In this paper we set-up a general framework for a formal deformation theory of Dirac structures. We give a parameterization of formal deformations in terms of two-forms obeying a cubic equation. The notion of equivalence is discussed in detail. We show that the obstruction for the construction of deformations order by order lies in the third Lie algebroid cohomology of the Dirac structure. However, the classification of inequivalent first order deformations is not given by the second Lie algebroid cohomology but turns out to be more complicated.
One way of reconciling classical and quantum mechanics is deformation quantization, which involves deforming the commutative algebra of functions on a Poisson manifold to a non-commutative, associative algebra, reminiscent of the space of quantum observables. This depends on a formal parameter $\hbar$, so that the original pointwise product is recovered when $\hbar=0$. In 1997 Kontsevich showed that a deformation quantization exists for every Poisson manifold. He furthermore gave a simple, combinatorial formula for producing a quantization of any Poisson structure on $\mathbb{R}^n$. The primar…
We develop the deformations theory of a Dirac--Jacobi structure within a fixed Courant--Jacobi algebroid. Using the description of split Courant--Jacobi algebroids as degree $2$ contact $\mathbb{N} Q$ manifolds and Voronov's higher derived brackets, each Dirac--Jacobi structure is associated with a cubic $L_\infty$ algebra for any choice of a complementary almost Dirac--Jacobi structure. This $L_\infty$ algebra governs the deformations of the Dirac--Jacobi structure: there is a one-to-one correspondence between the MC elements of this $L_\infty$ algebra and the small deformations of the Dirac-…
The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty$ algebra instead. We develop a simplified method for describing this $L_\infty$ algebra and use it to prove that the $L_\infty$ algebras corresponding to different transversals are canonically $L_\infty$-isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie grou…
Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones we propose a new twisted reality condition for the Dirac operator.
In this article, we use Harrison cohomology to provide a framework for commutative deformations. In particular, Kontsevich's result that formality of (the Hochschild complex of) an associative algebra implies its deformability is adapted for commutative algebras, with the Harrison complex.
Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds.
In these notes, we introduce formal hom-associative deformations of the quantum planes and the universal enveloping algebras of the two-dimensional non-abelian Lie algebras. We then show that these deformations induce formal hom-Lie deformations of the corresponding Lie algebras constructed by using the commutator as bracket.
In the framework of formal deformation quantization, we apply our formal moment map construction on the space of almost complex structures to recover the Donaldson-Fujiki moment map picture of the Hermitian scalar curvature. In the integrable case, it yields a formal moment map deforming the scalar curvature moment map.
In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak g,r)$ on a smooth manifold $M$. Using the formality of polydifferential operators on Lie algebroids we obtain a deformation quantization of $M$ together with a quantum group $\mathscr{U}_\hbar(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L_\infty$-morphisms which generalizes the one given by Drinfeld.
Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces
Ricci型辛流形是M. Cahen et al.本文描述了它们在分裂可解对称情形下的形变量子化。特别是，我们介绍了非形式的回火变形量子化的概念，这样的空间。我们证明了回火形变量子化的集合与实直线上的Schwartz算子乘子空间是一一对应的。证明了分裂Ricci型可解对称空间上的不变形式星积是调和非形式量子化的渐近展开。本文通过一个例子说明和部分评论了作者研究的一个关于存在大群对称的非形式量子化的问题。
We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over $R$. We apply this result to prove an annihilation theorem for obstructions of (derived) deformations of locally complete intersection ideal sheaves on projective schemes.
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 $E_2$-algebras 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 Hochschil…
