Mathematica (TM) packages for computing principal decompositions of simple Lie algebras and applications in extended conformal field theories

摘要

In this article, we propose two Mathematica(TM) packages for doing calculations in the domain of classical simple Lie algebras. The main goal of the first package, SimpleLieAlgebras.m, is to determine the principal three-dimensional subalgebra of a simple Lie algebra. The package provides several functions which give some elements related to simple Lie algebras (generators in fundamental and adjoint representation, roots, Killing form, Cartan matrix, etc.). The second package, PrincipalDecomposition.m, concerns the principal decomposition of a Lie algebra with respect to the principal three-dimensional embedding. These packages have important applications in extended two-dimensional conformal field theories. As an example, we present an application in the context of the theory of W-gravity. [References: 18]