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My question here is very computational. My problem is in mathematical physics, so I want to ask the community what kind of software they use to do the following computation if there is any? Let $$L_{n}=-\frac{n+1}{s} (u+v)n\frac{\partial}{\partial n} +\sum_{j=1}^{\infty} p_j (n+j)\frac{\partial}{\partial p_{n+j}}+\sum_{i+j=n}ij\frac{\partial^2}{\partial p_i \partial p_j} $$

For $n,m\geq 0$ it satisfies following equation $$[L_m,L_n]=(m-n)L_{m+n} $$ where $[]$ denote the commutation bracket in Weyl algebra. That is $[\frac{\partial}{\partial p_{j}}, p_j]=1$ other wise all other combination commute. Notice that there infinitely many $p_i$.

Andrews
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    You might have meant to ask this in [math.se]? This is a site for the computing software Mathematica, and so asking about other computing software is off-topic. –  Feb 27 '20 at 05:11
  • However, if you have tried to prove this identity using Mathematica software and encountered difficulties, please add your code to the question and ask for assistance. – bbgodfrey Feb 27 '20 at 16:28
  • You want a software in which you can evaluate commutators? Mathematica can do this with the NCAlgebra package. Otherwise you can do these commutators by hand quite easily (at least symbolically, the details will depend on the domain of the differential operators). – s.harp Feb 27 '20 at 16:56

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You can do this type of computation using the Maple Physics package; it supports commutator algebras, algebraic differential operators, tensorial non-commutative operators, etc. An example of a similar kind of calculation is presented in "Quantum Runge-Lenz Vector and the Hydrogen Atom, the hidden SO(4) symmetry", a step-by-step demonstration departing from basic principles using the kind of algebras and manipulation with differential operators you are asking about; at the end of it, there is a link to a PDF file with the steps visible. That post can give you an idea of how to formulate your problem.

Edgardo S. Cheb-Terrab
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