New internal symmetry found for Lie pair algebra.
problem Understanding Lie pair structures and their associated algebras.
method Introduced a Lie algebra action by Der(L) on the L_{≤3} algebra.
result Found internal symmetry of the L_{≤3} algebra.
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48 results for “Lie 3-algebras”
The paper categorifies Lie and Courant algebroids, establishing correspondences and new constructions.
problem Categorification of Lie and Courant algebroids to better understand geometric structures.
method Introducing and studying new algebraic structures like VB-Lie 2-algebroids and VB-LWX 2-algebroids.
result Established correspondences and new constructions between Lie and Courant algebroids.
The paper defines a new algebraic structure and shows its relation to existing ones.
problem Exploring new algebraic structures related to existing ones.
method Introducing CLWX 2-algebroids and showing their relation to QP-structures and Lie 3-algebras.
result QP-structures of degree 3 give rise to CLWX 2-algebroids, a higher analogue of Courant algebroids.
We study the concept of the fourth skein module of 3-manifolds, that is a skein module based on the skein relation b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 = 0 and a framing relation L^{(1)} = aL, where a, b_0, b_3 are invertible. We introdule the concept of n-algebraic tangles (and links) and analyze the skein module for 3-a…
We study -ary commutative superalgebras and -algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their -ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
We give a local analytic characterization that a minimal surface in the 3-sphere $\, \ES^3 \subset \R^4$ defined by an irreducible cubic polynomial is one of the Lawson's minimal tori. This provides an alternative proof of the result by Perdomo (\emph{Characterization of order 3 algebraic immersed minimal surfaces of $…
We study the boundary conditions in the topologically twisted Chern-Simons matter theories with the Lie 3-algebraic structure. We find that the supersymmetric boundary conditions and the gauge invariant boundary conditions can be unified as the complexified gauge invariant boundary conditions which lead to the supergro…
New invariant distinguishes lens spaces via categorified homotopy E_3-algebra.
problem Distinguishing lens spaces using categorified homotopy E_3-algebra.
method Categorification of LMO invariant using factorization homology and E_3-algebra structure of Jacobi diagrams.
result Constructs an invariant that distinguishes lens spaces.
This work connects knot invariants to Chern-Simons theories via factorization homology.
problem Understanding knot invariants in Chern-Simons theories.
method Constructing a filtered -algebra and proving an equality between factorization homology trace and Reshetikhin-Turaev link invariant.
result Established a connection between knot invariants and Chern-Simons theories.
From 3-algebras to -Groupsmath.GT
We introduce -groups and show how they fit in the context of lattice field theory. To a manifold we associate a -group . We define the symmetric cohomology of a group with coefficients in a -module . The -group is determined by the action of on and an el…
Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in . Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the e…
Classifies 3D non-degenerate left-symmetric algebras.
problem Classifying left-symmetric algebras in 3D.
method Using Nijenhuis geometry and algebraic independence of coefficients in characteristic polynomial.
result Classification of differentially non-degenerate LSA in dimension 3.
Study finite deformations from heterotic superpotential, leading to new complex effective action.
problem Finite deformations of the Hull--Strominger system.
method Expanding the heterotic superpotential around a supersymmetric vacuum, identifying complex coordinates, and using Maurer--Cartan equation.
result Generalizes complex effective action of Kodaira--Spencer and holomorphic Chern--Simons theory, with a supersymmetric locus described by an algebra.
We consider a parabolic-like systems of differential equations involving geometrical quantities to examine uniformization theorems for two- and three-dimensional closed orientable manifolds. We find that in the two-dimensional case there is a simple gauge theoretic flow for a connection built from a Riemannian structur…
The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (contro…
Differential Cohomotopy theory predicts brane interactions via chord diagrams.
problem Quantization of brane charges and moduli spaces.
method Differential refinement of Cohomotopy theory, configuration spaces, chord diagrams.
result Higher observables on brane moduli spaces are given by weight systems on chord diagrams.
In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2 and the associated action functional of this model is finite, then the generali…
The paper explores the formality of low-dimensional manifolds using algebraic structures.
problem Investigating the formality of low-dimensional manifolds.
method Introducing Poincaré DGCAs of Hodge type and small algebra/quotient algebras to study equivalence classes of manifolds.
result A connected Poincaré DGCA of Hodge type is -quasi-isomorphic to an -algebra, with the formality determined by a specific Harrison cohomology class.
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.
problem Lie's third theorem does not hold for Lie groupoids and Lie algebroids.
method Introducing a subcategory of diffeological spaces called quasi-etale, constructing a functor mapping singular Lie groupoids to Lie algebroids.
result Lie's third theorem is valid for Lie algebroids within the context of singular Lie groupoids.
Identifies groupoid analogue of symplectic Lie groups.
problem Understanding symplectic structures on Lie groupoids.
method Introduces -symplectic Lie groupoids and explores their properties.
result Establishes a correspondence between Lie algebroid structures and -symplectic Lie groupoid structures.
Introduces Hom-Lie groups and their integrability, defining Hexp map and adjoint representation.
problem Integrability of Hom-Lie algebras and associated Hom-Lie groups.
method Definition of Hom-Lie groups and algebras, integration of Hom-Lie algebras, Hexp map definition.
result Every regular Hom-Lie algebra is integrable, Hexp map is universal.
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
The paper integrates Lie-Leibniz triples into Lie group-rack triples.
problem Integrating Lie-Leibniz triples into Lie group structures.
method Defining Lie group-rack triples and integrating finite-dimensional Lie-Leibniz triples.
result Any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple.
Lie Algebroidsmath.DG
Lie algebroids are like infinitesimal Lie groupoids.
problem None explicitly stated in the abstract.
method Overview article.
result Lie algebroids are infinitesimal counterparts of Lie groupoids.
Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebrasmath-ph
The paper connects Lie bialgebras, Rota-Baxter Lie algebras, and their properties.
problem Exploring connections between Lie bialgebras and Rota-Baxter Lie algebras.
method Introducing quadratic Rota-Baxter Lie algebras, matched pairs, bialgebras, and Manin triples.
result Established a correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras.
A Lie 2-group's left-invariant vector fields are isomorphic to its Lie 2-algebra.
problem Understanding the relationship between Lie 2-groups and their associated Lie 2-algebras.
method Analyzing the Lie 2-group structure to derive left-invariant vector fields and comparing them to the Lie 2-algebra.
result The Lie 2-algebra of a Lie 2-group is isomorphic to the Lie 2-algebra of its left-invariant vector fields.
The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.
problem Integrating Rota-Baxter operators into Lie group structures and geometries.
method Introducing Rota-Baxter operators on Lie groups, Lie algebroids, and groupoids.
result Geometrization of Rota-Baxter Lie algebras and groups.
Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale L…
New Lie group structure on vertical bisections of Lie groupoids.
problem Constructing Lie group structure on vertical bisections of Lie groupoids.
method Construct Lie group structure on the group of vertical bisections of a regular Lie groupoid, identify Lie algebra, discuss regularity properties.
result Established Lie theoretic properties of vertical bisections of Lie groupoids over non-compact bases.
Constructing -Lie algebroids via connections
problem Constructing Lie algebroids and -Lie algebroids
method Using connections generated by finite families of differential operators and dual sections
result Providing sufficient conditions for generating families to determine Lie algebroid and -Lie algebroid structures
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each …
In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (…
In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-grou…
Investigate local Lie group structure of bisections over compact manifolds
problem Study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold.
method Investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid.
result Prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.
Lie algebras with -action are studied.
problem Lie algebra structures on symplectic Lie groups.
method Definition of -quasi-Frobenius Lie algebras.
result Induced -action on -quasi-Frobenius Lie algebras.
Current Lie groupoids generalize mappings to Lie groupoids.
problem Generalizing Lie group properties to mappings into Lie groupoids.
method Study of superposition operators and Lie groupoid properties.
result Current Lie groupoids inherit properties from underlying maps.
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the -category of curved Lie algebras using homotopy theory of algebras over a complete operad.
result Equivalence of -categories between Lie algebroids and certain kinds of curved Lie algebras.
The paper studies prolongations of Lie algebras associated with pseudo -type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo -type Lie algebras.
method Analyzing prolongations of associated fundamental graded Lie algebra and associated conformal pseudo-subriemannian fundamental graded Lie algebra.
result The prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under certain conditions.
Poisson-Nijenhuis groupoidsmath.DG
Defines new structures on Lie groupoids and Lie algebroids.
problem No specific problem stated; focuses on new definitions.
method Introduces multiplicative Poisson-Nijenhuis structures and P-N Lie bialgebroids.
result Establishes a correspondence between structures on Lie groupoids and Lie algebroids.
Lie Calculusmath.GR
Lie Calculus connects differential and Lie theory using groupoids.
problem Understanding the relationship between differential and Lie theories.
method Using groupoids to link differential and Lie theories.
result Higher order theory involves higher algebra (n-fold groupoids).
Study on simplicity of Lie skew braces, proving new results for compact cases.
problem Simplicity of Lie skew braces, focusing on compact connected cases.
method Reviewing correspondence, investigating ideals and rigidity, proving main result for compact Lie skew braces.
result Compact connected simple Lie skew braces are either trivial or have simple underlying Lie groups.
Generalizes Lie supergroups to Lie superheaps.
problem No specific problem stated; generalization of Lie supergroups.
method Uses functor of points to show isomorphism between categories of pointed Lie superheaps and Lie supergroups.
result There is an isomorphism between the categories of pointed Lie superheaps and Lie supergroups.
Directly constructs Lie groupoids from Lie algebroids.
problem Local integration of Lie algebroids.
method Explicit construction of Lie groupoids.
result Finite-dimensional proof of Lie theory equivalence.
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We furthe…