# 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 **?
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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 '13 at 15:23
There is of course Distribute[(a1 + a2 + a3) ** (b1 + b2 + b3)] as well. – chuy Feb 28 '13 at 16:20
For distribution you should consult the second application of NCM in Mathematica Documentation. – Stefan Feb 28 '13 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.