Derived Equivalences Of Derived Categories And Applications

The notion of derived categories (trangulated categories) was first introduced and studied by Grothendieck-Verdier in early sixties of last century, which marks a new starting point of the development of modern algebras. The theory of derived categories establish a beautiful relation between algebras and geometries. In recent years, derived categories and derived equivalences have been adapted extensively to a number of subjects beyond algebraic geometry, and have become one of the dominant subjects. Many profound results have been obtained and a lot of challenging problems have been put foreward. These reflect the interaction and development of different subjects of mathematical science of the new century. This dissertation cencentrates the derived categories and derived equivalences of finite-dimensional algebras and their applications, which includes five chapters altogether.In the first chapter, we give a detailed introduction of recent developments in derived categories, Morita's equivalence theory, derived equivalence (autoequivalence) theory, recollements and mutations, automorphisms of Lie algebras, and so on. In particular, we list some concepts and theorems which are closely related to this dissertation, and make a systematic exposition of our main results.The derived equivalences of extension algebras are studied in the second chapter. We prove that if two algebras are derived equivalence, then their repetitive algebras are also derived equivalence, furthermore, they are stable equivalence. Our results generalize the relavant results of Happel [Hap2], Rickard [Ric3], Du [Du] and Tachikawa-Wakamatsu [TW].The third chapter deals with the generalizations of derived equivalence-recollements. We show that if the unbound derived module category of a finite-dimensional algebra over arbitrary base field k admits symmetric recollement relative to unbounded derived categories of two finite-dimensional k-algebrasB and C, then so does their repetitive algebras. This result extends the main theorems in [CL1]. On the other hand, we study the recollement of the bounded derived module category of the triangle matrix algebra (especially the one-point extension algebra). Then, we point out that stratification algebras are not necessarily quasi-hereditary algebras.In the fourth chapter, using algebraic method, we prove that tubular mutations are auto equivalences of the derived category of a derived tubular algebra, and apply this method to the case of tame concealed algebras, which generalizes a result of Meltzer [Mel]. Then we give a computational formula for each tubular mutation.In the final chapter, we apply the tubular mutations of the derived category of a derived tubular algebra to Lie algebra. With the aid of the method established by Lin-Peng [LP] in the realization of 2-extended affine Lie algebras of type D4, E6, E7 and E8, and the computational formula for each tubular mutation given in the previous chapter, we obtain a class of automorphisms of 2-extended affine Lie algebras of type D4, E6, E7 and E8. Furthermore, we construct a class of braid groups such that the 2-extended affine Lie algebras afford their representations.