[en] We introduce the notion of double Courant–Dorfman algebra and prove that it satisfies the so-called Kontsevich–Rosenberg principle, that is, a double Courant–Dorfman algebra induces Roytenberg's Courant–Dorfman algebras on the affine schemes parametrizing finite-dimensional representations of a noncommutative algebra. The main example is given by the direct sum of double derivations and noncommutative differential 1-forms, possibly twisted by a closed Karoubi–de Rham 3-form. To show that this basic example satisfies the required axioms, we first prove a variant of the Cartan identity [LX,LY]=L[X,Y] for double derivations and Van den Bergh's double Schouten–Nijenhuis bracket. This new identity, together with noncommutative versions of the other Cartan identities already proved by Crawley-Boevey–Etingof–Ginzburg and Van den Bergh, establishes the differential calculus on noncommutative differential forms and double derivations and should be of independent interest. Motivated by applications in the theory of noncommutative Hamiltonian PDEs, we also prove a one-to-one correspondence between double Courant–Dorfman algebras and double Poisson vertex algebras, introduced by De Sole–Kac–Valeri, that are freely generated in degrees 0 and 1.
Disciplines :
Mathematics
Author, co-author :
Álvarez-Cónsul, Luis; Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid, Spain
Heluani, Reimundo; Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil
FERNANDEZ ALVAREZ, David ; University of Luxembourg > Faculty of Science, Technology and Medicine > Department of Mathematics > Team Sarah SCHEROTZKE
External co-authors :
yes
Language :
English
Title :
Noncommutative Poisson vertex algebras and Courant–Dorfman algebras
The first author is partially supported by the Spanish Ministry of Science and Innovation , through the ‘Severo Ochoa Programme for Centres of Excellence in R&D’ ( CEX2019-000904-S ) and grant PID2019-109339GB-C31 . The second author was supported by the Alexander von Humboldt Stiftung in the framework of an Alexander von Humboldt professorship endowed by the German Federal Ministry of Education and Research. During the initial stage of this work, he was supported by IMPA and CAPES through their postdoctorate of excellence fellowships at UFRJ. He deeply acknowledges their support and excellent working conditions. The third author is partially supported by CNPq grant number 305688/2019-7 .
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