In mathematics, a quantum groupoid is any of a number of notions in
noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
analogous to the notion of
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
*'' Group'' with a partial fun ...
. In usual geometry, the information of a groupoid can be contained in its
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
of representations (by a version of
Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topologi ...
), in its groupoid algebra or in the commutative
Hopf algebroid 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 i ...
of functions on the groupoid. Thus formalisms trying to capture quantum groupoids include certain classes of (autonomous)
monoidal categories
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left and ...
,
Hopf algebroid 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 i ...
s etc.
References
* Ross Street, Brian Day, "Quantum categories, star autonomy, and quantum groupoids", in "Galois Theory, Hopf Algebras, and Semiabelian Categories", Fields Institute Communications 43 (American Math. Soc. 2004) 187–226;
* Gabriella Böhm, "Hopf algebroids", (a chapter of) Handbook of algebra, Vol. 6, ed. by M. Hazewinkel, Elsevier 2009, 173–236
* Jiang-Hua Lu, "Hopf algebroids and quantum groupoids", Int. J. Math. 7, n. 1 (1996) pp. 47–70, , ,
Algebraic structures
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