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.
theory of Hopf algebras, a Hopfalgebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf k-algebroids. If k is...
algebraic topology and algebraic geometry, there is the notion of a Hopfalgebroid which encodes the information of a presheaf of groupoids whose object...
weak Hopf algebras are to be found in as well as See Hopfalgebroid Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra...
important in a certain noncommutative Galois theory, which generates Hopfalgebroids in place of the more classical Galois groups, whereas the notion of...
categories, Hopfalgebroids etc. Ross Street, Brian Day, "Quantum categories, star autonomy, and quantum groupoids", in "Galois Theory, Hopf Algebras, and...
comodules over Hopf-algebroids to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded Hopf-algebroid is used)....
org/nlab/show/image (Definition 3.1 and Remarks 3.2) in Gabriella Böhm, Hopfalgebroids, in Handbook of Algebra (2008) arXiv:0805.3806 paragraph 2-14 at page...
replaced by a possibly noncommutative k-algebra L {\displaystyle L} . Hopfalgebroids are associative bialgebroids with an additional antipode map which...
ℓ-adic formalism (theory of ℓ-adic sheaves) extends to algebraic stacks. Hopfalgebroid - encodes the data of quasi-coherent sheaves on a prestack presentable...
Quantum categories were introduced to generalize Hopfalgebroids and groupoids. A quantum groupoid is a Hopf algebra with several objects. 2004 Stephan Stolz-Peter...
arXiv:1709.09629 [math.AT]. Baker, A.; Jeanneret, A. (2002). "Brave new Hopfalgebroids and extensions of MU-algebras". Homology, Homotopy and Applications...
v2.n4.a3. S2CID 8898100. Crainic, Marius (2003). "Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes". Commentarii...
in Princeton in 1955. His other collaborators included; J. Frank Adams (Hopf invariant problem), Jürgen Berndt (projective planes), Roger Bielawski (Berry–Robbins...
Global Information