Isotropy quotients of Hopf algebroids and the fundamental groupoid of digraphs

Correspondence theorems for Hopf algebroids with applications to affine groupoids

“…Remark 4. 19 Notice that the coaction (4.6) corresponds to the action described in (4.1). Namely, denote by η the composition of A η → H (i) /I (i) ≅ H/I and, for a given ring R, consider the normal subgroupoid G H/I (R) of G H (R).…”

“…A more general notion of ideal for a Hopf algebroid would involve considering at the same time an ideal in A and an ideal in such that the canonical projections form a morphism of Hopf algebroids, but we do not discuss this case herein. See Section 3.2 of [19] for additional details.…”

“…Even for Hopf algebroids over noncommutative base algebras, forms a natural Hopf algebroid structure and is called the isotropy quotient of [19]. In particular, Lemma 4.10 was previously discussed in Example 3.13 of [19].…”

“…Let us first characterize the structures in the left-hand side of the bijection in Theorem 4.26. In Example 3.6 of [19], it was already shown that any Hopf ideal I of must be of the form for a subset of for which forms a subgroupoid of . If I is also normal, (NI1) implies that and therefore .…”

“…Similar to the classical case explained above, the motivation of this work is that it could lead ultimately to a better understanding of the exactness properties of the category of (abelian) affine -groupoids. Furthermore, our results have applications on a wide variety of topics, since the interest in Hopf algebroids is widespread in different branches of mathematics: from algebraic topology, algebraic geometry, and differential geometry (see [9, 16, 31]), to the study of linear differential equations (see [14, 17]), noncommutative differential calculus [18], and in the study of the fundamental groupoid of quivers [19], to mention only a few.…”