On the second cohomology of nilpotent orbits in exceptional Lie algebras

In [1], the second de Rham cohomology groups of nilpotent orbits in all the complex simple Lie algebras are described. In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations, we obtain upper bounds for the dimensions of the second cohomology groups.     

Restricted Kac modules for special contact Lie superalgebras of odd type

We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3: We give a suffcient and necessary condition in terms of typical or atypical weights for restricted Kac modules to be simple. In the process, we also determine the socle for each restricted Kac module and the length for each simple restricted module.     

三维幂零李超代数的Yang-Baxter算子

在复数域C上,利用三维幂零李超代数的分类,通过计算刻画了三维幂零李超代数的Yang-Baxter算子.     

Some Lie superalgebras associated to the Weyl algebras

Let be the Lie superalgebra . We show that there is a surjective homomorphism from to the Weyl algebra , and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of on and use this to show that if is made into a Lie superalgebra using its natural -grading, then . In addition, we show that if and are isomorphic as Lie superalgebras, then . This answers a question of S. Montgomery.

     

Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when g is classical. Here, the induced orbits play an important role instead of Richardson orbits.     

Completable nilpotent Lie superalgebras

We discuss a class of filiform Lie superalgebras L^n,m. From these Lie superalgebras, all the other filiform Lie superalgebras can be obtained by deformations. We have decompositions of Der0ˉ(L^n,m) and Der1 (L^n,m). By computing a maximal torus on each L^n,m, we show that L^n,m are completable nilpotent Lie superalgebras. We also view L^n,m as Lie algebras, prove that L^n,m are of maximal rank, and show that L^n,m are completable nilpotent Lie algebras. As an application of the results, we show a Heisenberg superalgebra is a completable nilpotent Lie superalgebra.     

Quasi-finite modules for Lie superalgebras of infinite rank

We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras , and , and determine the necessary and sufficient conditions for such modules to be unitarizable. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.

     

Kostant homology formulas for oscillator modules of Lie superalgebras

We provide a systematic approach to obtain formulas for characters and Kostant u-homology groups of the oscillator modules of the finite-dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite-dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned u-homology computation are worked out.     

Jordan李超代数的一些性质

代数A称为不可分解的,如果A不能分解成理想的直和.证明了满足C(L_o)=C(L)={0}的Jordan李超代数L一些重要性质.     

On structure and TKK algebras for Jordan superalgebras

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions reduce to one of two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions become nonequivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algebraically closed field of characteristic zero.     

Composition factors of Kac modules for the general linear Lie superalgebras

The composition factors of Kac modules for the general linear Lie superalgebras are explicitly determined. In particular, a conjecture of Hughes, King and van der Jeugt in [J. Math. Phys. 33 (1992), 470–491] is proved.     

On restricted simple modules of contact Lie superalgebras of odd type

For contact Lie superalgebras of odd type over an algebraically closed field of characteristic , the composition factors of Kac modules and the character formulas for all the restricted simple modules are explicitly determined. And the dimensions of simple modules in the restricted supermodule category are precisely calculated, thereby dimensions of all the restricted simple modules are given.     

四维Filiform李超代数的Yang-Baxter方程

利用复数域C上四维Filiform李超代数的分类,通过计算刻画了复数域C上四维Filiform李超代数上的所有Yang-Baxter方程的解.     

Super-biderivations of Lie superalgebras

In this paper, we attempt to investigate the super-biderivations of Lie superalgebras. Furthermore, we prove that all super-biderivations on the centerless super-Virasoro algebras are inner super-biderivations. Finally, we study the linear super commuting maps on the centerless super-Virasoro algebras.     

Filiform李超代数L_(1,2)上的Yang-Baxter方程

求解一般Filiform李超代数L_(n,m)的Yang-Baxter方程尚无一般方法.通过计算,刻画了特征零的代数闭域上四维Filiform李超代数L_(1,2)上的所有Yang-Baxter方程的解.