Free Courant and derived Leibniz pseudoalgebras

Abstract

We introduce the category of generalized Courant pseudoalgebras and show that it admits a free object on any anchored module over `functions'. The free generalized Courant pseudoalgebra is built from two components: the generalized Courant pseudoalgebra associated to a symmetric Leibniz pseudoalgebra and the free symmetric Leibniz pseudoalgebra on an anchored module. Our construction is thus based on the new concept of symmetric Leibniz algebroid. We compare this subclass of Leibniz algebroids with the subclass made of Loday algebroids, which were introduced in [12] as geometric replacements of standard Leibniz algebroids. Eventually, we apply our algebro-categorical machinery to associate a differential graded Lie algebra to any symmetric Leibniz pseudoalgebra, such that the Leibniz bracket of the latter coincides with the derived bracket of the former.