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? 2. Does this enable efficient solutions to other sorts of problems that SymPy might want to support? 3. What is likely to be the hardest part of developing this? Of what are you not yet certain? 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. 5. Convince me, a general member of the scientific community, that I should care about this project. On Fri, Apr 5, 2013 at 7:06 AM, Mary Clark <[email protected]>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]. > 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. > > > -- 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.
