# Courant algebroids, derived brackets and even symplectic supermanifolds

@article{Roytenberg1999CourantAD, title={Courant algebroids, derived brackets and even symplectic supermanifolds}, author={Dmitry Roytenberg}, journal={arXiv: Differential Geometry}, year={1999} }

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complicated properties of Courant algebroids, we interpret them by associating to each Courant algebroid a strongly homotopy Lie algebra in a natural way. Next, we propose an alternative…

## 322 Citations

### String Principal Bundles and Courant Algebroids

- MathematicsInternational Mathematics Research Notices
- 2019

Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids encode the infinitesimal symmetries of $S^1$-gerbes. At the same time, transitive…

### Courant algebroids from categorified symplectic geometry

- Mathematics
- 2009

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed nondegenerate (n+1)-form. The case…

### On LA-Courant Algebroids and Poisson Lie 2-Algebroids

- MathematicsMathematical Physics, Analysis and Geometry
- 2020

This paper reformulates Li-Bland’s definition for LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises…

### COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY: DRAFT VERSION

- Mathematics
- 2009

In categorified symplectic geometry, one studies the categorified algebraic and geometric structures that naturally arise on manifolds equipped with a closed non-degenerate n + 1-form. The case…

### On the structure of graded symplectic supermanifolds and Courant algebroids

- Mathematics
- 2002

This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles…

### Hierarchies and compatibility on Courant algebroids

- Mathematics
- 2011

We extend to the context of Courant algebroids several hierarchies that can be constructed on Poisson-Nijenhuis manifolds. More precisely, we introduce several notions (Poisson-Nijenhuis,…

### Quasiclassical Lian-Zuckerman Homotopy Algebras, Courant Algebroids and Gauge Theory

- Mathematics
- 2011

We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to “light modes”, i.e. the elements of zero conformal weight,…

### On higher analogues of Courant algebroids

- Mathematics
- 2010

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕ ∧nT*M for an m-dimensional manifold. As an application, we…

### Transitive Courant algebroids

- MathematicsInt. J. Math. Math. Sci.
- 2005

A class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection is described.

### Conformal Courant Algebroids and Orientifold T-duality

- Mathematics
- 2011

We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant…

## References

SHOWING 1-10 OF 28 REFERENCES

### Manin Triples for Lie Bialgebroids

- Mathematics
- 1995

In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does…

### Courant Algebroids and Strongly Homotopy Lie Algebras

- Mathematics
- 1998

Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the direct sum of tangent and cotangent bundles with the bracket introduced by T. Courant for the study…

### Strongly homotopy Lie algebras

- Mathematics
- 1994

The present paper can be thought of as a continuation of the paper "Introduction to sh Lie algebras for physicists" by T. Lada and J. Stasheff (International Journal of Theoretical Physics Vol. 32,…

### MOMENTUM MAPPINGS AND POISSON COHOMOLOGY

- Mathematics
- 1996

We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentum…

### Poisson Lie groups, dressing transformations, and Bruhat decompositions

- Mathematics
- 1990

A Poisson Lie group is a Lie group together with a compatible Poisson structure. The notion of Poisson Lie group was first introduced by Drinfel'd [2] and studied by Semenov-Tian-Shansky [17] to…

### Lie bialgebroids and Poisson groupoids

- Mathematics
- 1994

Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a…

### Quantum Groups

- Mathematics
- 1993

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions…

### The Geometry of the Master Equation and Topological Quantum Field Theory

- Mathematics
- 1997

In Batalin–Vilkovisky formalism, a classical mechanical system is specified by means of a solution to the classical master equation. Geometrically, such a solution can be considered as a QP-manifold,…

### Exact Gerstenhaber algebras and Lie bialgebroids

- Mathematics
- 1995

We show that to any Poisson manifold and, more generally, to any triangular Lie bialgebroid in the sense of Mackenzie and Xu, there correspond two differential Gerstenhaber algebras in duality, one…

### From Poisson algebras to Gerstenhaber algebras

- Mathematics
- 1996

© Annales de l’institut Fourier, 1996, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions…