Derived equivalence classification of self-injective algebras

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1. We obtained the following result on derived equivalences between self-injective algebras of the form Λ = A/<g> for some finite-dimensional algebras A over an algebraically closed field and some non-negative automorphism g of the repetition A of A with g^2 the Nakayama automorphism of A : A is expressed as a triangular matrix algebra (A_g__<A(g)> 0__<A_g>) ; and for another algebra Π = B/<h> of the same type, under a suitable codition on A and B, the algebras Λ and Π are derived equivalent if there is a tilting triple (Ag,T_0,B_h) such that (A,T,B) is also a tilting triple, where we put T = (T_0 【cross product】_<A_g> ε_1A) 【symmetry】 (T_0 【cross product】_<A_g> ε_2A), ε_1 := (1__0 0__0) ; and ε_2 := (0__0 0__1) ∈ A.2. Using the Hall algebra defined by the nilpotent modules over the path-algebra of a cyclic quiver, we realized special and general linear Lie algebras.3. We realized all types of simple complex Lie algebras as some factor Lie algebras of degenerate composition Lie algebras constructed from the Hall algebras of tame hereditary algebras.4. We realized simple complex Lie algebras with simply-laced Dynkin diagrams Δ as some factor Lie algebras L(A) of degenerate composition Lie algebras constructed from the Hall algebras of canonical algebras A of type Δ. In addition, we constructed a Lie algebra analogous to L(A) from the isoclasses of indecomposable objects of the quotient category D^b(mod A)/<T> of the bounded derived category by the shift T using triangles instead of exact sequences.5. We generalized the construction method of Lie algebras using canonical algebras in the above to construct a Lie algebra L(B) from an algebra B derived equivalent to a hereditary algebra. It is still under investigation whether this Lie algebra is invariant under derived equivalences.