What is real sense of the non-commutative algebra in the Nature?

Answer 1

Any binary operator operating on two 'things' with different results if the order of the 'things' is reversed. That is, for 'things' a and b and operator * a*b not equal to b*a.

Example: A thing called 'shoe' and a thing called 'sock' along with the operator 'and' which combiunes 'shoe' and 'sock' things. If you play around with it, you will find that the *order* is fairly important.

Another fairly obvious non-commutative operator is 'rotation' over multiple axis.


Doug

Answer 2

rotations of a sphere is a lie group, its associated lie algebra is a non-commutative algebra.

rotations on the plane is also a lie group, but its associated lie algebra is a commutative one.