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Connes' book is pretty tough to get through as a beginner. I would suggest as an alternative the book Elements of Noncommutative Geometry by Gracia-Bondia, Varilly, and Figueroa. Or for a more conci …

The $C^*$-algebra versions are treated in this paper by Piotr Soltan: http://arxiv.org/abs/0904.3019 The abstract reads: We prove that some well known compact quantum spaces like quantum tori and so …

I also think this question would benefit from some more detail, however... A very down-to-earth account of K-theory for operator algebras is in the book "K-Theory and C*-algebras" by N.E. Wegge-Olse …

This is a negative answer to your question, or at least a partially negative one. I don't know if you've thought about things in this way, but there is a naive way to formulate the idea of the fundam …

This is shown in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. The result you ask for is Proposition 63 in Chapter 11. I'd expand more upon this but I have to give a ta …

I don't know if you still care, but I think I found the answer to your question. Look at Proposition 1 in Chapter 14 of Quantum Groups and Their Representations by Klimyk and Schmudgen. It shows tha …

There is certainly a way to quantize the algebra of functions on a Lie group in a way that is compatible with the $q$-deformation of the universal enveloping algebra of its Lie algebra. The standard …

I suggest that you look into the paper Quantum Collections by Andre Kornell, available here on the arXiv. The abstract reads: We develop the viewpoint that the opposite of the category of $W^\ast …

See the paper Quantization of Multiply Connected Manifolds, by Eli Hawkins. arXiv link.

There is another way to see this than constructing an explicit resolution. This involves viewing $R$ as an iterated skew polynomial ring. I am assuming that you want $a_{ii} = 1$; otherwise $x_i^2 = …