arXiv research

In this note we prove that, for a vector bundle $E$ over a manifold $M$, a Dorfman bracket on $TM\oplus E^*$ anchored by $\operatorname{pr}_{TM}$ and with $E$ a vector bundle over $M$, is equivalent to a lift from $Γ(TM\oplus E^*)$ to linear sections of $TE\oplus T^*E\to E$, that intertwines the given Dorfman bracket w…

Using the technique of higher derived brackets developed by Voronov, we construct a homotopy Loday algebra in the sense of Ammar and Poncin associated to any symplectic $2$-manifold. The algebra we obtain has a particularly nice structure, in that it accommodates the Dorfman bracket of a Courant algebroid as the binary…

No direct generalized complex structure can be induced from $\mathbb S^6$'s nearly Kähler structure.

A new concept of Loday algebroid (and its pure algebraic version - Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are inter…

A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are con…

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus\wedge^nT^*M$ for an $m$-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an $(n+1)$-vector fi…

The paper extends a theorem for complex structures on Lie groups to Courant algebroids.

Generalized current algebras introduced by Alekseev and Strobl in two dimensions are reconstructed by a graded manifold and a graded Poisson brackets. We generalize their current algebras to higher dimensions. QP manifolds provide the unified structures of current algebras in any dimension. Current algebras give rise t…

We study generalized almost contact structures on odd-dimensional manifolds. We introduce a notion of integrability and show that the class of these structures is closed under symmetries of the Courant-Dorfman bracket, including T-duality. We define a notion of geometric type for generalized almost contact structures, …

We define solitons for the generalized Ricci flow on an exact Courant algebroid, building on the definitions of M. Garcia-Fernandez and J. Streets. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flo…

Motivated by the quest to understand the analog of non-geometric flux compactification in the context of M-theory, we study higher dimensional analogs of generalized Poisson sigma models and corresponding dual string and p-brane models. We find that higher generalizations of the algebraic structures due to Dorfman, Roy…

Study of multidifferential operators and Dorfman connections on Courant algebroids.

We define a new differential geometric structure, called Lie rackoid. It relates to Leibniz algebroids exactly as Lie groupoids relate to Lie algebroids. Its main ingredient is a selfdistributive product on the manifold of bisections of a smooth precategory. We show that the tangent algebroid of a Lie rackoid is a Leib…

Study polynomial structures on generalized tangent bundles and their compatibility with operators.

We show how the classical Moser Lemma from symplectic geometry extends to generalized complex structures (GCS) on arbitrary Courant algebroids. For this, we extend the notion of Lie derivative to sections of the tensor bundle $(\otimes^i E)\otimes(\otimes^j E^*)$ with respect to sections of the Courant algebroid $E$ us…

We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta-gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associat…

Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive …

Combines generalized and graded geometry to explore new structures.

In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Pois…

We define Dorfman connections, which are to Courant algebroids what connections are to Lie algebroids. Several examples illustrate this analogy. A linear connection $\nabla\colon \mathfrak{X}(M)\timesΓ(E)\toΓ(E)$ on a vector bundle $E$ over a smooth manifold $M$ is tantamount to a linear splitting $TE\simeq T^{q_E}E\op…

We prove that, for M theory or type II, generic Minkowski flux backgrounds preserving $\mathcal{N}$ supersymmetries in dimensions $D\geq4$ correspond precisely to integrable generalised $G_{\mathcal{N}}$ structures, where $G_{\mathcal{N}}$ is the generalised structure group defined by the Killing spinors. In other word…

Study on 3D Lie groups finds all generalized Einstein metrics.

Proposes a new geometric framework for M-theory algebroids.

This paper shows the equivalence of the categories of $N$-manifolds of degree $2$ with the category of double vector bundles endowed with a linear metric. Split Poisson $N$-manifolds of degree $2$ are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an …

We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion w…

New bracket unifies nonholonomic dynamics and Hamilton-Jacobi theory.

New bracket theory connects three nonholonomic dynamics models.

Flat connections derived from Poisson brackets on loop spaces.

It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…

The paper constructs compatible Poisson brackets on gl(N).

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp…

Researchers attempt to categorify biquandle brackets using Khovanov homology methods.

Explains biquandle brackets and quivers for a topology talk.

A new framework describes dissipation using a metriplectic 4-bracket.

Abstract: Poisson bracket on shear coordinates relates to Fenchel-Nielsen bracket on gluing parameters.

Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In th…

Extend Kauffman bracket skein module to homology theory using Heegaard splittings

Introduces a new bracket for multicontact geometry and applies it to field theories.

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves was discovered in stages passing through Thurston's earthquake deformations, Wolpert's corresponding calculations with Hamiltonian vector fields and Goldman's algebraic treatment of the latter leading to a Lie bracket on t…

We propose an extension of n-ary Nambu-Poisson bracket to superspace R^{n|m} and construct by means of superdeterminant a family of Nambu-Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace R^{n|m}. We prove in the case of …

Computes Kauffman bracket polynomial for specific 2-tangle shadows.

New Poisson bracket connects to logarithmic manifolds.

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure an…

Goldman bracket distinguishes surface homeomorphisms.

Paper derives explicit formulas for AJ-bracket of tied links.

The paper defines and proves equivalence of nonholonomic brackets in contact mechanical systems.

We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathem…

We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coefficients for moving strands over and under traces and identify a new stop condition for the recursive…