Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).     

Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Let be a connected semisimple Lie group with finite center. Let be the maximal compact subgroup of corresponding to a fixed Cartan involution . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary -module contains a -type with highest weight , then has infinitesimal character . Here is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary -modules with non-zero Dirac cohomology, provided has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant's cubic Dirac operator.

     

超Schrödinger代数J(1/1)的上同调

本文具体计算了系数在超Schrödinger代数J（1/1）的平凡模和有限维不可约模中的第一阶上同调群与第二阶上同调群,并给出了系数在通用包络代数U（J（1/1））中J（1/1）的第一阶与第二阶上同调群的维数是无限维的.     

Completeness of quasi-filiform Lie algebras

It is proved that for any quasi-filiform of non-zero rank the solvable Lie algebra obtained by adjoining a maximal torus of outer derivations is complete. Further, for any positive integer m, it is shown that there exist solvable complete Lie algebras with the second Chevalley–Eilenberg cohomology group of arbitrary dimension.     

李超代数gl_(2|1)到Witt超代数的低维上同调

Witt超代数在伴随表示的意义下是一般线性李超代数gl_(2|1^-)模.通过对Witt超代数进行子模分解和权空间分解,用简约的方法计算了gl_(2|1-)模.通过对Witt超代数进行子模分解和权空间分解,用简约的方法计算了gl_(2|1^-)到Witt超代数的低维上同调.     

Hochschild Cohomology of an Algebra Associated with a Circular Quiver

In this article we show that an algebra A = K Γ/(f(X ^s )) has a periodic projective bimodule resolution of period 2, where KΓ is the path algebra of the circular quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K and X is the sum of all arrows in KΓ. Moreover, by means of this projective bimodule resolution, we compute the Hochschild cohomology group of A, and we give a presentation of the Hochschild cohomology ring HH?(A) by the generators and the relations in the case K is a field.