We study Lie bialgebroid crossed modules which are pairs of algebroid in duality that canonically give rise to bialgebroids. A one-one correspondence between such and co-quadratic Manin triples [Formula: see text] is established, where a pair transverse Dirac structures text].

A bstract Recently Leutheusser and Liu [1, 2] identified an emergent algebra of Type III 1 in the operator $$ \mathcal{N} N = 4 super Yang-Mills theory for large N . Here we describe some 1/ corrections to this picture show that becomes II ? The is crossed product by its modular automorphism group. In context...