Related papers: Derived Algebraic Geometry III: Commutative Algebr…

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped…

This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…

This survey provides an elementary introduction to operads and to their applications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher…

We develop a theory of minimal models for algebras over an operad defined over a commutative ring, not necessarily a field, extending and supplementing the work of Sagave in the associative case.

We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.

Following our previous works on $C^*$-graph algebras and the associated Cuntz-Krieger graph families, in this paper we will try to have a look at the colored version of these structures and to see what a $C^*$-colored graph algebra might…

This is the third chapter in our "Toric Topology" book project. Further chapters are coming. Comments and suggestions are very welcome.

An operad describes a category of algebras and a (co)homology theory for these algebras may be formulated using the homological algebra of operads. A morphism of operads $f:\mathcal{O}\rightarrow\mathcal{P}$ describes a functor allowing a…

Involutive category theory provides a flexible framework to describe involutive structures on algebraic objects, such as anti-linear involutions on complex vector spaces. Motivated by the prominent role of involutions in quantum (field)…

Colour algebras over fields of odd characteristic are well-known noncommutative Jordan algebras. We define colour algebras more generally over a unital commutative associative ring with $\frac{1}{2}\in R$, and show that colour algebras can…

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.

In this paper, we prove some foundational results on the deformation theory of E-infinity ring spectra.

We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under…

In this paper, we introduce the representation of modified $\lambda$-differential $3$-Lie algebras and define the cohomology of modified $\lambda$-differential $3$-Lie algebras with coefficients in a representation. As applications of the…

Several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded $k$-algebra; these notions are justified by examples from noncommutative invariant theory.

We describe the modular operad structure on the moduli spaces of pointed stable curves equipped with an admissible $G$-cover. To do this we are forced to introduce the notion of an operad colored not by a set but by the objects of a…

This is a sequel to our paper "Permute, Graph, Map, Derange", involving decomposable combinatorial labeled structures in the exp-log class of type a=1/2, 1, 3/2, 2. As before, our approach is to establish how well existing theory matches…

We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.

The works of R. Descartes, I. M. Gelfand and A. Grothendieck have convinced us that commutative rings should be thought of as rings of functions on some appropriate (commutative) spaces. If we try to push this notion forward we reach the…