G-algebras, or Groebner bases algebras, were considered by Levandovsky, these algebras include very important families of algebras, like the Weyl algebras and the universal enveloping algebra of a finite dimensional Lie algebra. These algebras are not graded in a natural way, but they are filtered. Going to the associated graded algebra is the standard way to study their properties. In a previous paper we studied an homogeneous version of the G-algebras, which results naturally graded, and through a des homogenization property, we obtain an ordinary G-algebra. The main results in our first paper were the following: Homogeneous G-algebras are Koszul of finite global dimension, Artin Schelter regular, noetherian and have a Poincare Birkoff basis. We give the structure of their Yoneda algebras by generators and relations. In the second part of the paper we consider the quantum polynomial ring and prove, for the finitely generated graded modules, cohomology formulas analogous to those we have in the usual polynomial ring. In the third part we use the results of the previous part to study the cohomology of the homogeneous G-algebras. Part four was dedicated to the study of the relations between an homogeneous G-algebra B and its des homogenization B/(Z-1)B=A, Applications to the study of finite dimensional Lie algebras and Weyl algebras were given. In the last sections of the paper we study the relations, at the level of modules, among the homogenized algebra B, its graded localization B_{z}, the des homogenization A=B/(Z-1)B and the Yoneda algebra B^{!}. In this paper we investigate the relations among all these algebras at the level of derived categories.
Koszul and Gorenstein properties for homogeneous algebras
文献[5]将Koszul性质推广到了次数N>2的齐次代数，文献[7]将其与N-复形联系起来。我们证明了如果N-齐次代数A是广义Koszul，AS-Gorenstein且具有有限的整体维数，则可以将Van den Bergh对偶定理[23]应用于A，即，当N=2时，A的霍赫希尔德同调与上同调之间存在Poincare对偶。
Reductive homogeneous spaces of the compact Lie group $G_2$
第一提交人于2001年为她的博士论文《Espacios homogéneos reductivos y álgebras no associativas》进行答辩，导师是P. Benito和A.艾杜克本文研究了特征为零的域上具有标准包络代数$\mathfrak{g}_2$的Lie-Yamaguti代数的分类，特别是紧李群$G_2$的齐性约化空间的分类.在这项工作中，我们重新审视这种分类从一个更几何的方法。给出了相应的齐性空间的几何模型，并明确了它们之间的一些关系。
Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives
I. Panin在90年代证明了扭曲投射齐次簇的代数K理论可以用中心单代数代数来表示。后来，Merkurjev和Panin将复曲面簇的代数K理论描述为可分代数的代数K理论的直接和项。本文利用最近的非对易动机理论，将Panin和Merkurjev-Panin计算从代数K理论推广到所有的加法不变量.作为第一个应用，我们充分计算的循环同调（及其所有的变种）扭曲的投射齐次品种。作为第二个应用，我们证明了一个扭曲的投射齐次簇的非交换动机是平凡的当且仅当相关的中心单代数的Brauer类是平凡的。在此过程中，我们构造了一个完全忠实的张量函子，从Merkurjev-Panin的motivic范畴到Kontsevich的非交换Chow motives范畴，这是独立的兴趣。
