 and F 55 , and F 55 and F 59 ). Every BCI-algebra is naturally an F 54 BCI-algebra. The work is concluded with recommendations based on comparison between the behaviour of identities of Bol-Moufang (Fenyves’ identities) in quasigroups and loops and their behaviour in BCI-algebra. It is concluded that results of this work are an initiation into the study of the classification of finite Fenyves’ quasi neutrosophic triplet loops (FQNTLs) just like various types of finite loops have been classified. This research work has opened a new area of research finding in BCI-algebras, vis-a-vis the emergence of 540 varieties of Bol-Moufang type quasi neutrosophic triplet loops. A ‘Cycle of Algebraic Structures’ which portrays this fact is provided.

Finite irreducible modules of Lie conformal algebras 𝒲(a,b) and some Schrödinger–Virasoro type Lie conformal algebras

2019 ◽  
Vol 30 (06) ◽  
pp. 1950026 ◽  
Author(s):  
Lipeng Luo ◽  
Yanyong Hong ◽  
Zhixiang Wu
Keyword(s):  
Complete Classification ◽  
Conformal Algebra ◽  
Direct Sums ◽  
Conformal Algebras ◽  
Irreducible Modules ◽  
Rank One ◽  
Virasoro Type

Lie conformal algebras [Formula: see text] are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of [Formula: see text]. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schrödinger–Virasoro type Lie conformal algebras [Formula: see text] and [Formula: see text] are characterized.

scholarly journals Classification of finite irreducible conformal modules over Lie conformal algebra \mathcal{W}(a, b, r)

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Wenjun Liu ◽  
◽  
Yukun Xiao ◽  
Xiaoqing Yue
Keyword(s):  
Conformal Algebra

Computation over algebraic structures and a classification of undecidable problems

2016 ◽  
Vol 27 (8) ◽  
pp. 1386-1413
Author(s):  
CHRISTINE GAßNER
Keyword(s):  
Decision Problems ◽  
Algebraic Structures ◽  
Real Numbers ◽  
Arithmetical Hierarchy ◽  
Model Of Computation ◽  
Recursively Enumerable ◽  
Algebraic Properties ◽  
Uniform Model ◽  
Insight Into

We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.

Classification of algebraic structures by work space

1980 ◽  
Vol 11 (1) ◽  
pp. 320-333 ◽  
Author(s):  
Peter M. Winkler
Keyword(s):  
Algebraic Structures ◽  
Work Space
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