Definition of commutative and non-commutative algebra and algebra isomorphism

Question

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^?

Answer 1

$\mathbb C\langle\langle\ldots\rangle\rangle$ 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.