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The complete quantum theory of covariant closed strings is constructed in detail. The action is defined by elementary vertices satisfying recursion relations that give rise to Jacobi-like identities for an infinite chain of string field…
We extend M. Kontsevich's formality morphism to a homotopy braces morphism and to a homotopy Gerstenhaber morphism. We show that this morphism is homotopic to D. Tamarkin's formality morphism, obtained using formality of the little disks…
Using a Hamiltonian approach, we construct the classical and quantum theory of open WZW strings on a strip. (These are the strings which end on WZW branes.) The development involves non-abelian generalized Dirichlet images in an essential…
Operadic tangent cohomology generalizes the existing cohomology theories of Chevalley--Eilenberg, Hochschild, and Harrison to address the deformation theory of general types of algebras through gadgets known as deformation complexes. The…
We develop here a concept of deformed algebras through three examples and an application. Deformed algebras are obtained from a fixed algebra by deformation along a family of indexes, through formal series. We show how the example of…
The deformation theory of Lie-Yamaguti algebras is developed by choosing a suitable cohomology. The relationship between the deformation and the obstruction of Lie-Yamaguti algebras is obtained.
We develop a systematic functional-analytic framework for Hom--Lie Banach algebras, introducing bounded $\alpha$-twisted derivations and almost periodic elements. Under natural continuity and compactness assumptions, we establish a complete…
We describe two constructions giving rise to curved $A_{\infty}$-algebras. The first consists of deforming $A_{\infty}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures…
The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Hom-Alternative and Hom-Malcev algebras. We construct deformation cohomology groups in low dimensions. Using a composition construction, we give…
We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a PROP, induced by a strongly homotopy Lie structure on the defining cochain complex, such that the associated "quantum" master…
The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L_\infty-algebra which induces a graded Lie bracket on…
Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a…
A notion of meromorphic open-string vertex algebra is introduced. A meromorphic open-string vertex algebra is an open-string vertex algebra in the sense of Kong and the author satisfying additional rationality (or meromorphicity) conditions…
In the present paper, we aim to introduce the cohomology of $\mathcal{O}$-operators defined on the Hom-Lie conformal algebra concerning the given representation. To obtain the desired results, we describe three different cochain complexes…
A unified description of the relationship between the Hamiltonian structure of a large class of integrable hierarchies of equations and W-algebras is discussed. The main result is an explicit formula showing that the former can be…
In this review we give a detailed introduction to the theory of (curved) $L_\infty$-algebras and $L_\infty$-morphisms. In particular, we recall the notion of (curved) Maurer-Cartan elements, their equivalence classes and the twisting…
Let $X$ be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ of the free loop…
Classes of $G$-Hom-associative algebras are constructed as deformations of $G$-associative algebras along algebra endomorphisms. As special cases, we obtain Hom-associative and Hom-Lie algebras as deformations of associative and Lie…
We investigate two systematic constructions of inverse-closed subalgebras of a given Banach algebra or operator algebra A, both of which are inspired by classical principles of approximation theory. The first construction requires a closed…
There are various reasons why adding stubs to the vertices of open string field theory (OSFT) is interesting: Not only the stubs can tame certain singularities and make the theory more well-behaved, but also the new theory shares a lot of…