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Author: Burban, Igor × End date: 2009 × Has files: Yes × Type: Text × WGL Institute: MFO ×

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    Vector bundles on degenerations of elliptic curves and Yang-Baxter equations
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2007) Burban, Igor; Kreussler, Bernd
    In this paper we introduce the notion of a gemetric associative r-matrix 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.
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    Tilting on non-commutative rational projective curves
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2009) Burban, Igor; Drozd, Yuriy
    In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy2=x3+x2z.