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Question: Virasoro Algebra

Question:Virasoro Algebra

Mr.W00t 9 Maple
algebra example package integer linear
January 15 2009
1
Implement the Virasoro Algebra and it's representations. The Virasoro Algebra is an infinite dimensional Lie algebra with generators L_n, n::integer satisfying [L_m,L_n]=(m-n)*L_(m+n)+c/12*m*(m^2-1)*KroneckerDelta[m,-n] This is quite easy to implement with the "Physics" package: with(Physics); Setup(quantumop={L},algebrarule={ %Commutator=(m-n)*L[m+n]+c/12*m*(m^2-1)*KroneckerDelta[m,-n]}); What I'm having trouble with is defining vectors in my representation spaces. For example if I have a highest weight vector mu from which the entire representation is generated by the action of the Virasoro algebra. I define mu by mu:=Ket(h) where I'd like h to be the L_0 Eigenvalue. How do I define the action of the Virasoro algebra in mu? I tried define(L[0],'linear',L[0](mu)=h*mu); define(L[n::integer],'linear','conditional'(L[n](mu)=0,n>0)); which works for L_0, but the L_n with n>0 do not map mu to 0. What am I doing wrong.
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