Extension And Deformation Of Calabi-Yau Algebras

This Ph.D. thesis is on the extension and deformation of Calabi-Yau algebras. Since mathematician Shing-Tung Yau proved the Calabi conjecture, Calabi-Yau man-ifolds were extensively studied in Mathematics, mainnly in differential geometry, and Theoretical Physics. In the last nineties, mathematician M. Kontsevich proposed the Homological Mirroe Symmetry Conjecture. After that, Calabi-Yau categories and Calabi-Yau algebras appeared in some branchs of Mathematics and Physics, such as al-gebraic geometry, algebraic topology, noncommutative geometry and superstring, and show their importance gradually. This thesis consider the extension and deformation of Calabi-Yau algebras, and mainly consists of the following three parts.1. We consider the Calabi-Yau property of skew group algebras by using of ho-mological determinant (refer to [JoZ], or see section 1 chapter 3 in this thesis). For a p-Koszul Calabi-Yau algebra A and a finite subgroup G of its graded automorphism group, skew group algebra A#G is Calabi-Yau if and only if that the homological de-terminant of the elements in G equal to 1. This generalizes the result that skew group algebra of a polynomial algebra with a finite subgroup of special linear group is a Calabi-Yau algebra.By the A?algebraic structure over Yoneda Ext-algebra of p-Koszul algebras, we proved that any p-Koszul Calabi-Yau algebra can be derived from a superpotential. And on this basis, we construct a superpotential for A#G. Our result is a generalization of the result in [BSW] about the Koszul Calabi-Yau algebras and skew group algebras of polynomial ring.2. More generally, we consider the Calabi-Yau property of smash product of a Hopf algebra acting on a Calabi-Yau algebra. By using of the homological determinant of the Hopf action on an Artin-Schelter Gorenstein algebra(refer to [KKZ2], or see section 1 chapter 4 in this thesis), we generalized the above result to smash product, proved the following result:Suppose that H is be an involutory Calabi-Yau Hopf algebra and A be a left graded H-module algebra. If A is a p-Koszul (p?2) Calabi-Yau algebra, then A#H is graded Calabi-Yau if and only if that the homological determinant of the Hopf-action on A is trivial. And we construct a superpotential for Calabi-Yau algebra A#H when H is semisimple.3. Finally, we discuss the following problem:(1) the Calabi-Yau properties of central regular extensions of Koszul Calabi-Yau al-gebras;(2) the Calabi-Yau properties of PBW deformation of Koszul Calabi-Yau algebras.By using of the Rees algebra methods and the relation between the central regular extension and PBW deformation [CS], we proved that the PBW deformation and central regular extension of a Koszul Calabi-Yau algebra is still a Calabi-Yau algebra if and only if the socalled Jacobi-type identity is satisfied. These results partially generalized the result of R. Berger and R. Taillefer about the the Calabi-Yau properties of PBW deformations of 3-dimensional Calabi-Yau algebras [BT]. Apply the above result to the universal enveloping algebras and Sridharan enveloping algebras of finite dimensional Lie algebras, we reproved the results of J. W. He, V. van Oystaeyen and Y. H. Zhang [HOZ1].