Related papers: Flag Partial Differential Equations and Representa…
Let k be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a…
In this paper we investigate how a typical, large-dimensional representation looks for a complex Lie algebra. In particular, we study the family $\mathfrak{sl}_{r+1}(\mathbb{C})$ of Lie algebras for $r \geq 2$ and derive asymptotic…
We study the affine variety $L_{n}(\mathfrak{g})$ of Lie algebra representations, the collection of all homomorphisms from an arbitrary $n$-dimensional Lie algebra into a fixed real semi-simple Lie algebra $\mathfrak{g}$. Using techniques…
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie…
The objective of this paper is to derive analytical solutions of fractional order Laplace, Poisson and Helmholtz equations in two variables derived from the corresponding standard equations in two dimensions by replacing the integer order…
Unlike the heat equation or the Laplace equation, solutions of the wave equation on general domains have no known stochastic representation. This short note gives a simple solution to this well known problem in arbitrary dimensions. The…
A representation of solutions of the wave equation with two spatial coordinates in terms of localized elementary ones is presented. Elementary solutions are constructed from four solutions with the help of transformations of the affine…
Let g be a finite-dimensional complex semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra z g[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra…
A probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived. Two different strategies have been proposed, both requiring generating suitable random trees combined with a…
We provide the details of an implementation of Fourier techniques for solving second-order linear partial differential equations (with constant coefficients) using a computer algebra system. The general Sturm-Liouville problem for the heat,…
Several classes of self-similar, spherically symmetric solutions of relativistic wave equation with nonlinear term of the form sign(\phi) are presented. They are constructed from cubic polynomials in the scale invariant variable t/r. One…
We construct representation theory of Lie algebras with filtrations. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.
Invariants of the coadjoint representation of two classes of Lie algebras are calculated. The first class consists of the nilpotent Lie algebras $T(M)$, isomorphic to the algebras of upper triangular $M\times M$ matrices. The Lie algebra…
Lie-theoretic structures of type $E_8$ (e.g., Lie groups and algebras, Hecke algebras and Kazhdan-Lusztig cells, ...) are considered to serve as a `gold standard' when it comes to judging the effectiveness of a general algorithm for solving…
We study irreducible *-representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold. All irreducible *-representations are classified for a subclass of flag…
We present a systematic study of symmetries, invariants and moduli spaces of classes of coframes. We introduce a classifying Lie algebroid to give a complete description of the solution to Cartan's realization problem that applies to both…
Many important physical situations such as fluid flows, marine environment, solid-state physics and plasma physics have been represented by shallow water wave equation. In this article, we construct new solitary wave solutions for the…
Reduction algebras (also known as generalized Mickelsson algebras, Zhelobenko algebras, or transvector algebras) are well-studied associative algebras appearing in the representation theory of Lie algebras. In the 1990s, Zhelobenko noted…
By identifying each standard flag with a trivalent Feynman diagram, the corresponding propagators can be read directly from the flag itself. Within the flag representation, the kinematic Jacobi identity (equivalently, the residue theorem on…
The class of the free relativistic covariant equations generated by the fractional powers of the D'Alambertian operator $(\square^{1/n})$ is studied. Meanwhile the equations corresponding to n=1 and 2 (Klein-Gordon and Dirac equations) are…