Forests may fall, But not the dusk they shield. H.P. Lovecraft Contents 1. Coherent sheaves on elliptic curves 3 2. Hall algebra of an elliptic curve 6 3. Drinfeld double of H X 11 4. The algebra U X 17 5. The algebra E σ,σ 26 6. Further results : integral form and central extension 31 7. Summary 38 Appendix A 39 Appendix B 40 References 46

In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite-dimensional symmetric and satisfy τ 2 = id. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.

In this paper we introduce the notion of a geometric associative rmatrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstraß cubic curves.Since the complex manifold M (n,d) E is a homogeneous space over the algebraic group J = Pic 0 (E), it turns out that r(v 1 , v 2 ; y 1 , y 2 ) ∼ r(v 1 − v 2 ; y 1 , y 2 ) = r(v; y 1 , y 2 ),

In this article we classify indecomposable objects of the derived categories of finitely-generated modules over certain infinite-dimensional algebras. The considered class of algebras (which we call nodal algebras) contains such well-known algebras as the complete ring of a double nodal point k[[x, y]]/(xy) and the completed path algebra of the Gelfand quiver. As a corollary we obtain a description of the derived category of Harish-Chandra modules over SL 2 (R). We also give an algorithm, which allows to construct projective resolutions of indecomposable complexes. In the appendix we prove the Krull-Schmidt theorem for homotopy categories.  2004 Elsevier Inc. All rights reserved.