Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:50:47Z http://mathoverflow.net/feeds/question/85400 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85400/symbolic-computations-with-differential-operators-universal-envelopings-i-e-non Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ? Alexander Chervov 2012-01-11T10:37:30Z 2012-01-11T10:48:23Z <p>Please give suggestions about soft to make symbolic computations with NON-commutative variables.</p> <p>Typical examples I am interesting - Capelli identities <a href="http://en.wikipedia.org/wiki/Capelli" rel="nofollow">http://en.wikipedia.org/wiki/Capelli</a>'s_identity</p> <p>For example let 2x2 matrix X be defined:</p> <p>$(x_{11}~~~ x_{12})$</p> <p>$(x_{21}~~~ x_{22})$</p> <p>and D is defined:</p> <p>$(\partial_{x_{11}}~~~ \partial_{x_{12}})$</p> <p>$(\partial_{x_{21}}~~~ \partial_{x_{22}})$</p> <p>Then there is identity: $det^{column}(XD^t+diag(1,0)) = det(X)det(D)$</p> <p><strong>Question</strong> Is there soft which can easily check it ?</p> <p>The ideal would be if some one can provide example of code checking this thing. (I mean it is true, of course, just to understand how code such things).</p> <p>More general things I would like to do - some computations in universal enveloping of Lie algebras - like check two expressions commute. e.g. check that ef+h^2+h is Casimir for sl(2).</p> <hr> <p>I am familiar with MatLab and Mathematica - but it seems they cannot do this. May be I am wrong ?</p> <p>I know that MatLab can differentiate Diff( p,x) - will give symbolic derivative of symbolic function "p" in x. But it seems MatLab cannot do things like d*x-x*d =1...</p> <p>I have heard that Macaulay2 can do such things - but once I had trouble just installing it, so I am quite afraid of it... May be I am wrong ? Does it have some graphic interface ?</p> http://mathoverflow.net/questions/85400/symbolic-computations-with-differential-operators-universal-envelopings-i-e-non/85402#85402 Answer by Igor Rivin for Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ? Igor Rivin 2012-01-11T10:48:23Z 2012-01-11T10:48:23Z <p>There is a boat-load of mathematica packages for Lie Algebra computations. Some examples are:</p> <p><a href="http://www.equaonline.com/math/SuperLie/SuperLie.pdf" rel="nofollow">SuperLie</a></p> <p><a href="http://library.wolfram.com/infocenter/MathSource/7622/?affilliate=1" rel="nofollow">Quantum Mathematica.</a></p>