Matrix factorization categories for ADE singularities

What is known about the matrix factorization categories of singularities of type ADE? Any references on this would be greatly appreciated.

Background: For ADE singularities, see for example this. For matrix factorizations, see for example this.

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See: "Matrix Factorizations and Representations of Quivers II: type ADE case" (math/0511155) by Kajiura, Saito, and Takahashi for a recent account.

Older references include: "Construction geometrique de la correspondance de McKay" Gonzalez-Sprinberg,and Verdier (1983) Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings (1990)

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Matrix factorization categories for these singularities depend on a grading that you consider. If you consider the maximal grading for ADE singularities in a standard form like

$X^{l+1}+ Y^2+\cdots$(sum of squares) for $A_l$ and so on till $X^3+Y^5+\cdots$(sum of squares) for $E_8$, then the category will be equivalent to the derived category of representations of the corresponding Dynkin quiver. (see paper math/0511155 and especially Appendix A for a short proof)

If you consider non-graded case then for A-type singularities the category is described in the end of the paper math/0302304. I am sure that these non-graded categories can be obtained from the graded versions as orbit categories with respect to a related autoequivalence in Definition of Bernhard Keller math/0503240. But it seems that this fact is not written yet.

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Sorry for the stupid question, but I don't understand what is meant by grading here. –  Kevin H. Lin Apr 7 '10 at 21:01
In non graded case it is known that the matrix factorization category is equivalent to a triangulated category of singularities of the singular fiber. There is a so called a graded version of triangulated category of singularities. By definition a triangulated category of singularities is the quotient of the bounded derived category of f.g modules by the subcategory of perfect complexes. Now if we have an algebra with a grading then we can consider a graded version, i.e. the quotient of bounded derived category of graded modules by the subcategory of perfect complexes. –  user2464 Apr 8 '10 at 10:42

@ploughshare: If M is a CM A-module, then M and M(n) are isomorphic in the completion \hat{A}. this is how one goes from the graded to the ungraded.

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