Non analytic ideals on $\omega$

Question

I would like to gather examples of NON analytic ideals on $\omega$.

However, I have found nothing in the books and papers I have consulted.

Could anyone tell me some reference/s?

Answer 1

Every non principal prime ideal over $\omega$ is both Lebesgue non measurable and does not have Baire property so it cannot be analytic.

Answer 2

We just submitted a paper, http://arxiv.org/pdf/1510.05699v1.pdf, in Section 2 you can find examples of quite natural coanalytic ideals (and more).