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In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf k-algebroids. If k is a field, a commutative k-algebroid is a cogroupoid object in the category of k-algebras; the category of such is hence dual to the category of groupoid k-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory. The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000). They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The antipode axioms have been changed by G. Böhm and K. Szlachányi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions.

Definition

A left Hopf algebroid (H, R) is a left bialgebroid together with an antipode: the bialgebroid (H, R) consists of a total algebra H and a base algebra R and two mappings, an algebra homomorphism s: R → H called a source map, an algebra anti-homomorphism t: R → H called a target map, such that the commutativity condition s(r1) t(r2) = t(r2) s(r1) is satisfied for all r1, r2 ∈ R. The axioms resemble those of a Hopf algebra but are complicated by the possibility that R is a non-commutative algebra or its images under s and t are not in the center of H. In particular a left bialgebroid (H, R) has an R-R-bimodule structure on H which prefers the left side as follows: r1 ⋅ h ⋅ r2 = s(r1) t(r2) h for all h in H, r1, r2 ∈ R. There is a coproduct Δ: H → H ⊗R H and counit ε: H → R that make (H, R, Δ, ε) an R-coring (with axioms like that of a coalgebra such that all mappings are R-R-bimodule homomorphisms and all tensors over R). Additionally the bialgebroid (H, R) must satisfy Δ(ab) = Δ(a)Δ(b) for all a, b in H, and a condition to make sure this last condition makes sense: every image point Δ(a) satisfies a(1) t(r) ⊗ a(2) = a(1) ⊗ a(2) s(r) for all r in R. Also Δ(1) = 1 ⊗ 1. The counit is required to satisfy ε(1H) = 1R and the condition ε(ab) = ε(as(ε(b))) = ε(at(ε(b))).

The antipode S: H → H is usually taken to be an algebra anti-automorphism satisfying conditions of exchanging the source and target maps and satisfying two axioms like Hopf algebra antipode axioms; see the references in Lu or in Böhm-Szlachányi for a more example-category friendly, though somewhat more complicated, set of axioms for the antipode S. The latter set of axioms depend on the axioms of a right bialgebroid as well, which are a straightforward switching of left to right, s with t, of the axioms for a left bialgebroid given above.
Examples

As an example of left bialgebroid, take R to be any algebra over a field k. Let H be its algebra of linear self-mappings. Let s(r) be left multiplication by r on R; let t(r) be right multiplication by r on R. H is a left bialgebroid over R, which may be seen as follows. From the fact that H ⊗R H ≅ Homk(R ⊗ R, R) one may define a coproduct by Δ(f)(r ⊗ u) = f(ru) for each linear transformation f from R to itself and all r, u in R. Coassociativity of the coproduct follows from associativity of the product on R. A counit is given by ε(f) = f(1). The counit axioms of a coring follow from the identity element condition on multiplication in R. The reader will be amused, or at least edified, to check that (H, R) is a left bialgebroid. In case R is an Azumaya algebra, in which case H is isomorphic to R ⊗ R, an antipode comes from transposing tensors, which makes H a Hopf algebroid over R. Another class of examples comes from letting R be the ground field; in this case, the Hopf algebroid (H, R) is a Hopf algebra.

References
Further reading

Böhm, Gabriella (2005). "An alternative notion of Hopf algebroid". In Caenepeel, Stefaan (ed.). Hopf algebras in noncommutative geometry and physics. Proceedings of the conference on Hopf algebras and quantum groups, Brussels, Belgium, May 28–June 1, 2002. Lecture Notes in Pure and Applied Mathematics. 239. New York, NY: Marcel Dekker. pp. 31–53. ISBN 978-0-8247-5759-5. Zbl 1080.16034.
Böhm, Gabriella; Szlachányi, Kornél (2004). "Hopf algebroid symmetry of abstract Frobenius extensions of depth 2". Commun. Algebra. 32 (11): 4433–4464. arXiv:math/0305136. doi:10.1081/AGB-200034171. Zbl 1080.16036.
Jiang-Hua Lu, "Hopf algebroids and quantum groupoids", Int. J. Math. 7, n. 1 (1996) pp. 47-70, https://arxiv.org/abs/q-alg/9505024, http://www.ams.org/mathscinet-getitem?mr=95e:16037, https://dx.doi.org/10.1142/S0129167X96000050

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