Graph products of spheres, associative graded algebras and Hilbert series

Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view

In this paper we establish a faithfulness result, in a homotopical sense, between a subcategory of the model category of augmented differential graded commutative algebras CDGA and a subcategory of the model category of augmented differential graded algebras DGA over the field of rational numbers $\mathbb{Q}$.

A differential graded Lie algebra approach to non abelian extensions of associative algebras

In this paper we show that non abelian extensions of an associative algebra $\mathcal{B}$ by an associative algebra $\mathcal{A}$ can be viewed as Maurer-Cartan elements of a suitable differential graded Lie algebra $L$. In particular we show that $\mathcal{MC}(L)$, the Deligne groupoid of $L$, is in 1-1 correspondence with the non-abelian cohomology $H^2_{nab}(\mathcal{B},\mathcal{A})$.

On the NonKoszulity of (2p+1)-ary partially associative Operads

We present a study of quadratic operads for n-ary algebras and their dual for n odd. We will focus on the ternary case (i.e n=3). The aim is to underline the problem of computing the dual operad and the fact that this last is in general defined in the graded differential operad framework. We prove that the operad associated to 3-ary partially associative algebra is not Koszul. Recall that, if n is even, the operad of n-ary partially associative algebras is Koszul.

On the Associative Algebra Kernels and Obstruction

The theory of abstract kernels in non-trivial extensions for many kinds of algebraical objects, such as groups, rings and graded rings, associative algebras, Lie algebras, restricted Lie algebras, DG-algebras and DG-Lie algebras, has been widely studied since 1940's. Gerhard Hochschild firstly treats associative algebra as an generic type in the series of kernel problems. He proves the theorem of constructing kernel by presenting many tedious relations that may lost the readers today. In this paper, we shall illustrate the formulation and recast it for Lie algebra(-oid) kernels. We also prove …

李、结合与交换拟同构

Lie, associative and commutative quasi-isomorphism

本文证明特征零下交换 dg 代数的交换与结合拟同构等价，并给出同伦完备 dg 李代数的 Koszul 对偶结果，从而确定有理同伦型并推出幂零李代数由泛包络代数识别。

Disconnected rational homotopy theory

We construct two algebraic versions of homotopy theory of rational disconnected topological spaces, one based on differential graded commutative associative algebras and the other one on complete differential graded Lie algebras. As an application of the developed technology we obtain results on the structure of Maurer-Cartan spaces of complete differential graded Lie algebras.