On Friday, 5 April 2013 13:26:56 UTC+1, Matthew wrote: > > I know nothing about Lie Algebras (well, maybe I could make a loose > definition). From an outsider's perspective I'd like to know the > following. These questions are general to any project. They reflect my > personal interests, not necessarily the interests of the community > > 1. What is the computational crux of this problem? How are you planning > to address this? > It will probably be the implementation of the Weyl group;
> 2. Does this enable efficient solutions to other sorts of problems that > SymPy might want to support? > Well, Lie algebras are intimately related to lie groups through the exponential action. It is much easier to work with Lie algebras than with Lie groups. Lie algebras and algebras have many applications in symmetry (differential geometry, quantum physics, differential equations, etc). Have a module to work with Lie algebras would be invaluable in many areas of SymPy. > 3. What is likely to be the hardest part of developing this? Of what are > you not yet certain? > Likely implementing the Weyl groups and a group action on the weight lattice. This would also have applications to SymPy's work on groups. 4. Are there more general forms of this problem that we haven't yet > developed? If so how could you refactor your design to address these > general issues first and then build your system on top of them. > Well, I'd like to include an abstract bracket for general computations with lie algebras. One of the books I'm reading has several algorithms that would be useful for that (computing the normaliser and centraliser, determining if it is simple, etc). I think that the rest of the work would allow for the computation of the root system of a user-given Lie algebra, and then compare it with the root systems of exception Lie algebras and look for a match. > 5. Convince me, a general member of the scientific community, that I > should care about this project. > Lie algebras are everywhere, and they frequently come up in mathematical research. My research explicitly uses them (and I do a mix of mathematical physics and algebra). Recently, it's been shown that finite cluster algebras are in a 1-1 correspondence with the exception Lie algebras. Have a module that allows users to work with the properties unique to each Lie algebra (root system, Cartan matrix, Weyl group, etc) will be useful in computations and in general research. > > > On Fri, Apr 5, 2013 at 7:06 AM, Mary Clark <[email protected]<javascript:> > > wrote: > >> >> >> On Friday, 5 April 2013 01:26:10 UTC+1, David Joyner wrote: >>> >>> A couple of questions: >>> How would you represent the Weyl gp? As a permutation group? >>> >> >> Yes, as a permutation group. They are most easily expressed this way; >> for example the Weyl group of sln is just Sn, and for the other exceptional >> Lie algebras it easy enough to express them as permutation groups. Also in >> the WeylGroup class, I'd like to make the weight lattice accessible. >> >> How would you represent an element of a root system? As both >>> a dict or a vector (or just one or the other)? >>> >> >> At this point, I'm not 100% sure. The entire root system would either be >> a dict or a vector. If it was a vector, it would be easy to do >> calculations (i.e. like ARootSystem[0] + ARootSystem[1]), but a dict would >> be more valuable in displaying the roots for the user to see (ARootSystem = >> [1: (1,-1,0,0), 2: (0,1,-1,0), 3:(0,0,1,-1)], etc). >> >> Would you implement a group action on the root lattice? >>> >> >> Yes; this would help with functionality for the Weyl group. >> >> >>> Would you implement a Lie algebra as a vector space over QQ >>> with a bracket operation? If not, how? >>> >> >> Ideally, yes. I'd like to implement an abstract bracket operation which >> the user could specify, adn then sympy would check if it satisfies >> bilinearity, the Jacobi identity, etc and output if it is a Lie bracket as >> a boolean. >> >>> >>> >>> >>> On Thu, Apr 4, 2013 at 7:37 PM, Mary Clark <[email protected]> wrote: >>> >>>> Hello all, >>>> >>>> I've been working on and thinking about my proposal for a Lie Algebra >>>> module. Ideally I'd like to have the following classes: >>>> >>>> >>>> - Cartan Type (eg A4, B3, etc) >>>> - subclasses would implement Dynkin diagrams and the Cartan >>>> matrix >>>> - also include functions to return the rank of the given Lie >>>> algebra, and whether or not it is finite >>>> - Weyl group >>>> - This would return the Weyl group of a given classical Lie >>>> algebra, as well as be able to return the simple reflections >>>> - RootSystem >>>> - Output the root system of a given Cartan type >>>> - Keep the roots either in a dictionary or an array to allow the >>>> user to perform operations with the roots >>>> >>>> I'd also like to have files with the basic information about for the >>>> different classical Lie algebras (A,B,C,D,E,F,G). One source that is >>>> particularly useful is Lie Algebras: Theory and Algorithm by W.A. de >>>> Graaf. I also own several other books on Weyl groups and Lie algebras in >>>> general that I think would be quite useful for this project. >>>> >>>> I was hoping others could provide feedback on what I have so far for >>>> this project! >>>> >>>> -- >>>> You received this message because you are subscribed to the Google >>>> Groups "sympy" group. >>>> To unsubscribe from this group and stop receiving emails from it, send >>>> an email to sympy+un...@**googlegroups.com. >>>> To post to this group, send email to [email protected]. >>>> >>>> Visit this group at >>>> http://groups.google.com/**group/sympy?hl=en-US<http://groups.google.com/group/sympy?hl=en-US> >>>> . >>>> For more options, visit >>>> https://groups.google.com/**groups/opt_out<https://groups.google.com/groups/opt_out> >>>> . >>>> >>>> >>>> >>> >>> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected] <javascript:>. >> To post to this group, send email to [email protected] <javascript:> >> . >> Visit this group at http://groups.google.com/group/sympy?hl=en-US. >> For more options, visit https://groups.google.com/groups/opt_out. >> >> >> > > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy?hl=en-US. For more options, visit https://groups.google.com/groups/opt_out.
