Derived equivalence classification of self-injective algebras

自注入代数的导出等价分类

• 批准号: 14540038
• 负责人: SUMIOKA Takeshi
• 金额: $ 1.86万
• 依托单位: Osaka City University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540038/
• 关键词: algebras; Hall algebras; canonical algebras; simple Lie algebras; Derived categories; quotient categories; repetition; self injective; 代数; 中山自己同型; Hall多元環

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.

1.得到了如下结果：对于代数闭域上的有限维代数A，得到了形式为Λ=A/&lt；g&gt；的自射代数与A的具有g^2的重复A的非负自同构g之间的导出等价：A的Nakayama自同构表示为三角矩阵代数(A_g_&lt；A(G)&gt；0_g&lt；A_g&gt；)；另一个代数Π=B/&lt；h&gt；对于相同类型的代数，在A和B的适当条件下，如果存在倾斜三元组(Ag，T_0，B_h)使得(A，T，B)也是倾斜三元组，则代数Λ和Π是等价的，其中T=(T_0[叉积]_&lt；A_g&gt；ε_1A)[对称性](T_0[叉积]_&lt；A_g&gt；ε_2A)，ε_1：=(1_0_0_0)；和ε_2：=(0_0 0_1)∈A.2。利用循环箭图的路代数上的幂零模所定义的Hall代数，实现了特殊的和一般的线性李代数。我们将所有类型的单复李代数都实现为由驯服遗传代数的Hall代数构造的退化复合李代数的因子李代数。我们将具有简单花边动态图Δ的单复李代数实现为由Δ类型的正则代数A的霍尔代数构造的退化复合李代数的因子李代数L(A)。此外，我们利用有界导范畴的商范畴D^b(Moda A)/&lt；T&gt；的不可分解对象的同构类，利用三角形而不是正则序列的移位T，构造了一个类似于L(A)的李代数。在上面，我们推广了用正则代数构造李代数的方法，由一个与遗传代数等价的代数B构造了一个李代数L(B)。这个李代数在导出等价下是否不变仍在研究中。

项目成果

On a left H-ring with Nakayama Automorphism

在具有中山自同构的左 H 形环上

• 发表时间: 2005
• 期刊: Advance in Ring Theory (Proceedings of the 4-th China-Japan-Korea International Conference)
• 作者: Hashimoto Y.;兼田正治;Rumynin;D;河田成人;河田 成人;浅芝秀人;河田成人;浅芝 秀人;河田 成人;加戸 次郎
• 通讯作者: 加戸 次郎

On simple-injective modules

关于简单内射模

• 发表时间: 2004
• 期刊: Mathematical J. of okayama University 46
• 作者: 住岡 武;渡嘉敷 尚
• 通讯作者: 渡嘉敷 尚

河田成人: "群環の自明なソースをもつ加群とAuslander-Reiten列について"京都大学数理解析研究所講究録. 1251. 130-138 (2002)

Masato Kawata：“关于群环和 Auslander-Reiten 序列的平凡源的模”京都大学数学科学研究所 Kokyuroku 1251. 130-138 (2002)。

On some SR(H)-blocks for the symmetric groups

关于对称群的一些 SR(H)-块

• 发表时间: 2003
• 期刊: Journal of Algebra 270
• 作者: 住岡 武;渡嘉敷 尚;浅芝 秀人;津島 行男
• 通讯作者: 津島 行男

住岡 武: "On simple-injective modules"Mathematical Journal of Okayama University. 発表予定.

Takeshi Sumioka：“论简单内射模”冈山大学数学杂志预定发表。