We will discuss a bar/coalgebra construction of strong homotopy Leibniz algebras. We will give a generalized framework of derived bracket construction. We will prove that a deformation derivation of differential graded Leibniz algebra induces a strong homotopy Leibniz algebra by derived bracket method.

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. When modules are two-dimensional, we categorical action blob algebra on derived categories these dg-algebras which intertwines $\mathfrak{sl}_2$. From above derive categorification algebra.

After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎wit...

let a be a banach algebra and e be a banach a-bimodule then s = a  e, the l1-direct sum of a and e becomes a module extension banach algebra when equipped with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). in this paper, we investigate △-amenability for these banach algebras and we show that for discrete inverse semigroup s with the set of idempotents es, the module extension bana...

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of as those space and time, first order differential equations over all quantities proves to encompass Maxwell equations, leads natural extension incorporating rest mass spin, allows an integration with quantum mechanics. Although is not division algebra, it parallels reality...

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the alge...

We establish tools to facilitate the computation and application of the ChekanovEliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots in standard contact three-space. More specifically, we reformulate the DGA in terms of front projections, and introduce the characteristic algebra, a new invariant derived from the DGA. We use the characteristic algebra ...

Given the nilpotent cone of a complex reductive Lie algebra, we consider its equivariant constructible derived category of sheaves with coefficients in an arbitrary field. This category and its subcategory of perverse sheaves play an important role in Springer theory and the theory of character sheaves. We show that the composition of the Fourier–Sato transform on the Lie algebra followed by re...

In order to realize the quantum group in a global way, the Hall algebra of a root category (which is a orbit category of the derived category of a hereditary algebra) is constructed. The PBW-basis, ltration structure and integral form are investigated. Their degeneration at v = 1 actually provide a realization of the universal enveloping algebra of semisimple Lie algebras.

We construct a basis for the right quantum algebra introduced by Garoufalidis, Lê and Zeilberger and give a method making it possible to go from an algebra subject to commutation relations (without the variable q) to the right quantum algebra by means of an appropriate weight-function. As a consequence, a strong quantum MacMahon Master Theorem is derived. Besides, the algebra of biwords is syst...