Graded Lie algebras and regular prehomogeneous vector spaces with one-dimensional scalar multiplication

Differential Calculi on Some Quantum Prehomogeneous Vector Spaces

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quan…

Mackey 对离散分次代数的障碍映射

Mackey's obstruction map for discrete graded algebras

本文为离散群分次代数（不要求饱和）构造了 Mackey 型障碍映射，将其定义为等变类到第二上同调类的同态，并证明该障碍类决定了 Bahturin-Zaicev-Sehgal 分解中的精细部分。

Realisability and Localisation

Let $A$ be a differential graded algebra with cohomology ring $H^*A$. A graded module over $H^*A$ is called \emph{realisable} if it is (up to direct summands) of the form $H^*M$ for some differential graded $A$-module $M$. Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the $A_{\infty}$-algebra structure of $H^*A$. In this thesis we mainly consider differential graded algebras $A$ with graded-commutative cohomology ring. We show that a finitely presen…

$A$-无穷双模的障碍理论

Obstruction theory for $A$-infinity bimodules

本文构建截断最小 $A$-无穷双模扩张的障碍理论，以新型边缘谱序列和上同调刻画障碍，并推出微分分次双模的内在形式性与近乎形式性结果。

Lax operator algebras

In this paper we develop a general concept of Lax operators on algebraic curves introduced in [1]. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct the orthogonal and symplectic analogs of Lax operators, prove that they constitute almost graded Lie algebras and construct local central extensions of those Lie algebras.