If [Formula: see text] is a directed graph and field, the Leavitt path algebra of over naturally graded by group integers We formulate properties which are equivalent with being crossed product, skew ring, or ring respect to this natural grading. state main result so that also characterized in terms pre-ordered Grothendieck text]-group text]. has finitely many vertices, we characterize when str...

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...