Related papers: On the eigenvalue problem for arbitrary odd elemen…

In this study, singular diffusion operator with jump conditions is considered. Integral representations have been derived for solutions that satisfy boundary conditions and jump conditions. Some properties of eigenvalues and eigenfunctions…

We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for…

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& \lambda g(x) f(u) \qquad \text{in}\ \Omega \hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $ \lambda >0$ is a parameter and $…

Using the projective oscillator representation of sl(n+1) and Shen's mixed product for Witt algebras, Zhao and the second author (2011) constructed a new functor from sl(n)-Mod to sl(n+1)-Mod. In this paper, we start from n = 2 and use the…

Let G be a connected reductive group over a non-archimedean local field K, and assume that G splits over an unramified extension of K. We establish a local Langlands correspondence for irreducible unipotent representations of G. It comes as…

We derive explicit integral formulas for eigenfunctions of quantum integrals of the Calogero-Sutherland-Moser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack's symmetric functions. To…

An algebraic method is introduced for an analytical solution of the eigenvalue problem of the Tavis-Cummings (TC) Hamiltonian, based on polynomially deformed su(2), i.e. su_n(2), algebras. In this method the eigenvalue problem is solved in…

For any complex reflection group $G=G(m,p,n)$, we prove that the $G$-invariants of the division ring of fractions of the $n$:th tensor power of the quantum plane is a quantum Weyl field and give explicit parameters for this quantum Weyl…

We study various categories of Whittaker modules over the queer Lie superalgebras $\mathfrak q(n)$. We formulate standard Whittaker modules and reduce the problem of composition factors of these standard Whittaker modules to that of Verma…

We discuss a quartic eigenvalue problem arising in the context of an optical waveguiding problem involving atomically thick 2D materials. The waveguide configuration we consider consists of a gradient-index (spatially dependent) dielectric…

A weight basis for each finite-dimensional irreducible representation of the orthogonal Lie algebra o(2n) is constructed. The basis vectors are parametrized by the D-type Gelfand--Tsetlin patterns. Explicit formulas for the matrix elements…

This paper is the detailed version of math.QA/0403477 (T. Arakawa, Quantized Reductions and Irreducible Representations of W-Algebras) with extended results; We study the representation theory of the W-algebra $W_k(g)$ associated with a…

The Lie algebra gl(lambda) dependent on the complex parameter lambda is a continuous version of the Lie algebra gl(inf) of infinite matrices with only finite number of nonzero entries. The gl(lambda) was first introduced by B.L.Feigin in…

In this paper, we investigate finite-dimensional irreducible representations of the quantum affine general linear superalgebra $\mathrm{U}_q\big(\widehat{\mathfrak{gl}}_{m|n,\mathbf{s}}\big)$ for arbitrary 01-sequences $\mathbf{s}$, using…

This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n,…

In this work we obtain the exact solution of quantum integrable system associated with the Lie superalgebra $\mathfrak{gl}(1|1)$, both for periodic and for generic open boundary conditions. By means of the fusion technique we derive a…

We develop an algebraic approach to the branching of representations of the general linear Lie superalgebra $\mathfrak{gl}_{p|q}({\mathbb C})$, by constructing certain super commutative algebras whose structure encodes the branching rules.…

In this paper, we study the existence and uniqueness of solutions to the weighted eigenvalue problem for $k$-Hessian equation. To achieve this, we establish the uniform a priori estimates for gradient and second derivatives of solutions to…

We present a simple algebraic procedure that can be applied to solve a range of quantum eigenvalue problems without the need to know the solution of the Schr\"odinger equation. The procedure, presented with a pedagogical purpose, is based…

The semi-classical limit of the algebraic Bethe Ansatz method is used to solve the theory of Gaudin models. Via the off-shell method we find the spectra and eigenvectors of the N-1 independent Gaudin Hamiltonians with symmetry osp(2|1). We…