Absolute algebras, contramodules, and duality squares

IF 0.7 2区 数学 Q2 MATHEMATICS
Victor Roca i Lucio
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引用次数: 0

Abstract

Absolute algebras are a new type of algebraic structures, endowed with a meaningful notion of infinite sums of operations without supposing any underlying topology. Opposite to the usual definition of operadic calculus, they are defined as algebras over cooperads. The goal of this article is to develop this new theory. First, we relate the homotopy theory of absolute algebras to the homotopy theory of usual algebras via a duality square. It intertwines bar-cobar adjunctions with linear duality adjunctions. In particular, we show that linear duality functors between types of coalgebras and types of algebras are Quillen functors and that they induce equivalences between objects with finiteness conditions on their homology. We give general comparison results between absolute types of algebras and their classical counterparts. We work out examples of this theory such as absolute associative algebras and absolute Lie algebras, and show that it includes the theory of contramodules. Finally, in [9], the authors showed that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic. As an application of our results, we generalize their theorem to the setting of absolute Lie algebras and absolute L-algebras.
绝对代数,控制模和对偶平方
绝对代数是一种新型的代数结构,它被赋予了无限运算和的有意义的概念,而不假设任何底层拓扑。与通常的操作演算定义相反,它们被定义为协同上的代数。本文的目的就是要发展这一新理论。首先,我们通过对偶平方将绝对代数的同伦理论与一般代数的同伦理论联系起来。它将条形和线性对偶连接在一起。特别地,我们证明了在代数类型和代数类型之间的线性对偶函子是Quillen函子,并且它们在它们的同调上推导出具有有限条件的对象之间的等价。给出了绝对代数与经典代数的一般比较结果。我们给出了绝对结合代数和绝对李代数的例子,并证明了它包含了控制模理论。最后,在[9]中,作者证明了两个泛包络代数同构为结合代数的幂零李代数必然是同构的。作为结果的一个应用,我们将其定理推广到绝对李代数和绝对L∞-代数的集合中。
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来源期刊
CiteScore
1.70
自引率
12.50%
发文量
225
审稿时长
17 days
期刊介绍: The Journal of Pure and Applied Algebra concentrates on that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications.
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