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In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in its monoidal category of representations (by a version of Tannaka–Krein duality), in its groupoid algebra or in the commutative Hopf algebroid of functions on the groupoid. Thus formalisms trying to capture quantum groupoids include certain classes of (autonomous) monoidal categories, Hopf algebroids etc.
In mathematics, a quantumgroupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the...
theoretical physicists and mathematicians focused on quantum group and quantumgroupoid applications in quantum theories; the proceedings of the meeting are published...
{\displaystyle T^{*}M} is not always integrable to a Lie groupoid. A symplectic groupoid is a Lie groupoid G ⇉ M {\displaystyle {\mathcal {G}}\rightrightarrows...
are also generalizations of Hopf algebras. Weak Hopf algebras, or quantumgroupoids, are generalizations of Hopf algebras. Like Hopf algebras, weak Hopf...
isotropy group is the group of isomorphisms from any object to itself in a groupoid.[dubious – discuss] An isotropy representation is a representation of an...
Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably...
including but not limited to computer science, physics (in particular quantum mechanics), natural language processing, control theory, probability theory...
diffeomorphisms. An orbifold groupoid is given by one of the following equivalent definitions: a proper étale Lie groupoid; a proper Lie groupoid whose isotropies...
out the gauge group to obtain the gauge groupoid as the closest description of the gauge connection in quantum field theory. For ordinary Lie algebras...
using categories, including algebraic topology, categorical topology, quantum topology, low-dimensional topology; Categorical logic and set theory in...
the fundamental groupoid of a topos E in the general theory of topoi, and also in their physical applications in nonabelian quantum theories, and recent...
automaton groupoid. Therefore, in the most general case, categories of variable automata of any kind are categories of groupoids or groupoid categories...
is a generalization of universal algebra to partial operations. partial groupoid field — the multiplicative inversion is the only proper partial operation...
notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories...
he collaborated with Vladimir Voevodsky on ∞ {\displaystyle \infty } -groupoids, following the proposal made by Alexander Grothendieck in Esquisse d'un...
groupoids" EMS Tracts in Mathematics Vol 15 (2011), A Bibliography for Categories and Algebraic Topology Applications in Theoretical Physics Quantum Algebraic...
because of the dagger. A discrete category is trivially a dagger category. A groupoid (and as trivial corollary, a group) also has a dagger structure with the...
J. Hiley. "Towards a Quantum Geometry, Groupoids, Clifford algebras and Shadow Manifolds" (PDF). Schönberg, M. (1957). "Quantum kinematics and geometry"...
mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional...
S2CID 119162795. Zbl 1080.16036. Jiang-Hua Lu, "Hopf algebroids and quantumgroupoids", Int. J. Math. 7, n. 1 (1996) pp. 47–70, https://arxiv.org/abs/q-alg/9505024...
quantization of Poisson manifolds as well as a description of the symplectic groupoid integrating a Poisson manifold as an infinite-dimensional symplectic quotient...
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