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In this paper, we introduce the concepts of endomorphism operator, left averaging operator, differential operator and Rota-Baxter Operator, and we construct examples of these linear maps on associative algebras with a left identity, a…
We establish the Composition-Diamond lemma for non-associative algebras over a free commutative algebra. As an application, we prove that every countably generated non-associative algebra over an arbitrary commutative algebra $K$ can be…
Given a non-negative integer $q$, we study two different notions of the $q$-capability of Lie algebras via the non-abelian $q$-exterior product of Lie algebras. The first is related to the $q$-crossed modules and inner $q$-derivations, and…
We study commutative associative polynomial operations $\mathbb{A}^n\times\mathbb{A}^n\to\mathbb{A}^n$ with unit on the affine space $\mathbb{A}^n$ over an algebraically closed field of characteristic zero. A classification of such…
Recently, we have shown that von Neumann algebras form a model for Selinger and Valiron's quantum lambda calculus. In this paper, we explain our choice of interpretation of the duplicability operator "!" by studying those von Neumann…
We develop a general theory of pushforward operations for principal $G$-bundles equipped with a certain type of orientation. In the case $G=BU(1)$ and orientations in twisted K-theory we construct two pushforward operations, the projective…
Curved algebras are algebras endowed with a predifferential, which is an endomorphism of degree -1 whose square is not necessarily 0. This makes the usual definition of quasi-isomorphism meaningless and therefore the homotopical study of…
We study generalizations of pre-Lie algebras, where the free objects are based on rooted trees which edges are typed, instead of usual rooted trees, and with generalized pre-Lie products formed by graftings. Working with a discrete set of…
We consider non-selfadjoint operator algebras $\mathfrak{L}(G,\lambda)$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative generalizations of…
Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of…
We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical…
Crawley-Boevey introduced the definition of a noncommutative Poisson structure on an associative algebra A that extends the notion of the usual Poisson bracket. Let V be a symplectic manifold and G be a finite group of symplectimorphisms of…
The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that…
Let $A$ be a not necessarily commutative monoid with zero such that projective $A$-acts are free. This paper shows that the algebraic K-groups of $A$ can be defined using the +-construction and the Q-construction. It is shown that these two…
We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local…
This paper contains an elementary proof of the existence of the classical model structure on the category of unbounded DG-Lie algebras over a field of characteristic zero, with an emphasis on the properties of free and semifree extensions,…
We introduce the notion of pie algebra for a 2-monad, these bearing the same relationship to the flexible and semiflexible algebras as pie limits do to flexible and semiflexible ones. We see that in many cases, the pie algebras are…
For any cocommutative Hopf algebra $H$ and a left $H$-module $V$, we construct an operad $\mathcal{P}^{cl}_H(V)$, which in the special case when $H$ is the algebra of polynomials in one variable reduces to the classical operad…
There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group on three elements. The first…
Infinitesimal deformations are governed by partition Lie algebras. In characteristic $0$, these higher categorical structures are modelled by differential graded Lie algebras, but in characteristic $p$, they are more subtle. We give…