Representations Of Yangian Research Articles

Fundamental representations of Yangians and singularities of R-matrices.

Article Fundamental representations of Yangians and singularities of R-matrices. was published on January 1, 1991 in the journal Journal für die reine und angewandte Mathematik (volume 1991, issue 417).

Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras

New combinatorial formulas are obtained for the multiplicities in the decomposition of the tensor product of the representations of simple Lie algebras into irreducible components.

DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS

The quantum nonlinear Schrödinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painlève transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrödinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.