scholarly journals An algorithm for analysis of the structure of finitely presented Lie algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Vladimir V. Kornyak
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nonlinear Partial Differential Equations ◽  
Practical Importance ◽  
Physical Models ◽  
Generators And Relations ◽  
Defining Relations ◽  
Number Of Generators ◽  
Finite Set ◽  
Finitely Presented

International audience We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.

scholarly journals Finite presentability of some metabelian Hopf algebras

2005 ◽  
Vol 72 (1) ◽  
pp. 109-127 ◽  
Author(s):  
Dessislava H. Kochloukova
Keyword(s):  
Abelian Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Abelian Ideal ◽  
Finite Presentability ◽  
Finitely Presented ◽  
Homological Type

We classify the Hopf algebras U (L)#kQ of homological type FP2 where L is a Lie algebra and Q an Abelian group such that L has an Abelian ideal A invariant under the Q-action via conjugation and U (L/A)#kQ is commutative. This generalises the classification of finitely presented metabelian Lie algebras given by J. Groves and R. Bryant.

scholarly journals The word problem for lattice ordered groups

1975 ◽  
Vol 19 (3) ◽  
pp. 217-219 ◽  
Author(s):  
A. M. W. Glass
Keyword(s):  
Word Problem ◽  
Decision Procedure ◽  
Recursively Enumerable Set ◽  
Finitely Presented Groups ◽  
Defining Relations ◽  
Number Of Generators ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Finite Set ◽  
Naive Approach

The purpose of this note is to show the existence of a lattice ordered group G with a finite set of generators and a recursively enumerable set of defining relations such that there is no decision procedure to determine whether or not an arbitrary word in the generators reduces to the identity in G. In addition to the usual group-theoretic words, we may also use the two lattice operations ∨ and ∧ ; for example, a−l(b∨c) is a word in the generators a, b and c. At first sight it might appear that since we have an even greater harvest of words than in group theory and there exist finitely presented groups H (H has a finite number of generators and defining relations) with an insoluble word problem (no decision procedure to determine whether an arbitrary word in the generators reduces to the identity)—see (1), (2), (4) or (6)—the same would be true of lattice ordered groups. Unfortunately, such a naïve approach overlooks two salient points. First, the class of lattice ordered groups is strictly smaller than the class of all groups; second, there are certain relations connecting the lattice operations with the group operations which hold true for all lattice ordered groups. For example, a(b∧c) = ab∨ac and (a∨b)−1 = a−1∧b−1.

THE WORD PROBLEM FOR SOME VARIETIES OF SOLVABLE LIE ALGEBRAS

1994 ◽  
Vol 04 (03) ◽  
pp. 481-491
Author(s):  
O. KHARLAMPOVICH ◽  
D. GILDENHUYS
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Word Problem ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebras ◽  
Solvable Lie Algebras ◽  
Finitely Presented

The word problem is said to be solvable in a variety of Lie algebras if it is solvable in every algebra, finitely presented in this variety. Let [Formula: see text] denote the variety of (2-step nilpotent)-by-abelian Lie algebra and [Formula: see text] the variety of abelian-by-(2-step nilpotent) Lie algebras. It is proved that the word problem is unsolvable in the “interval” of varieties containing the variety [Formula: see text] (of centre-by-[Formula: see text] Lie algebras over a field of characteristic zero), and contained in the variety [Formula: see text].

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

scholarly journals Structure of n-Lie Algebras with Involutive Derivations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Ruipu Bai ◽  
Shuai Hou ◽  
Yansha Gao
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional

We study the structure of n-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1  ∔  A-1 such that dim A1=2 or dim A-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dim A1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.

scholarly journals COMMUTATORS OF SKEW-SYMMETRIC MATRICES

2005 ◽  
Vol 15 (03) ◽  
pp. 793-801 ◽  
Author(s):  
ANTHONY M. BLOCH ◽  
ARIEH ISERLES
Keyword(s):  
Optimal Control ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Integration ◽  
Symmetric Matrices ◽  
Frobenius Norm

In this paper we develop a theory for analysing the "radius" of the Lie algebra of a matrix Lie group, which is a measure of the size of its commutators. Complete details are given for the Lie algebra 𝔰𝔬(n) of skew symmetric matrices where we prove [Formula: see text], X, Y ∈ 𝔰𝔬(n), for the Frobenius norm. We indicate how these ideas might be extended to other matrix Lie algebras. We discuss why these ideas are of interest in applications such as geometric integration and optimal control.

WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA

2004 ◽  
Vol 15 (10) ◽  
pp. 987-1005 ◽  
Author(s):  
MAHMOUD BENKHALIFA
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Integral Domain ◽  
Free Lie Algebra ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Equivalent Class

Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

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