A Criterion For A Connected Cochain DG Algebra To Be Calabi-Yau And Its Applications

In this thesis,we study the Calabi-Yau properties of connected cochain DG algebras,and give a criterion to judge whether a DG algebra is Calabi-Yau.By using the criterion,we prove that some connected cochain DG algebras satisfying certain conditions are Calabi-Yau DG algebras.In the first part,we show A is a Calabi-Yau connected cochain DG algebra if the trivial DG algebra?H?A?,0?is a Calabi-Yau DG algebra.We apply the criterion to study the Calabi-Yau property of a connected cochain DG algebra A when H?A?belongs to certain types of 3-Calabi-Yau graded algebras.In the second part,we introduce the definition of 3-dimensional DG Sklyanin algebras.In a word,a 3-dimensional DG Sklyanin algebra is a connected cochain DG algebra whose underlying graded algebra A^#is a 3-dimensional Sklyanin algebra S _a,b,c.And then we figure out all possible differential structures of a 3-dimensional DG Sklyanin algebra A.We show?_A=0 if either a² b²or c 0.If a=-b and c=0,then the 3-dimensional DG Sklyanin algebra is actually a DG polynomial algebra,which has been systematically studied in[20].If a=b and c=0,then the DG Sklyanin algebra is uniquely determined by a 3×3 matrix M.In this case,the DG Sklyanin algebra is just the DG algebra A_-1?k?3?M?in[25].By using the classification of 3-dimensional Sklyanin algebras and the above criterion,we prove that3-dimensional Sklyanin DG algebras are all Calabi-Yau DG algebras.</sub>