Definition of commutative and non-commutative algebra and algebra isomorphism I am not sure of the meaning of the notation C<<...>> used to define a commutative algebra A and non commutative algebra A^ in the image attached. I do understand the meaning of the ideal. Also if the commutator in the denominator of A in (2.84) is zero what does C[[...]] mean?
This is a picture with the definitions I am referring to.
Also what does it mean for W to be an algebra isomorphism between A and A^?
 A: $\mathbb C\langle\langle\ldots\rangle\rangle$ denotes the ring of power series in noncommuting variables.
$\mathbb C[[\ldots]]$ denotes the ring of power series in commuting variables.
Frankly I don't know what else could be meant by "$W$ is an isomorphism between $A$ and $\widehat{A}$" other than "$W$ is a bijective ring homomorphism between $A$ and $\widehat{A}$".
That's all I can say given the limited context.

I do understand the meaning of the ideal.

"$\mathcal I$ is the ideal generated by the commutation relations of the coordinate functions"
I would take that to mean the ideal generated by all the relations necessary to describe the difference between $\hat{x}_i\hat{x}_j$ and $\hat{x}_j\hat{x}_i$. I don't know what that is in your particular case (lack of context again). If you include $\hat{x}_i\hat{x}_j-\hat{x}_j\hat{x}_i$, that says the two coordinate functions commute.  But it could also be something like $\hat{x}_i\hat{x}_j-\hat{x}_j\hat{x}_i-i\hbar$ depending on their relationship.
