Robert Dodier wrote:
> On 11/22/10, Leo Butler <l.butler at ed.ac.uk> wrote:
>
>> A finite
>> dimensional Lie algebra L is a class in this sense, with a number of
>> methods:
>> -a method to determine if x is a basis element (*);
>> -a method that computes the commutator of two basis elements;
>> -a method that determines what a scalar is (**);
>>
>> One can build on top of this. In particular, one would like to add
>> -a method that computes the commutator of two elements (linear
>> combinations of basis elements).
>>
>> And, then add the tensor algebra of L, and that of L^*, and extend
>> the commutator. This can be done with rules, but I think it best to
>> avoid tellsimp and friends (as your example does).
>
> OK. Can you show how you would like to work with such objects?
> I'd like to see what you would consider a convenient, natural
> notation for working with Lie groups. What are some operations
> you'd like to represent?
>
> I gather that, among other things, you'd like to have two or
> more Lie groups to work with at the same time.
> What is a notation that can distinguish them?
>
> best
>
> Robert Dodier
May i bring some comments. Your method builds on the fact that the
commutator acts as a derivation on the universal envelopping algebra (*),
so you expand the commutator of high degree objects using this fact.
The problem i see in this approach is that at the end you get a bunch of
terms each containing at least one commutator (which is fine, perhaps) but
there is absolutely no unicity of such an expression, so that you may miss
many identities that you are searching. If you bring everything to a
canonical form in terms of ordered monomials (PBW) then you know that there
is unicity of such presentation, hence you can ascertain if two apparently
different expressions are equal or different. The price you pay is that you
don't see commutators in the end result. The second problem, as expressed by
Leo is that you don't use the expression of the commutator in terms of
degree one objects (the expression with structure constants) so this
computation doesn't know anyhing about the Lie algebra structure in fact.
Finally may i venture another comment, this bunch of posts clearly show how
the inline documentation of the rules system is deficient. It would be
invaluable if people like you who know how the rules system works in Maxima
really explained the things in the inline doc. At the moment it is complete
gibbersish for the uninitiated, that one at most try to decipher by trial
and error. This is a domain where Mathematica shines, it has a nice rules
system, very well explained in the book, and uses a much simpler and clearer
notation.
(*) For the people who are not used to this concept, this algebra is spanned
by monomials in the elements of the Lie algebra (that is X^nY^m ...) where
X, Y, etc. are in the Lie algebra, but one assumes that there is
multilinearity (one can expand naturally (X+X')^n(Y+Y')^m ...) etc. and
moreover that one can replace XY-YX by the degree 1 element [X,Y].
Technically this means that one takes the quotient of the tensor algebra by
the ideal generated by this relation. Since this relation mixes degree 2 and
degree 1 elements it is not obvious to see what one gets at the end, this is
why it is a theorem (PBW) to characterize the result.
The Lie algebra itself can be characterised by generators and relations
instead of by a basis and the structure constants of the commutators of
basis elements. The description by generators and relations (called Serre
presentation) is more economical and less arbitrary, i suppose it is what
Leo has in mind.
--
Michel Talon