Re: structure constants depend on the basis chosen?
- From: "galathaea" <galathaea@xxxxxxxxx>
- Date: 21 Mar 2007 08:12:37 -0700
On Mar 21, 6:03 am, "Barrow" <GRsemi...@xxxxxxxxx> wrote:
Dear all,
The structure constant of a Lie group should be independent of
representations.
But in the book "Lie algebras in particles physics", it says: "The
structure constants depend on what basis we choose in the vector space
of the generators"
There seems to be some conflict?
A representation means that we realize the multiplication table of an
abstract group on a certain vector space. So, an abstract group
element becomes the transformation(or operator) on vectors in the
vector space. Thus, if the structure constants are independent of the
representations, it should be independent of the basis we choose in
the vector space of the generators, right?
but they are not independent
the quote you give
states clearly there is a dependence
you have a lot of options in choosing a basis
any particular choice
will give a new set of structure constants
because the basis vectors will be different
that doesn't mean the multiplication will give new results
it simply means that the elements will have new representations
there is
however
an almost canonical basis
for finite dimensional semisimple lie algebras
the "cartan-weyl" basis
( modulo some normalisation and an arbitrary choice )
which gives a canonical set of structure constants
such work does not apply to solvable algebras
for instance
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