Eigenvalues Of Casimir Operators Research Articles

A group method solving many-body systems in intermediate statistical representation

The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediate statistical representation. We give the relation between the conjugacy-class operator of permutation group and the Casimir operator of unitary group in the intermediate statistical representation, called the Gentile representation. Bose and Fermi cases are two limitations of the Gentile representation. We also discuss the representation space of symmetric and unitary group in the Gentile representation and give an example of the Heisenberg model to demonstrate this method. It is shown that this method is effective to solve interacting many-body problems.

Polynomial Realization of slq(2) and Fusion Rules at Exceptional Values of q

Representations of the $s\ell_q(2)$ algebra are constructed in the space of polynomials of real (complex) variable for $q^N=1$. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated.

Canonical realizations of the lie algebras gl( n, R) and sl( n, R) II. Casimir operators

The eigenvalues of Casimir operators in the canonical realization of gl( n, R) and sl( n, R) presented in the first part of this paper are expressed by concrete ploynomials in parameters characterizing the realization and some of their properties are derived.