Differentiable families of traceless matrix triples

Abstract

Analysis of spectra of families of sets of matrices verifying certain properties is not simple because phenomena as singularities and bifurcations appear. An excellent tool for the analysis can be making use of versal deformations because of the spectrum of the family coincides with the spectrum of its versal deformation. Disposing of a versal deformation is advantageous since any perturbation of an element can be described up to equivalence by its versal deformation, and it gives the possibility to calculate bifurcation diagrams of families of elements in general position. V.I. Arnold constructed versal deformations, of a differentiable family of square matrices under conjugation and his techniques have been generalised to different cases as to matrix pencils under the strict equivalence, for example.

In this paper, we present versal deformations of elements of the Lie algebra consisting of triples of traceless matrices to coefficients on $\mathbb{F} =\mathbb{ C}$ or $\mathbb{R}$, which are simultaneously diagonalizable.

Study families of traceless matrix triples have great interest because the Lie algebra is related to gauge fields because they appear in the Lagrangian describing the dynamics of the field, then they are associated to 1-forms that take values on a certain Lie algebra. It is also of interest to note that triples of traceless matrices have some relevance for supergravity theories. Another application is found when we must give the instanton solution of Yang-Mills field can be presented in an octonion form, and it can be represented by triples of traceless matrices.