Edwin Woollett wrote:
> I view the jacobi identity as a way to re-express
> a particular simplification. If different
> simplification schemes reduce the starting
> expression down to series to the
> general form:
>
> expr1 = a1*comm(a,b)+a2*comm(b,c) +....
> expr2 = b1*comm(a,b) + b2*comm(b,c) + ....
>
> then, if one asks whether the
> results are equivalent, one should make use somehow
> of the jacobi identity to try to work expr1 into
> expr2, and if one cannot, then the results of the
> simplification are not consisten.
>
>
> ted
In fact the computations you want to do are really computations in the
enveloping algebra of a Lie algebra and express the commutator using
structure constants of the Lie algebra.
A basis of enveloping algebra is given by ordered monomials A_1^{n_1]
A_2^{n_2}.... where A_1, A_2, ... is a basis of the Lie algebra with a
chosen order. This is the Poincar? Birkhoff Witt theorem.
To get any monomial in this form one
has to commute disordered terms such as A_2 A_1 in the form
A_1A_2+[A_2,A_1] and express the commutator using the structure constants.
Of course if one goes this way, that is working in an algebra where
[A,B]=AB-BA, then the Jacobi identity is a triviality which doesn't even
need checking.
As i said last time, if you are working in a particular representation of
the Lie algebra and thus of the enveloping algebra, there are further
relations. For example a nxn matrix necessarily obeys a degree n polynomial
equation, the characteristic polynomial vanishes when plugging the matrix in
it. More simply, the Pauli matrices are of square 1, the gamma matrices of
square 1 or-1, and so on.
Anyways these are the relations that have to be implemented in a rules based
system for doing the computations you mentioned.
--
Michel Talon