Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
The L∞-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one L∞-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We a…
Study on deformations of pre-symplectic structures using an L-infinity algebra.
problem Deformation theory of pre-symplectic structures.
method Parametrization of deformations using Koszul L-infinity algebra.
result A quotient of the Koszul L-infinity algebra is isomorphic to the L-infinity algebra controlling foliations.
Study YB operators and their deformations, finding integrable and nontrivial cases.
problem Understanding deformations of Yang-Baxter operators and their integrability.
method Relating deformations to Lie algebra deformations, analyzing cohomology groups.
result Existence of integrable YB deformations and nontrivial cases not arising from SD deformations.
2-compatible Lie algebras are quadratic deformations of Lie algebras with specific constraints.
problem Classifying contact Lie algebras using quadratic deformations.
method Defining 2-compatible Lie algebras as quadratic deformations of Lie algebras and studying the constraints on these deformations.
result Any (2p+1)-dimensional contact Lie algebra is isomorphic to a quadratic deformation of the Heisenberg algebra.
The paper studies deformations of Lie ideals in Lie algebras.
problem Understanding deformations of Lie ideals in Lie algebras.
method Develops deformation theory, compares cohomologies, enriches deformation complex.
result Deformation cohomology classes differentiate smooth deformations of ideals.
Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.
problem Deformation theory of Lie groupoids and algebroids.
method Defining a morphism between deformation complexes and Hochschild complexes, applying to adiabatic groupoids.
result Induced van Est map from geometric to algebraic deformation cohomology.
In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.
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 deformed algebra used in \cite{Ma2013} is only an example among others, and how they o…
Geometrically deforms L∞ algebras to Lie algebroids, revealing new invariants.
problem Classifying geometric invariants of L∞ algebras arising from vector bundles. method Define geometric deformations of curved L∞ algebras and show they correspond to Lie algebroid structures. result Geometric deformations of L∞ algebras classify new geometric invariants. Characterizes deformability of maps into projective space using Lie algebra forms.
problem Deformability of maps into projective space across different geometries.
method Characterization via Lie algebra valued 1-forms.
result Unified approach to known results in deformability.
Simplified method for L∞ algebra of Dirac structures.
problem Deformation of Dirac structures.
method Simplified method for L∞ algebra. result Canonical L∞-isomorphism of L∞ algebras. Geometric deformations preserve post-Lie algebra structure in regularity structures.
problem Deriving geometric deformations of post-Lie algebras.
method Extending geometrical notions of torsion and curvature, deriving compatibility conditions.
result Derives a pre-Lie structure for regularity structures, isomorphic to a post-Lie algebra.
This research classifies deformations of Yang-Baxter operators using cohomology of n-Lie algebras.
problem Classifying deformations of Yang-Baxter operators via cohomology of n-Lie algebras. method Introducing a cohomology theory for n-ary self-distributive objects, showing natural injections and isomorphisms, and constructing deformation theories. result The self-distributive deformations classify the Yang-Baxter operator deformations, with nontrivial examples provided.
Unified approach to deform Lie-Hamilton systems using Poisson-Hopf algebra.
problem Deforming Lie systems with quantum algebras.
method Poisson-Hopf algebra deformations applied to Lie-Hamilton systems.
result Unified approach to deformations of Lie-Hamilton systems on the real plane.
The paper quantizes Hessian structures on R^2 using KV-algebras.
problem Quantizing Hessian structures on a 2D space.
method Deformation quantization within Koszul-Vinberg algebras.
result Established links between deformation theory and Hessian geometry.
Study on deformations of symmetric spaces using Jordan algebras.
problem Deformability of symmetric Einstein metrics on compact Lie algebras.
method Developed sandwich operators and quadratic Casimir operators for compact Lie algebras; calculated obstruction integrals from invariant polynomials; explored relation to simple Jordan algebras.
result Proved the nonlinear instability of most infinitesimally deformable irreducible compact symmetric spaces.
Deforms orbits in Lie algebras to Lagrangian submanifolds.
problem Deforming orbits in semisimple Lie algebras.
method Coadjoint orbit deformation and Hermitian symplectic form.
result Constructs Lagrangian submanifolds.
The paper models and deforms A-infinity structures for bordered knot algebras.
problem Understanding A-infinity structures for bordered knot algebras.
method Combinatorial model and weighted deformation of A-infinity structures.
result Explicit combinatorial model for bordered knot algebras' A-infinity structure.
Develops deformation theory for symplectic foliations using L∞-algebras.
problem Deformation of symplectic foliations.
method Uses L∞-algebras to control deformation problems. result Establishes a correspondence between small deformations and Maurer-Cartan elements of L∞-algebra. Develops deformed algebras and groups from formal series over groupoids, with applications to cobordism.
problem Formal series over groupoids for algebras and groups.
method Deformation along a family of indexes, formal series.
result Regular Frölicher Lie groups and sometimes Fréchet Lie groups.
Study infinitesimal deformations of Lie algebroid pairs.
problem Infinitesimal deformations of Lie algebroid pairs.
method Investigate isomorphism classes of infinitesimal deformations of (L,A) modulo automorphisms from exponentials of derivations of L and those from the exponentials of inner derivations of L. result Find the associated governing L∞-algebras in the sense of extended deformation theory. The paper studies deformations of Nijenhuis structures in Lie algebras and algebroids.
problem Deformations of Nijenhuis structures in Lie algebras and algebroids.
method Operadic study, introduction of homotopy Nijenhuis Lie algebras, construction of L∞-algebras for deformations. result The Poincaré Lemma holds for certain Nijenhuis operators, confirming a conjecture.
Homotopy operators help describe structures in equivariant deformation problems.
problem Equivariant deformation problems in algebraic structures.
method Use homotopy operators for an L∞-algebra associated with the problem. result Smooth parametrization of the space of structures around a given one.
We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.
In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold (C∞ or complex analytic).
Nijenhuis forms help understand deformations of Lie algebroids and Poisson structures.
problem Understanding deformations of Lie algebroids and related structures.
method Introducing Nijenhuis forms on Lie-infinity algebras.
result Nijenhuis forms provide a new perspective on Poisson-Nijenhuis and quasi-Nijenhuis structures.
Defines Courant pairs and studies their deformations via cohomology.
problem Deforming Courant pairs and understanding their structure.
method Constructs a cohomology bicomplex with coefficients in a module.
result Establishes a connection between Hochschild and Leibniz cohomologies.
Study coisotropic submanifolds in Jacobi manifolds with algebraic invariants.
problem Deformations of coisotropic submanifolds in Jacobi manifolds.
method Attach algebraic invariants (L-infinity[1] algebra and BFV-complex) to coisotropic submanifolds.
result Control formal and non-formal coisotropic deformation problems.
A deformation of the Orlik-Solomon algebra of a matroid M is defined as a quotient of the free associative algebra over a commutative ring R with 1. It is shown that the given generators form a Groebner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as R-…
New L∞ algebra governs deformations of Dirac-Jacobi structures.
problem Deformation theory of Dirac-Jacobi structures.
method Using higher derived brackets and split Courant-Jacobi algebroids, an L∞ algebra is associated with each Dirac-Jacobi structure. result There is a one-to-one correspondence between MC elements of the L∞ algebra and small deformations of the Dirac-Jacobi structure. Deform symplectic structures using moment maps and Lie algebra elements.
problem Dealing with symplectic structure deformations on manifolds.
method Using quasi-Poisson theory and Lie algebra elements to deform symplectic structures.
result Concrete examples of symplectic structure deformations on complex projective and Grassmannian spaces.
Study complex structure deformations on Lie algebras and Dolbeault cohomology.
problem Deformations of complex structures on Lie algebras and their associated Dolbeault cohomology.
method Construct a complete deformation of complex structures similar to the Kuranishi family, showing extension isomorphism validity.
result Analytic open subset of deformations where Dolbeault cohomology can be computed by left invariant tensor fields.
Explains how pre-symplectic structures can be changed.
problem Understanding how pre-symplectic structures can be deformed.
method Uses Dirac geometry to explain the geometric origin of L∞-algebra controlling deformations. result Discovers the geometric origin of the L∞-algebra controlling deformations of pre-symplectic structures. A differential calculus, differential geometry and the E-R Gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the defo…
New method deforms function algebras on manifolds using spectral decomposition.
problem Deforming function algebras on compact Riemannian manifolds.
method Introducing a bilinear product on the finite spectral core of smooth functions using unimodular phases.
result The product extends to a Sobolev algebra and admits iteration under certain conditions.
Study Lie algebras with complex structures, focusing on degenerations and deformations.
problem Understanding the space of Lie algebras with complex structures and their transformations.
method Identifying invariants that remain consistent under degenerations and applying to four-dimensional case.
result Found invariants that help in understanding the behavior of Lie algebras under complex structures.
We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain A∞ algebra structures and some canonically defined deformations of s…
We investigate Nijenhuis deformations of L∞-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie n-algebras and n-plectic manifolds, are included.
Goto proved deformation smoothness for special geometric structures.
problem Deformation smoothness of geometric structures.
method Used L∞-algebra and homotopy abelian properties. result Unified and provided new proofs of deformation smoothness.
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an L∞-algebra to each submanifold. result Controls the deformation theory of Lagrangian NQ-submanifolds using an L∞-algebra. Necessary and sufficient conditions for some deformation algebras to provide formal Frobenius structures are given. Also, examples of formal Frobenius structures with fundamental tensor that is not of the deformation type and examples of symmetric non-metric connections are presented.
Abstract proposes a new categorical approach to quantization of Poisson algebras.
problem Quantization of Poisson algebras.
method Defining quantization categories as subcategories of R-module categories with classical limits.
result Categories of strict deformation quantization, prequantization, and matrix regularization are equivalent, while Poisson enveloping algebra is not.
Defines Killing (super)algebras for spin manifolds, including gauge transformations.
problem Understanding deformations of spin structures on manifolds.
method Introduces a new algebraic structure, studies its deformations using Spencer cohomology.
result Identifies subclasses of deformations and reconstructs supersymmetric backgrounds.
We classify nontrivial deformations of the standard embedding of the Lie algebra $\Vect(S^1)$ of smooth vector fields on the circle, into the Lie algebra~$\PD(S^1)$ of pseudodifferential symbols on S1. This approach leads to deformations of the central charge induced on $\Vect(S^1)$ by the canonical central extensio…
The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.
problem Characterizing and studying deformations and cohomologies of relative Rota-Baxter operators.
method Constructing graded Lie algebras and studying their Maurer-Cartan elements, cohomology, and deformations.
result Homomorphisms between cohomology groups of relative Rota-Baxter operators and deformation cohomology groups of left-symmetric algebroids.
Proves super-version of index theorem from algebraic cobordism invariants.
problem Cobordism invariants in supersymmetric quantum mechanics.
method Trace methods for deformation quantization.
result Recovery of cobordism invariant using trace methods.
The paper studies deformations of submanifolds using a new algebraic structure.
problem Deformations of submanifolds in geometric contexts.
method Introduces strongly homotopy Lie algebras to govern deformations of submanifolds.
result Deformations of submanifolds form an analytic variety under certain assumptions.