# Symbolic computations with differential operators (universal envelopings i.e. non-commutative variables) ?

Please give suggestions about soft to make symbolic computations with NON-commutative variables.

Typical examples I am interesting - Capelli identities http://en.wikipedia.org/wiki/Capelli's_identity

For example let 2x2 matrix X be defined:

$(x_{11}~~~ x_{12})$

$(x_{21}~~~ x_{22})$

and D is defined:

$(\partial_{x_{11}}~~~ \partial_{x_{12}})$

$(\partial_{x_{21}}~~~ \partial_{x_{22}})$

Then there is identity: $det^{column}(XD^t+diag(1,0)) = det(X)det(D)$

Question Is there soft which can easily check it ?

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).

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).

I am familiar with MatLab and Mathematica - but it seems they cannot do this. May be I am wrong ?

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...

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 ?

-