Departamento de Matemáticas UAM

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.