Projective completions of Jordan pairs. Part I: The generalized projective geometry of a Lie algebra

有限维代数的终端歧义几何

Terminal Ambiguity Geometry for Finite-Dimensional Algebras

本文针对有限维代数的根截断导致模信息丢失的问题，引入终端歧义几何，通过格拉斯曼轨道空间分类终端完备化，并给出刚性判据、维数及有限域计数，同时建立重构与对称性结果。

交换诺特环上的大型纯投射模：与完备化的比较

Big pure projective modules over commutative noetherian rings: comparison with the completion

该研究针对交换诺特环上的纯投射模分类问题，利用有限生成数据刻画无限生成直和项，拓展经典理论至无限生成情形，还研究了相关幺半群结构并得到子幺半群结论与实现结果。

算子Kantor对

Operator Kantor Pairs

本文引入了算子Kantor对的概念，将其推广至任意交换含单位元环，构造了相关的5-分次Lie代数，并在特定环上建立了其与Kantor对的一一对应关系。

Structurable algebras and groups of type E_6 and E_7

It is well-known that every algebraic group of type F_4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type ^1E_6 with trivial Tits algebras arise as the isometry groups of norm forms of such Jordan algebras. We describe a similar relationship between groups of type E_6 and groups of type E_7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type E_7 with trivial Tits algebras. The underlying algebraic structure for the relationship considered here are a sort of 56-dimensional structurable …

The periplectic Brauer algebra II: decomposition multiplicities

We determine the Jordan-Holder decomposition multiplicities of projective and cell modules over periplectic Brauer algebras in characteristic zero. These are obtained by developing the combinatorics of certain skew Young diagrams. We also establish a useful relationship with the Kazhdan-Lusztig multiplicities of the periplectic Lie superalgebra.

A Geometrical Interpretation of Okubo Spin Group

In this work we define, for the first time, the affine and projective plane over the real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. Okubo algebra is a flexible, composition algebra which is also a not unital division algebra. Even though Okubo algebra has been known for more than 40 years, we believe that this is the first time the algebra was used for affine and projective geometry. After showing that all axioms of affine geometry are verified, we define a projective plane over Okubo algebra as completion of the affine plane and directly through the use o…

Associative Geometries. I: Torsors, linear relations and Grassmannians

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.

Associative Geometries. I: Torsors, linear relations and Grassmannians

We define and investigate a geometric object, called an associative geometry, corresponding to an associative algebra (and, more generally, to an associative pair). Associative geometries combine aspects of Lie groups and of generalized projective geometries, where the former correspond to the Lie product of an associative algebra and the latter to its Jordan product. A further development of the theory encompassing involutive associative algebras will be given in subsequent work.

射影Banach李双代数、射影Yang-Baxter方程与射影Poisson-Lie群

Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson-Lie groups

本文引入射影Banach李双代数、射影Yang-Baxter方程及射影Poisson-Lie群，证明拟三角射影r-矩阵产生射影Banach李双代数，射影Poisson-Lie群微分具有该结构，并给出三角射影r-矩阵的等价描述。

The Nakayama functor and its completion for Gorenstein algebras

Duality properties are studied for a Gorenstein algebra that is finite and projective over its center. Using the homotopy category of injective modules, it is proved that there is a local duality theorem for the subcategory of acyclic complexes of such an algebra, akin to the local duality theorems of Grothendieck and Serre in the context of commutative algebra and algebraic geometry. A key ingredient is the Nakayama functor on the bounded derived category of a Gorenstein algebra, and its extension to the full homotopy category of injective modules.

An Essay on the Completion of Quantum Theory. I: General Setting

We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of understanding the axioms of geometry, we complete the spaces of observables and of states in a similar way as in classical geometry linear or affine spaces are completed by projective spaces. In this sense, our theory can be considered as a "completion of usual linear quantum theory" , such that the usual theory appears as the special case where a reference frame is…