Matrix Lie Algebra Research Articles

An equation for disentangling time-ordered exponentials with arbitrary quadratic generators

In many quantum-mechanical constructions, it is necessary to disentangle an operator-valued time-ordered exponential with time-dependent generators quadratic in the creation and annihilation operators. By disentangling, one understands the finding of the matrix elements of the time-ordered exponential or, in a more general formulation. The solution of the problem can also be reduced to calculation of a matrix time-ordered exponential that solves the corresponding classical problem. However, in either case the evolution equations in their usual form do not enable one to take into account explicitly the symmetry of the system. In this paper the methods of Weyl analysis are used to find an ordinary differential equation on a matrix Lie algebra that is invariant with respect to the adjoint action of the dynamical symmetry group of a quadratic Hamiltonian and replaces the operator evolution equation for the Green's function.

Identities in matrix Lie algebras

Identities in matrix Lie algebras

The homology of matrix Lie algebras over rings and the Hochschild homology

The homology of matrix Lie algebras over rings and the Hochschild homology

A hyperbolic GCM Lie algebra and the Fibonacci numbers

The Fibonacci numbers are found to be involved with the Weyl-Macdonald-Kac denominator formula for a certain rank 2 Generalized Cartan Matrix Lie algebra.

Lie Algebra Multiplicities

Exact formulas for root space multiplicities in Cartan matrix Lie algebras and their universal enveloping algebras are computed. We go on to determine the number of free generators of each degree of the radicals defining these algebras.

Lie algebra multiplicities

Exact formulas for root space multiplicities in Cartan matrix Lie algebras and their universal enveloping algebras are computed. We go on to determine the number of free generators of each degree of the radicals defining these algebras.

Lie algebras and classical partition identities

In this paper we interpret Macdonald's unspecialized identities as multivariable vector partition theorems and we relate the well-known Rogers-Ramanujan partition identities to the Weyl-Kac character formula for an infinite-dimensional Euclidean generalized Cartan matrix Lie algebra.