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Can you give me examples of $1$-Calabi-Yau triangulated categories $D$ different from the bounded derived category of coherent sheaves on an elliptic curve? I would like moreover the numerical Grothendieck group of $D$ to be of rank $2$ (by numerical Grothendieck group i mean the Grothendieck group modulo the pairing $$X(A,B)= \sum_i (-1)^i{\rm dim} \ {\rm Hom}_{D}(A,B[i])$$). Thank you.

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Finite type ones are even classified!

https://arxiv.org/abs/math/0612141

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There are some results about that question in the works of Roosmalen (http://arxiv.org/pdf/math/0703457.pdf)

He states that the only other example is given by the finite dimensional representations of k[[t]].

EDIT: This is for Abelian 1-Calabi-Yau categories, not exactly the question.

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