Larger Lie Algebra Research Articles

Representations of large Mackey Lie algebras and universal tensor categories

Representations of large Mackey Lie algebras and universal tensor categories

Классификация трехмерных алгебр Ли над по­лем комплексных чисел

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation. Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

Enhanced group analysis of a semi linear generalization of a general bond-pricing equation

The enhanced group classification of a semi linear generalization of a general bond-pricing equation is carried out by harnessing the underlying equivalence and additional equivalence transformations. We employ that classification to unearth the particular cases with a larger Lie algebra than the general case and use them to find non trivial invariant solutions under the terminal and the barrier option condition.

Infinite-dimensional Lie algebras, classical r-matrices, and Lax operators: Two approaches

For each finite-dimensional simple Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g, starting from a general \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}\otimes \mathfrak {g}$\end{document}g⊗g-valued solutions r(u, v) of the generalized classical Yang-Baxter equation, we construct infinite-dimensional Lie algebras \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g-valued meromorphic functions. We outline two ways of embedding of the Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− into a larger Lie algebra with Kostant-Adler-Symmes decomposition. The first of them is an embedding of \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− into Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}(u^{-1},u))$\end{document}g̃(u−1,u)) of formal Laurent power series. The second is an embedding of \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{-}_r$\end{document}g̃r− as a quasigraded Lie subalgebra into a quasigraded Lie algebra \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r$\end{document}g̃r: \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r=\widetilde{\mathfrak {g}}^{-}_r+\widetilde{\mathfrak {g}}^{+}_r$\end{document}g̃r=g̃r−+g̃r+, such that the Kostant-Adler-Symmes decomposition is consistent with a chosen quasigrading. We construct dual spaces \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^*_r$\end{document}g̃r*, \documentclass[12pt]{minimal}\begin{document}$(\widetilde{\mathfrak {g}}^{\pm }_r)^*$\end{document}(g̃r±)* and explicit form of the Lax operators L(u), L±(u) as elements of these spaces. We develop a theory of integrable finite-dimensional hamiltonian systems and soliton hierarchies based on Lie algebras \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}_r$\end{document}g̃r, \documentclass[12pt]{minimal}\begin{document}$\widetilde{\mathfrak {g}}^{\pm }_r$\end{document}g̃r±. We consider examples of such systems and soliton equations and obtain the most general form of integrable tops, Kirchhoff-type integrable systems, and integrable Landau-Lifshitz-type equations corresponding to the Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathfrak {g}$\end{document}g.

Computational Modelling and Similarity Reduction of Equations for Transient Fluid Flow and Heat Transfer with Variable Properties

We consider a system of coupled partial differential equations describing transient fluid flow and heat transfer with variable flow properties. Classical Lie point symmetry analysis of this system resulted in admitted large Lie algebras for some special cases of the arbitrary constants and the source term. Symmetry reductions are performed and as such the system of partial differential equations is reduced to the system of ordinary differential equations. Some reduced ordinary differential equation could be solved exactly with restrictions on the parameters appearing in it. In addition, shooting quadrature is employed to numerically tackle the nonlinear model boundary value problem and pertinent results are presented graphically and discussed quantitatively.

Primordials and primordial Lie algebras

Primordials \({d \in \mathcal{P}}\) are generalizations of ordinals \({\sigma \in \mathcal{O}}\) . Primordials are governed by their succession and precession. Primordials with their succession and precession are of interest in their own right. Remarkably, they also lead directly to certain primordial Lie algebras of set theory. Among these is the large primordial Lie algebra of set theory, whose basis is a class and not a set. The large primordial Lie algebra of set theory generalizes naturally to the large primordial Lie algebras of characteristic p ≥ 2. The simple primordial Lie algebras are the natural primordial Lie algebra \({\mathcal{L}^\natural}\) , the free primordial Lie algebras \({\mathcal{L}^c}\) for r ≥ 1 and r-tuples C of denumerable sequences Cj (1 ≤ j ≤ r) of elements of k, and, for p > 2, the normal sub Lie algebras of the \({\mathcal{L}^\natural,\mathcal{L}^c}\) as well. The split simple primordial Lie algebras are the Lie algebras L of type W—those which may be built directly from the natural primordial Lie algebra \({\mathcal{L}^\natural}\)—except when p = 2 and L is not free. Consequently, they are, up to isomorphism, the purely inseparable forms of the finite and infinite dimensional Lie algebras of type W. This sheds new light on, and adds interest to, the structure of these purely inseparable forms.

Symmetry reductions and solutions for pollutant diffusion in a cylindrical system

This study is devoted to investigating transient coupled fluid flow and mass transfer partial differential equations (PDEs) describing pollutant transport in cylindrical coordinates. Symmetry analysis of the system of coupled PDEs is performed and some large Lie algebras are obtained for some special cases of the arbitrary and special choices of constants, and the source term. Optimal systems are constructed for all the admitted symmetries. We perform reductions for different choices of the source term. In some cases invariant solution is sought, however some cases resulted in coupled systems of highly nonlinear ordinary differential equations (ODEs). Imposing realistic boundary conditions and considering a constant source term, we then use the Adomain decomposition techniques to solve the boundary value problem.

Analysis of a class of potential Korteweg-de Vries-like equations

We analyze a class of third-order evolution equations, i.e. ut = f(x, ux, uxx) uxxx+g(x, ux, uxx) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd.

Large restricted Lie algebras

We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application, we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which can serve as counterexamples for various Burnside-type questions.

Similarity reductions of equations for river pollution

We consider a system of coupled partial differential equations describing pollutant transport in a river system. Symmetry analysis of this system resulted in admitted large Lie algebras for a some special cases of the arbitrary constants and the source term. Furthermore, we construct the one-dimensional optimal systems of the admitted symmetries. However, similarity (invariant) solutions for the system are constructed for some more realistic source term.

A REVIEW OF SYMMETRY ALGEBRAS OF QUANTUM MATRIX MODELS IN THE LARGE N LIMIT

This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and the other for the closed string sector. Physical observables of quantum matrix models in the large N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient algebras of a larger Lie algebra. We will also discuss some properties of these Lie algebras not published elsewhere yet, and briefly review their relationship with well-known algebras like the Cuntz algebra, the Witt algebra and the Virasoro algebra. We will also review how the Yang–Mills theory, various low energy effective models of string theory, quantum gravity, string-bit models, and the quantum spin chain models can be formulated as quantum matrix models. Studying these algebras thus help us understand the common symmetry of these physical systems.

MULTISTRING VERTICES AND HYPERBOLIC KAC-MOODY ALGEBRAS

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac-Moody algebras, in particular E10. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the N-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds more generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain “decoupling polynomials.” This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a nontrivial root space of E10· Because the N-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as E10 by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras.

On the automorphism groups of certain Lie algebras

We fix a ground field k and a finite separable extension K of k. To a Lie algebra L over k is associated the Lie algebra KL = K ⊗kL over K. If we forget the action of K, we can think of KL as a larger Lie algebra over k; in particular we can ask what is the automorphism group Autk KL of KL as a k-algebra. There does not seem to be any simple answer to this question in general; the purpose of this note is to give a simple condition on L which makes Autk KL quite easy to determine. Examples of algebras which satisfy this condition include the free nilpotent Lie algebras and the algebras of all n × n triangular nilpotent matrices.

New exact nonlinear filters with large Lie algebras

The ideas previously used (Stochastics. Vol. 5, pp. 65–92, 1981) to construct some finite-dimensional nonlinear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations. In engineering language, the new filters are obtained from the old by smoothing the output through an n-pole linear system before adding observation noise in the usual way; this adds 2 n to the Lie algebra dimension. In the simplest case (drift = tanh x, observation = x) we put x through a one-pole described by a new variable ζ, and observe ζ + noise instead of x + noise; the new Lie algebra has additional generators ζ and ∂/∂ξ besides the four from the oscillator algebra, to give dimension 6. The filter, which gives a recursive construction of the conditional density, can be ‘derived’ by any of three (here equivalent methods: (i) direct integration of the Kallianpur-Striebel formula as a Gaussian integral; (ii) solution of a parabolic PDE with quadratic potential; and (iii) the Wei-Norman procedure.

New exact non-linear filters by large Lie algebras

The ideas previously used (Beneš (1981)) to construct some finite-dimensional non-linear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations.

New exact non-linear filters by large Lie algebras

The ideas previously used (Beneš (1981)) to construct some finite-dimensional non-linear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations.