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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

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)