By Year By Month By Week Today Search Jump to month January February March April May June July August September October November December 2016 2020 2021 2022 2023 2024 2025 2026 2027
LECTURA DE TESIS DOCTORAL

LECTURA DE TESIS DOCTORAL

Título: Non-commutative symplectic NQ-geometry and Courant algebroids

Día:   Martes, 26/01/2016

Hora:  12:00

Lugar: Aula Naranja, Instituto de Ciencias Matemáticas (ICMAT)

Doctorando: David Fernández (UAM-ICMAT)

Director: Luis Álvarez Cónsul

Abstract:

In this thesis we propose a notion of non-commutative Courant
algebroid that satisfies the Kontsevich–Rosenberg principle, whereby a
structure on an associative algebra has geometric meaning if it
induces standard geometric structures on its representation spaces.
Replacing vector fields on varieties by Crawley-Boevey’s double
derivations on associative algebras, this principle has been
successfully applied by Crawley-Boevey, Etingof and Ginzburg to
symplectic structures, and by Van den Bergh to Poisson structures.

A direct approach to define non-commutative Courant algebroids fails,
because the Cartan identities are unknown in the calculus of
non-commutative differential forms and double derivations, so in this
thesis we follow an indirect method. Following ideas of Ševera,
Roytenberg proved that symplectic NQ-manifolds of weights 1 and 2 are
in 1-1 correspondence with Poisson manifolds and Courant algebroids,
respectively. Our method to construct non-commutative Courant
algebroids is to adapt this result to a graded version of the
formalism of Crawley-Boevey, Etingof and Ginzburg.

We start generalizing to graded associative algebras the theories of
bi-symplectic forms and double Poisson brackets of
Crawley-Boevey–Etingof–Ginzburg and Van den Bergh, respectively,
obtaining a notion of bi-symplectic NQ-algebra. In this framework, we
prove suitable Darboux theorems and prove a 1-1 correspondence between
appropriate bi-symplectic NQ-algebras of weight 1 and Van den Berg’s
double Poisson algebras. We then use suitable non-commutative Lie and
Atiyah algebroids to describe bi-symplectic N-graded algebras of
weight 2 defined by means of graded quivers, in terms of Van den
Berg’s pairings on projective bimodules. Using non-commutative derived
brackets, we calculate the algebraic structure that corresponds to
symplectic NQ-algebras of this type. By the analogy with Roytenberg’s
correspondence, we call this structure a double Courant–Dorfman algebra.



Location Día: Martes, 26/01/2016 Hora: 12:00 Lugar: Aula Naranja, Instituto de Ciencias Matemáticas (ICMAT)