ترغب بنشر مسار تعليمي؟ اضغط هنا

Derived deformation theory of algebraic structures

297   0   0.0 ( 0 )
 نشر من قبل Gr\\'egory Ginot
 تاريخ النشر 2019
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of algebras (associative, commutative, Poisson...) or bialgebras (associative and coassociative, Lie, Frobenius...), that is algebraic structures parametrized by props. A central aspect is that we define and study moduli spaces of deformations of algebraic structures up to quasi-isomorphisms (and not just isotopies or isomorphisms). To do so, we implement methods coming from derived algebraic geometry, by encapsulating these deformation theories as classifying (pre)stacks with good infinitesimal properties and %derived formal geometry, by means of derived formal moduli problems and derived formal groups. In particular, we prove that the Lie algebra describing the deformation theory of an object in a given $infty$-category of dg algebras can be obtained equivalently as the tangent complex of loops on a derived quotient of this moduli space by the homotopy automorphims of this object. Moreover, we provide explicit formulae for such derived deformation problems of algebraic structures up to quasi-isomorphisms and relate them in a precise way to other standard deformation problems of algebraic structures. This relation is given by a fiber sequence of the associated dg-Lie algebras of their deformation complexes. Our results provide simultaneously a vast generalization of standard deformation theory of algebraic structures which is suitable (and needed) to set up algebraic deformation theory both at the $infty$-categorical level and at a higher level of generality than algebras over operads. In addition, we study a general criterion to compare formal moduli problems of different algebraic structures and apply our formalism to $E_n$-algebras and bialgebras.



قيم البحث

اقرأ أيضاً

175 - J. Scott Carter 2010
Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frob enius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.
285 - Gregory Ginot , Sinan Yalin 2016
A first goal of this paper is to precisely relate the homotopy theories of bialgebras and $E_2$-algebras. For this, we construct a conservative and fully faithful $infty$-functor from pointed conilpotent homotopy bialgebras to augmented $E_2$-algebra s which consists in an appropriate cobar construction. Then we prove that the (derived) formal moduli problem of homotopy bialgebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of $E_2$-algebra structures on this cobar construction. We show consequently that the $E_3$-algebra structure on the higher Hochschild complex of this cobar construction, given by the solution to the higher Deligne conjecture, controls the deformation theory of this bialgebra. This implies the existence of an $E_3$-structure on the deformation complex of a dg bialgebra, solving a long-standing conjecture of Gerstenhaber-Schack. On this basis we solve a long-standing conjecture of Kontsevich, by proving the $E_3$-formality of the deformation complex of the symmetric bialgebra. This provides as a corollary a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which extends to homotopy dg Lie bialgebras and is independent from the choice of an associator. Along the way, we establish new general results of independent interest about the deformation theory of algebraic structures, which shed a new light on various deformation complexes and cohomology theories studied in the literature.
616 - Amnon Yekutieli 2011
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-li
221 - Adeel A. Khan 2020
These are some notes on the basic properties of algebraic K-theory and G-theory of derived algebraic spaces and stacks, and the theory of fundamental classes in this setting.
We give a general treatment of deformation theory from the point of view of homotopical algebra following Hinich, Manetti and Pridham. In particular, we show that any deformation functor in characteristic zero is controlled by a certain differential graded Lie algebra defined up to homotopy, and also formulate a noncommutative analogue of this result valid in any characteristic.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا