Some remarks on Bridgeland’s Hall algebras

We study the functorial properties of Bridgeland’s Hall algebras. Specifically, let 𝒜 and ? be two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and let F:𝒜→? be a fully faithful exact functor, which preserves projectives, then F induces an embedding of algebras from the Bridgeland’s Hall algebra of 𝒜 to the one of ?. In addition, let A be a finite-dimensional algebra over a finite field and B some special quotient algebra of A, then the Bridgeland’s Hall algebra of B is the quotient algebra of the one of A. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras.     

On categories associated to a Quasi-Hopf algebra

A quasi-Hopf algebra H can be seen as a commutative algebra A in the center 𝒵(H-Mod) of H-Mod. We show that the category of A-modules in 𝒵(H-Mod) is equivalent (as a monoidal category) to H-Mod. This can be regarded as a generalization of the structure theorem of Hopf bimodules of a Hopf algebra to the quasi-Hopf setting.     

The superalgebra of W(2,2) and its modules of the intermediate series

In this paper, we study a new Lie superalgebra constructed by a 2|2-dimensional Balinsky–Novikov superalgebra, which is called the superalgebra of W(2,2). It can be realized from semi product of the W-algebra W(2,2) and its module of the intermediate series. Finally, we determine all modules of the intermediate series over this superalgebra. Since it is di?cult to do so directly, we make it by using modules of the intermediate series over the trivial super extension of the Witt algebra.     

Resolvent estimates and smoothing for homogeneous partial differential operators on graded Lie groups

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators H on graded groups, including Rockland operators, sublaplacians, and many others. Left or right invariance is not required. Typically the globally smooth operator has the form T = V|H|^1∕2, where V only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group ?^N some new classes of partial differential operators are treated.     

Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.     

经典N=2李共形超代数的导子和第二上同调群

研究了经典N=2李共形超代数的导子和第二上同调群的结构,并应用第二上同调群的结果确定了该李共形超代数的泛中心扩张.     

限制pre-李代数

研究限制pre-李代数的结构,给出了限制pre-李代数和可限制pre-李代数的定义,得到了限制pre-李代数的p-映射性质和可限制pre-李代数的性质,还讨论了带有半单元的限制pre-李代数及拟环面限制pre-李代数和分解唯一性定理.     

保持拟可逆性或拟零因子的可加映射

设X和Y是维数大于1的复Banach空间,A和B分别是B(X)和B(Y)中包含有限秩算子的范数闭子代数.A,B∈A,定义A。B=A+B-AB,称。为A,B的拟积.刻画了从A到B的双边保持算子的(左,右)拟可逆性或(左,右,半)拟零因子的可加满射的结构.