Derived equivalence classification of self-injective algebras

1. 对于代数闭域上的有限维代数A和A的重复A的非负自同构g与g^2的自同构g，我们得到了形式为Λ = A/<g>的自射代数之间的推导等价的结果：A表示为三角形矩阵代数（A_g__<A(g)> 0__<A_g>）；对于另一个相同类型的代数Π = B/<h>，在a和B上的适当条件下，如果存在一个倾斜三重体（Ag,T_0,B_h）使得（a，T，B）也是一个倾斜三重体，则代数Λ和Π是等价的，其中我们设T = （T_0【叉积】_<A_g> ε_1A）【对称】（T_0【叉积】_<A_g> ε_2A）， ε_1:= (1_0 0_0)；和ε_2:=(0__0 0__1)∈A.2。利用循环颤振路径代数上幂零模所定义的霍尔代数，实现了特殊线性李代数和一般线性李代数。我们将所有类型的简单复李代数实现为由普通遗传代数的霍尔代数构造的退化复合李代数的一些因子李代数。我们将具有简带Dynkin图Δ的简单复李代数实现为由类型为Δ的正则代数A的Hall代数构造的简并复合李代数的若干因子李代数L(A)。此外，我们利用移位T的有界派生范畴的商范畴D^b(mod a)/<T>的不可分解对象的等类构造了一个类似于L(a)的李代数，使用三角形代替精确序列。我们利用上面的正则代数推广了李代数的构造方法，从等价于遗传代数的代数B构造李代数L(B)。这个李代数在派生等价下是否不变还有待研究。

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.