We determine the \emph{$L_\infty$-algebra} that controls deformations of a relative Rota-Baxter Lie algebra and show it is an extension dg controlling underlying LieRep pair by operator. Consequently, we define {\em cohomology} algebras relate to their infinitesimal deformations. A large class obtained from triangular bialgebras construct map between corresponding deformation complexes. Next, notion \emph{homotopy} introduced. homotopy intimately related \emph{pre-Lie$_\infty$-algebras}.

We show that generalised Calabi-Yau dg (co)algebras are Koszul dual to symmetric (co)algebras, without needing assume any smoothness or properness hypotheses. Similarly, we Gorenstein and Frobenius properties. As an application, give a new characterisation of Poincar\'e duality spaces, which extends theorem F\'elix- Halperin-Thomas the non-simply connected setting.

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A 1 -, C -and L -algebras.This is based on noncommutative geometry as expounded by Connes and Kontsevich.The developed machinery then used establish form Hodge decomposition Hochschild cyclic generalises puts conceptual previous work Loday Gerstenhaber-Schack.

We construct an explicit minimal model for algebra over the cobar-construction of a differential graded operad.The structure maps this are expressed in terms sums decorated trees.We introduce appropriate notion homotopy equivalence operadic algebras and show that our is equivalent to original algebra.All generalizes gives conceptual explanation well-known results A∞-algebras.Further, we these carry case modular operads; trees get replaced by general Feynman graphs.As by-product work prove...

Homotopical algebra and higher category theory play an increasingly important role in pure mathematics, methods have seen tremendous development the last couple of decades. The talks delivered at workshop described some latest progress this area applications to various problems algebra, geometry, combinatorics.

The "Workout" module is included in the content of model program physical education for students grades 5-9 New Ukrainian School (NUS) from 2024-2025 academic year. educational material basic elements this presented compliance with peculiar terminology used by workouts outdoor training. Our research has identified problem unpreparedness teachers to teach due difficulties understanding terms. purpose publication develop an explanatory dictionary terms workout exercises set out variable...

We define and study the degeneration property for BV-infinity algebras show that it implies underlying L-infinity are homotopy abelian. The proof is based on a generalisation of well-known identity \Delta(e^x)=e^x(\Delta(x)+[x,x]/2) which holds in all BV algebras. As an application we higher Koszul brackets cohomology manifold supplied with generalised Poisson structure vanish.

Using the theory of extensions L ∞ algebras, we construct rational homotopy models for classifying spaces fibrations, giving answers in terms classical homological functors, namely Chevalley–Eilenberg and Harrison cohomology. We also investigate algebraic structure complexes algebras show that they possess, along with Gerstenhaber bracket, an is abelian.

We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a S -algebra R in sense of Elmendorf, Kriz, Mandell, May and Strickland.The formulation this spectral sequence is similar to classical case calculation its E 2 -term involves cohomology certain 'brave new Hopf algebroids' * .In working out details we resurrect Adams' original approach Universal Coefficient Sequences modules an spectrum.We describe some examples motivating = M U .

We study the structure of formal groups associated to Morava K-theories integral Eilenberg-Mac Lane spaces.The main result is that every group in collection {K(n) * K(Z, q), q = 2, 3, . ..} for a fixed n enters it together with its Serre dual, an analogue principal polarization on abelian variety.We also identify isogeny class each these over algebraically closed field.These results are obtained help Dieudonné correspondence between bicommutative Hopf algebras and modules.We extend P....

We give a construction of an L ∞ map from any algebra into its truncated Chevalley-Eilenberg complex as well cyclic and A analogues.This fits with the inclusion full (or respective analogues) to form homotopy fiber sequence algebras.Applications deformation theory graph homology are given.We employ machinery Maurer-Cartan functors in algebras associated twistings which should be independent interest.

We introduce and study the notion of a dual Feynman transform smodular operad.This generalizes gives conceptual explanation Kontsevich's construction producing graph cohomology classes from contractible differential graded Frobenius algebra.The modular operad is indeed linear to introduced by Getzler Kapranov when evaluated on vacuum graphs.In marked contrast transform, admits an extremely simple presentation via generators relations; this leads explicit easy description its algebras.We...

Motivated by ideas from stable homotopy theory we study the space of strongly associative multiplications on a two-cell chain complex.In simplest case this moduli is isomorphic to set orbits group invertible power series acting certain space.The Hochschild cohomology rings resulting A ∞ -algebras have an interpretation as totally ramified extensions discrete valuation rings.All are supposed be unital and give detailed analysis structures which independent interest.