A python module for computations with Lie groups, Lie algebras, representations, root systems, and more.
lie is based on the computer algebra package LiE, written by M. A. A. van Leeuwen, A. M. Cohen and B. Lisser in the early 90's. LiE implemented a proprietary scripting language as a wrapper for its mathematical algorithms. While this language is useful for interactive computations and short scripts, python is more expressive and powerful.
lie provides python interfaces for the following objects in the LiE language:
- grp: a Lie group
- vec: a vector of integers
- mat: a 2-dimensional matrix of integers
- pol: a (multivariable) polynomial with integer coefficients
In addition, lie wraps some of these into abstract objects which are more mathematically natural:
- AbstractVec: a vector in the rational span of a root system of some Lie group. This includes roots, weights, and linear combinations.
- weyl: a Weyl group element.
- toral: a toral element.
- rep : a Lie group representation.
lie implements most functions from LiE (see the documentation).
This example calculation shows how the 10 + 5bar + 1 representation of SU(5) contains a single standard model generation.
Python 3.1 (r31:73578, Jun 27 2009, 21:49:46) >>> from lie import *
A4 is SU(5) in Cartan's classifiation, as we can see when we pretty-print.
>>> g = A4; g.pprint() 'SU(5)'
A representation is shown as "group(highest-weight-vector)", But physicists usually label representations by their dimension (the b is for "bar", or "dual").
>>> r = g.fund().dual() + g.alt_tensor(2) + g.trivial() >>> r A4(0,0,0,0) + A4(0,0,0,1) + A4(0,1,0,0) >>> r.pprint() '(1) + (5b) + (10)'
We can see that SU(2)xSU(3) is a subgroup of SU(5). Next let's get the restriction matrix for SU(2)xSU(3) in SU(5), and add the U(1) part by hand:
>>> [h.pprint() for h in g.max_subgrps()] ['SU(4)', 'SO(5)', 'SU(2)xSU(3)'] >>> m = g.res_mat(A1*A2) >>> m = m.transpose().concat(mat([[-2,1,-1,2]])).transpose()
Finally, we decompose r under the group SU(2)xSU(3)xU(1). The first two numbers are the SU(2) and SU(3) rep dimensions, and the third is the U(1) charge.
>>> r.branch(A1*A2*T1, m).pprint() '(1,1,0) + (1,1,6) + (1,3b,-4) + (1,3b,2) + (2,1,-3) + (2,3,1)'
Users of Mac OS 10.6 and above should be able to just type "make" in the root directory. I have not had time to tune the Makefiles for other systems, so other users should try to make and be prepared for some debugging.
Written by David Simmons-Duffin in 2009 for use in:
- D. Poland and D. Simmons-Duffin, "Superconformal Flavor Simplified," arXiv:0910.4585 [hep-ph].
This module basically wraps the C-source code from:
- M. A. A. van Leeuwen, A. M. Cohen and B. Lisser, "LiE, A Package for Lie Group Computations", Computer Algebra Nederland, Amsterdam, ISBN 90-74116-02-7, 1992
C-python bridge made possible by Cython