# Non-commutative algebra

I'm constantly dealing with non-commutative algebras. ** is inbuilt, non-commutative and associative. That's good :-) But it is not distributive. Rats.

• What is a simple way (I probably won't need much more) to have, say, (a1 + a2 + a3)**(b1 + b2 + b3) always expand to a1**b1 + ... + a3**b3, on the fly?
• And if I like to add (also executed on the fly) laws like a1**b1 = c1 + d1?
• And, last question, if I did and have a2**a1**b1 (with, say, a2**a1 = e1 forced), do (a2**a1)**b1 and a2**(a1**b1) substitute to e1**b1 and a2**(c1+d1), respectively, or both to e1**b1 due to flatness of **?
• Are you familiar with UpValues? The easiest thing to do would be to create your own symbol (to which you can set your own rules and infix). However, if it is important to still use NonCommutativeMultiply you can in theory unprotect it, then add UpValues, but that is frowned upon and a bit dangerous.
– VF1
Feb 28, 2013 at 15:23
• There is of course Distribute[(a1 + a2 + a3) ** (b1 + b2 + b3)] as well.
– chuy
Feb 28, 2013 at 16:20
• For distribution you should consult the second application of NCM in Mathematica Documentation. Feb 28, 2013 at 18:20

I recommend you try the NCAlgebra package if you are going to do these kinds of computations. It is a mature package that has been under development for many years. The function you are looking for is NCExpand.

• I will never grow tired of upvoting an answer suggesting this package. +1
– gpap
Feb 28, 2013 at 15:45
• THX! Frankly, the existence of such a package doesn't surprise me in the least, but still being a MATHEMATICA n00b, I would have googled minutes for it :-) I'll give it a try. Mar 2, 2013 at 15:20

If you want something lighter-weight than NCAlgebra you can do things like

NonCommutativeMultiply[H___, Plus[a_, addends_], T___] :=  NonCommutativeMultiply[H, a, T] +

NonCommutativeMultiply[H___, a1, b1, T___] := NonCommutativeMultiply[H, c1 + d1, T]


Then (a1 + a2 + a3)**(b1 + b2 + b3) evaluates to

NonCommutativeMultiply[c1] + NonCommutativeMultiply[d1] + a1 ** b2 +
a1 ** b3 + a2 ** b1 + a2 ** b2 + a2 ** b3 + a3 ** b1 + a3 ** b2 + a3 ** b3


where the only "issue" is that the singletons are wrapped with a NonCommutativeMultiply.

NonCommutativeMultiply[H___, a2, a1, T___] := NonCommutativeMultiply[H, e1, T]

then, at least with \$Version="12.3.0 for Mac OS X x86 (64-bit) (May 10, 2021)", your first paren example gives e1**b1 and the other gives a2 ** c1 + a2 ** d1.