We develop the deformation theory of a Dirac-Jacobi structure within a fixed Courant-Jacobi algebroid. Using the description of split Courant-Jacobi algebroids as degree 2 contact NQ manifolds and Voronov's higher derived brackets, each Dirac-Jacobi structure is associated with a cubic L-infinity algebra for any choice of a complementary almost Dirac-Jacobi structure. This L-infinity algebra governs the deformations of the Dirac-Jacobi structure: there is a one-to-one correspondence between the MC elements of this L-infinity algebra and the small deformations of the Dirac-Jacobi structure. Further, by Cattaneo and Schatz's equivalence of higher derived brackets, this L. algebra does not depend (up to L-infinity-isomorphisms) on the choice of the complementary almost Dirac-Jacobi structure. These same ideas apply to get a new proof of the independence of the L-infinity algebra of Dirac structure from the choice of a complementary almost Dirac structure (a result proved using other techniques by Gualtieri, Matviichuk and Scott). (C) 2021 Elsevier B.V. All rights reserved.
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