Algebraic structure and Poisson's theory of mechanico-electrical systems

This paper is a contribution to the structure and representation theory of finitedimensional cylindric and polyadic equality algebras.The main result is that every singulary algebra is representable, where an algebra is singular y if it is generated by elements supported by singletons.In particular, all prime algebras are representable (a prime algebra has no proper subalgebra, and every cylindric or polyadic equality algebra has a prime subalgebra).Since every polyadic algebra can be embedded in a polyadic equality algebra the results also apply to them.Every infinite-dimensional singulary algebra is locally finite and hence is known to be representable, so we shall be concerned just with the finite-dimensional case, where previously very little was known about representation.It may be mentioned that for any dimension greater than one there are nonrepresentable cylindric and polyadic equality algebras(2).For some special kinds of singulary algebras we obtain a direct construction of the representation, which brings out the structure of the algebra clearly.The methods used essentially constitute an algebraic version of Behmann's solution of the decision problem for the singulary predicate calculus with equality(3).Since only finitely many variables are available, the algebraization has some novel features.The essential ideas can be seen in the case of prime algebras, for which no advanced results about cylindric or polyadic algebras are needed.In the general case, however, use is made of the fact that a cylindric algebra is representable if every finitely generated subalgebra of it is.In the last section of the paper we discuss the logical counterpart of the methods and results of the algebraic part of the paper.1. Introduction.A cylindric algebra of dimension a (a CAa) is a system 21 = <[A, +,-,-, cK,dKXyKtX<x such that (A, +,-, -> is a Boolean algebra, a is an ordinal, cK is a singulary Presented to the Society, November 17,1962 under the title On finite dimensional cylindric algebras; received by the editors April 2, 1963.0) The research herein reported on was supported by

The basic structure and representation theory of finite-dimensional simple Lie algebras over \(\mathbb{C} \), as developed by W. Killing, E. Cartan and H. Weyl, is a well-known and frequently used branch of mathematics. It is a striking fact that very significant new results in this classical area of mathematics have come to light during the past decade. These new results have originated from the development of the theory of quantum groups. The purpose of this article is to survey briefly the classical results on the representation theory of simple Lie algebras and then to describe the recent developments primarily due to G. Lusztig, M. Kashiwara and P. Littelmann.

This book presents an introduction to the structure and representation theory of modular Lie algebras over fields of positive characteristic. It introduces the beginner to the theory of modular Lie algebras and is meant to be a reference text for researchers.

We have studied the algebraic structure of the dynamical equations of a rotational relativistic Birkhoff system. It is proven that autonomous and semi-autonomous rotational relativistic Birkhoff equations possess consistent algebraic structure and Lie algebraic structure. In general, non-autonomous rotational relativistic Birkhoff equations possess no algebraic structure, but a type of special non-autonomous rotational relativistic Birkhoff equation possesses consistent algebraic structure and consistent Lie algebraic structure. Then, we obtain the Poisson integrals of the dynamical equations of the rotational relativistic Birkhoff system. Finally, we give an example to illustrate the application of the results.

Algebraic structure and Poisson method for a weakly nonholonomic system

In this paper a criterion of undecidability of theories based on algorithmic logic [1,8] is formulated. By the application of this criterion we are able to assert undecidability of algorithmic theory finite fields, theories of date structures e.g. dictionaries, storage management system and others theories of structures with finite universes. It is also proved that every decidable algorithmic theory admits elimination of iteration quantifiers and examples of decidable algorithmic theories are given.

Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras

Maximal subalgebras and chief factors of Lie algebras

This paper focuses on the relevance of a certain class of left-invariant Hamiltonians (affine-quadratic) on a reductive semi-simple Lie algebra g for the theory of integrable systems and the equations of applied mathematics.Any semi-simple Lie group G that contains a closed subgroup K is reductive, in the sense, that the orthogonal complement p in g of the Lie algebra k of K , relative to the Killing form, satisfies [k, p] p.This implies that K acts (by adjoint action) on p and therefore induces the semi-direct product p K .Consequently, g , as a vector space, carries two Lie algebra structures: semi-simple, and semi-direct.Hence, the dual g * carries two Poisson structures as well.Any affine-quadratic function H on g can be simultaneously viewed as a Hamiltonian for either Poisson structure.We will show that certain coadjoint orbits relative to the semi-direct action are the cotangent bundles of SO(n).This implies that the equations of an n -dimensional top can be represented on such coadjoint orbits.In this situation there is a canonical affine-quadratic Hamiltonian whose Hamiltonian equations on these coadjoint orbits coincide with the equations of the top.This implies that the integrable cases of the top correspond to the the integrable cases of the overseeing affine Hamiltonian.More generally, we will identify a subclass of affine-quadratic Hamiltonians, called isospectral, that provides new insights into the theory of integrable systems based on the contributions of S. V. Manakov, A. T. Fomenko, A. S. Mischenko and O. Bogoyavlensky listed in the references.

One fruitful motivating principle of much research on the family of integrable systems known as “Toda lattices” has been the heuristic assumption that the periodic Toda lattice in an affine Lie algebra is directly analogous to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper shows that the analogy is not perfect. A discrepancy arises because the natural generalization of the structure theory of finite-dimensional simple Lie algebras is not the structure theory of loop algebras but the structure theory of affine Kac-Moody algebras. In this paper we use this natural generalization to construct the natural analog of the nonperiodic Toda lattice. Surprisingly, the result is not the periodic Toda lattice but a new completely integrable system on the periodic Toda lattice phase space. This integrable system is prescribed purely in terms of Lie-theoretic data. The commuting functions are precisely the gauge-invariant functions one obtains by viewing elements of the loop algebra as connections on a bundle over S.

The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.

The purpose of this paper is to provide a general algebraic and geometric structure for the set of matrices. First, the identity equivalence is introduced to the set. Then two operations STA and STP are introduced, which are the generalizations of conventional matrix addition and matrix product respectively. Using them, the set of matrices is partitioned into several vector spaces. A topological structure is proposed to make each vector space a second countable, Hausdorff space. Then a differential structure is posed to make each vector space a differential manifold with multi-dimensions. Finally, the multi-dimensional Lie group and Lie algebra structures are obtained for the set of all square matrices. Some applications are also explored.

In (2009), Towers [10] presented the notion of c-ideality of a subalgebra of a Lie algebra, and gave some characterizations of solvable and supersolvable Lie algebras. In this article, we further investigate the influence of c-ideality of some subalgebras on the structure of Lie algebras. We also obtain some equivalent conditions for supersolvability of a finite dimensional Lie algebra.

Non-Euclidean geometry and Weierstrassian mathematics: The background to Killing's work on Lie algebras

Let $$\mathfrak{g }$$ be a Lie algebra, $$E$$ a vector space containing $$\mathfrak{g }$$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $$E$$ such that $$\mathfrak{g }$$ is a Lie subalgebra of $$E$$ . A general product, called the unified product, is introduced as a tool for our approach. Let $$V$$ be a complement of $$\mathfrak{g }$$ in $$E$$ : the unified product $$\mathfrak{g } \,\natural \, V$$ is associated to a system $$(\triangleleft , \, \triangleright , \, f, \{-, \, -\})$$ consisting of two actions $$\triangleleft $$ and $$\triangleright $$ , a generalized cocycle $$f$$ and a twisted Jacobi bracket $$\{-, \, -\}$$ on $$V$$ . There exists a Lie algebra structure $$[-,-]$$ on $$E$$ containing $$\mathfrak{g }$$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $$(E, [-,-]) \cong \mathfrak{g } \,\natural \, V$$ . All such Lie algebra structures on $$E$$ are classified by two cohomological type objects which are explicitly constructed. The first one $$\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$$ will classify all Lie algebra structures on $$E$$ up to an isomorphism that stabilizes $$\mathfrak{g }$$ while the second object $$\mathcal{H }^{2} (V, \mathfrak{g })$$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $$\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$$ and $$\mathcal{H }^{2} (V, \mathfrak{g })$$ are worked out in detail in the case of flag extending structures.