Here I present my programs in MATHEMATICA-5.0
where some basic objects in differential geometry of Lie groups are calculated for SU(2) and SU(3) groups in Euler angles parametrization.
Math.rar (168KB) :
- SU(2).nb -- material which concerns SU(2) group,
- aSU3-left.nb -- left invariant vector fields and Maurer-Cartan forms,
- aSU3-right.nb -- right invariant vector fields and Maurer-Cartan forms,
- aSU3-R-matrix.nb -- the adjoint matrix which rotates left
invariant vector fields and Maurer-Cartan forms into the right
ones,
- aSU3-metric.nb -- the invariant metric and some tensors constructed using metric like Riemann curvature tensor, Casimir operators, harmonic forms and so on.
- commutator.nb -- some identities for Gell-Mann matrices and structure constants for SU(3) group, traces of antisymmetrized products of matrices.
All files may be run independently. Warning: some calculations (e.g. the adjoint matrix for SU(3) group) take a long time (a few hours on Celeron-500). Used formulas are simple, they mainly contain exponentiation of matrix, matrix multiplications, taking traces, differentiation and so on. The programs themselves represent protocols of consequent calculations and may be realized, of course, not only in MATHEMATICA, but in every computer system for analytical computation, for example MAPLE, REDUCE and others. The main difficulty is simplification due to the huge amount of trigonometric functions. And this takes the main part of time.
Needed definitions, used formulas for calculation and testing, and results are collected in
PDF-file (257 KB)
Books and articles which cover used material on differential geometry of Lie groups are pointed out in this text.
I thank Mark Byrd (Harvard University) and Arsen Khvedelidze for their help and consultations during this work.