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Constructs in non-commutative geometry
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
Quantum_groupoid
Study of categorified structures
categories of groupoids. Quantum Algebraic Topology Quantum geometry Quantum gravity Quantum group Topological quantum field theory Local quantum field theory
Higher-dimensional_algebra
theoretical physicists and mathematicians focused on quantum group and quantum groupoid applications in quantum theories; the proceedings of the meeting are published
Timeline_of_quantum_mechanics
ISSN 1432-0916. Chang, Liang (2014-04-01). "Kitaev models based on unitary quantum groupoids". Journal of Mathematical Physics. 55 (4): 041703. arXiv:1309.4181
Quantum_double_model
Construction in algebra
are also generalizations of Hopf algebras. Weak Hopf algebras, or quantum groupoids, are generalizations of Hopf algebras. Like Hopf algebras, weak Hopf
Hopf_algebra
Ukrainian and French mathematician
algebras and quantum groups, as well as quantum hypergroups and quantum groupoids. Vainerman's research advanced the theory of quantum groups, including
Leonid_I._Vainerman
Duality between a group and its representations
for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual Hopf algebroids
Tannaka–Krein_duality
History of maths
using categories, including algebraic topology, categorical topology, quantum topology, low-dimensional topology; Categorical logic and set theory in
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Generalized manifold
diffeomorphisms. An orbifold groupoid is given by one of the following equivalent definitions: a proper étale Lie groupoid; a proper Lie groupoid whose isotropies
Orbifold
Type of order at absolute zero
groupoids" EMS Tracts in Mathematics Vol 15 (2011), A Bibliography for Categories and Algebraic Topology Applications in Theoretical Physics Quantum Algebraic
Topological_order
Quantum mechanics posed in terms of category theory
Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably
Categorical_quantum_mechanics
Hypothesis in mathematical category theory
homotopy hypothesis states, homotopy-theoretically speaking, that the ∞-groupoids are spaces. One version of the hypothesis was claimed to be proved in
Homotopy_hypothesis
Mathematical structure in differential geometry
{\displaystyle T^{*}M} is not always integrable to a Lie groupoid. A symplectic groupoid is a Lie groupoid G ⇉ M {\displaystyle {\mathcal {G}}\rightrightarrows
Poisson_manifold
Romanian–born American mathematician
-L. Chau and W. Nahm, Eds., Plenum Press, 1990. From Poisson Groupoids to Quantum Groupoids, and Back, in Proceedings of the XIX International Conference
Meinhard_E._Mayer
Branch of mathematics
graded algebras; and constructions related to deformation quantization, groupoid C*-algebras, cyclic homology, and K-theory. A standard example is the noncommutative
Noncommutative_geometry
Uniformity in all orientations
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
Isotropy
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
Nonabelian_algebraic_topology
S2CID 119162795. 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
Hopf_algebroid
Branch of mathematics
and Groupoids. Booksurge. ISBN 978-1-4196-2722-4. (Provides a well-motivated, geometric account of general topology, and shows the use of groupoids in
Topology
Branch of mathematics
algebra K-theory Lie algebroid Lie groupoid Ramification theory Serre spectral sequence Sheaf Topological quantum field theory Hatcher, Allen. "Algebraic
Algebraic_topology
Graphical representation of a morphism
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
String_diagram
Study of abstract machines and automata
automaton groupoid. Therefore, in the most general case, categories of variable automata of any kind are categories of groupoids or groupoid categories
Automata_theory
Applications of category theory
including but not limited to computer science, physics (in particular quantum mechanics), natural language processing, control theory, probability theory
Applied_category_theory
Algebraic structure
subsets, even if the subsets are not closed under all operations." partial groupoid field — the multiplicative inversion is the only proper partial operation
Partial_algebra
Field of mathematical analysis
Geometric Analysis Atiyah–Singer index theorem Geometric analysis Lie groupoid Pseudogroup Morse theory Structural stability Harmonic map Smale, S. (January
Global_analysis
Generalisation of a sheaf; a fibered category that admits effective descent
with image V. A stack is called a stack in groupoids or a (2,1)-sheaf if it is also fibered in groupoids, meaning that its fibers (the inverse images
Stack_(mathematics)
Russian mathematician (born 1962)
he collaborated with Vladimir Voevodsky on ∞ {\displaystyle \infty } -groupoids, following the proposal made by Alexander Grothendieck in Esquisse d'un
Mikhail_Kapranov
Derivative used in gauge theories
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
Gauge_covariant_derivative
arrangements: Weyl groupoids and simplicial arrangements, Bull. London Math. Soc. 43 (2011), no.4, 734-744. Cuntz, Heckenberger: Finite Weyl groupoids, J. Reine
Nichols_algebra
Type of category in category theory
categories, whose internal language, linear type systems, are suitable for both quantum and classical computation. Named after René Descartes (1596–1650), French
Cartesian_closed_category
British quantum physicist (1935–2025)
J. Hiley. "Towards a Quantum Geometry, Groupoids, Clifford algebras and Shadow Manifolds" (PDF). Schönberg, M. (1957). "Quantum kinematics and geometry"
Basil_Hiley
Symmetric monoidal category with a special involution
categorical quantum mechanics, an area that now also considers dagger symmetric monoidal categories when dealing with infinite-dimensional quantum mechanical
Dagger symmetric monoidal category
Dagger_symmetric_monoidal_category
Concept in category theory
a symmetric monoidal functor is the mathematical model of topological quantum field theory. Let B o r d ⟨ n − 1 , n ⟩ {\displaystyle \mathbf {Bord} _{\langle
Monoidal_functor
French mathematician (born 1947)
(functional analysis) Higgs boson C*-algebra Noncommutative quantum field theory M-theory Groupoid Spectral triple Criticism of non-standard analysis Riemann
Alain_Connes
Category equipped with involution
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
Dagger_category
Concept in algebraic geometry
over U where every morphism is an isomorphism (i.e., each fiber of p is a groupoid). moduli stack of principal bundles Behrend 2002, Example 20.2. Behrend
Moduli stack of vector bundles
Moduli_stack_of_vector_bundles
Branch of mathematics
{\text{Sets}}} which can be generalized further to have targets of higher groupoids (which are expected to be modelled by homotopy types). These derived stacks
Derived_algebraic_geometry
Set with associative invertible operation
x)\simeq G} . More generally, a groupoid is any small category in which every morphism is an isomorphism. In a groupoid, the set of all morphisms in the
Group_(mathematics)
Jpn. 29, 459–492, 1977 Lu, Jiang-HUA (1996), "Hopf Algebroids and Quantum Groupoids", International Journal of Mathematics, 07: 47–70, arXiv:q-alg/9505024
Associative_bialgebroid
Symbols for constants, special functions
factorial the complete elliptic integral of the third kind the fundamental groupoid osmotic pressure π {\displaystyle \pi } represents: Archimedes' constant
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Group that is also a differentiable manifold with group operations that are smooth
also to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement
Lie_group
Italian mathematician and physicist (born 1967)
2003 by Maxim Kontsevich), as well as a description of the symplectic groupoid integrating a Poisson manifold as an infinite-dimensional symplectic quotient
Alberto_Cattaneo
English mathematician (1935–2024)
fundamental groupoid (1998), Topology and Groupoids (2006) and Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids (EMS
Ronald_Brown_(mathematician)
Combination of higher category theory with Chern–Simons theory
concepts defined in any cohesive ∞-topos, for example that of smooth ∞-groupoids. Principal bundles on which Lie groups act are for example replaced by
∞-Chern–Simons_theory
American mathematician
15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry. Mackey earned
George_Mackey
Category admitting tensor products
matter physics. Braided monoidal categories have applications in quantum information, quantum field theory, and string theory. A monoidal category is a category
Monoidal_category
French mathematician (1928–2014)
Agamben and Hervé Le Tellier. Gallimard. p. 64. ISBN 978-2-07-316366-0. ∞-groupoid λ-ring AB5 category Abelian category Accessible category Algebraic geometry
Alexander_Grothendieck
*-algebra of bounded operators on a Hilbert space
Neumann algebras of a measurable equivalence relation and a measurable groupoid can be defined. These examples generalise von Neumann group algebras and
Von_Neumann_algebra
Cellular automaton that can be run backwards
generalize the defining axiom (for a single binary operation) of a central groupoid. As Boykett argues, any one-dimensional reversible cellular automaton is
Reversible_cellular_automaton
French mathematician (1869–1951)
made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians
Élie_Cartan
Mathematical objects that generalise the notion of Hilbert spaces
role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras. Let A {\displaystyle A} be a C*-algebra (not
Hilbert_C*-module
English mathematician (1937–2020)
have deep connections to string theory. Conway introduced the Mathieu groupoid, an extension of the Mathieu group M12 to 13 points. As a graduate student
John_Horton_Conway
Ordered chemical structure with no repeating pattern
instead of lattices, quasilattices must be used. Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the
Quasicrystal
Swiss physicist and mathematician
quantization of Poisson manifolds as well as a description of the symplectic groupoid integrating a Poisson manifold as an infinite-dimensional symplectic quotient
Giovanni_Felder
German mathematician
Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "Re-gauging groupoid, symmetries and degeneracies for Graph Hamiltonians and applications to
Ralph_Kaufmann
Nichols algebra there is by attached a generalized root system and a Weyl groupoid. These are classified in. In particular several Dynkin diagrams (for inequivalent
List of finite-dimensional Nichols algebras
List_of_finite-dimensional_Nichols_algebras
Canadian mathematician
S2CID 119749421. Joyal, André; Tierney, Myles (2000). "On the theory of path groupoids". Journal of Pure and Applied Algebra. 149: 69–100. doi:10.1016/S0022-4049(98)00164-9
André_Joyal
Connects set theory with category theory
Crane, Louis; Frenkel, Igor B. (1994-10-01). "Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases". Journal of Mathematical
Categorification
Structure in group theory (in mathematics)
inductive groupoid can always be constructed from an inverse semigroup, and conversely. More precisely, an inverse semigroup is precisely a groupoid in the
Inverse_semigroup
Dutch mathematician
foliations and Lie groupoids. Recently Moerdijk pursues, among other topics, research on the theory of operads, on the logic structure of quantum information
Ieke_Moerdijk
Polish mathematician and physicist
and physicist. The main areas of his research were the theory of Banach groupoids and algebroids related to the structure of W*-algebras, quantization of
Anatol_Odzijewicz
Variant of the notion of the center of a monoid, group, or ring to a category
"Drinfeld double for orbifolds", Israel mathematical conference proceedings. Quantum groups. Proceedings of a conference in memory of Joseph Donin, Haifa, Israel
Center_(category_theory)
algebra. Groupoid algebra. Suppose G = ( G 0 , G 1 ) {\displaystyle G=(G_{0},G_{1})} is a groupoid and let K [ G ] {\displaystyle K[G]} be the groupoid algebra
Weak_Hopf_algebra
Type of topological space
Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids. European Mathematical Society Tracts in Mathematics Vol 15. ISBN 978-3-03719-083-8
CW_complex
Greek and French mathematician (born 1955)
works on operator algebras, K-theory of operator algebras, groupoids, locally compact quantum groups and singular foliations. In 2002 with Nigel Higson
Georges_Skandalis
Infinite-dimensional group in topology
higher group. It can be thought of the topological realization of the groupoid B U ( 1 ) {\displaystyle \mathbf {B} U(1)} whose object is a single point
String_group
Semigroup in abstract algebra
e = ett* for some projection e. In a *-semigroup, PI(S) is an ordered groupoid with the partial product given by s⋅t = st if s*s = tt*. In terms of examples
Semigroup_with_involution
American mathematician
(2009). "Canonical extensions of the Johnson homomorphisms to the Torelli groupoid". Advances in Mathematics. 221 (2): 627–659. doi:10.1016/j.aim.2009.01
Robert_Penner
British-Australian mathematician
"Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs". Advances in Mathematics. 331: 268–325. arXiv:1610.00343. doi:10
Jacqui_Ramagge
Dutch mathematician (1942–2010)
will turn out to be crucial for proving the analogous theorem for Lie groupoids and for its applications to Poisson geometry. His work with Alberto Grünbaum
Hans_Duistermaat
Generalization of associativity properties
include, for example, modules over a commutative ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc. Given a
Operad
Branch of mathematics
Although in general, it is more convenient/required to work with functors of groupoids instead of sets. This is true for moduli of curves. Infinitesimals have
Deformation_(mathematics)
Algebraic structure
Generalized inverse Identity element Light's associativity test Principal factor Quantum dynamical semigroup Semigroup action Semigroup ring Weak inverse The closure
Semigroup
Polish mathematician
Grabowska (11 November 2015). "Graded Bundles in the Category of Lie Groupoids". Symmetry, Integrability and Geometry: Methods and Applications. 11 (11 ed
Janusz_Grabowski
QUANTUM GROUPOID
QUANTUM GROUPOID
Boy/Male
Hindu, Indian
Calm
Male
English
English surname transferred to forename use, derived from the Norman baronial name Cuinchy, a derivative of Roman Quintus, QUINCY means "fifth."
Surname or Lastname
English
English : nickname from Middle English cointe, quointe ‘known’ (via Old French, from Latin cognitus ‘known’). The Middle English word was used in various senses, any of which could have given rise to the surname: ‘cunning’, ‘crafty’, ‘knowledgeable’ (especially about dress, hence ‘elegant’), ‘attractive’. The sense development continued with ‘odd’ or ‘unusual’, the normal meaning of the modern English word ‘quaint’.German and Dutch : variant of Quandt.
Boy/Male
Latin Biblical
Born fourth.
Biblical
fourth
Girl/Female
Biblical
Fourth.
Surname or Lastname
South German
South German : occupational name for an official in charge of the legal auction of property confiscated in default of a fine; such a sale was known in Middle High German as a gant (from Italian incanto, a derivative of Late Latin inquantare ‘to auction’, from the phrase In quantum? ‘To how much (is the price raised)?’).German : metonymic occupational name for a cooper, from Middle High German ganter, kanter ‘barrel rack’.German : variant of Gander 3.English : occupational name for a glover, from Old French gantier, an agent derivative of gant ‘glove’ (see Gant).
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : habitational name from any of several places in France deriving their names from the Gallo-Roman personal name Quintus, meaning ‘fifth(-born)’ + the locative suffix -acum. The earliest bearers of the name in England were from Cuinchy in Pas-de-Calais, but other stocks may be from Quincy-sous-Sénard in Seine-et-Oise or Quincy-Voisins in Seine-et-Marne.The American Quincy family were established in MA by Edmund Quincy in 1633. Fifth in descent was Josiah Quincy (1744–75), a leading patriot, who was sent to England to argue the colonists’ case in 1774. His son Josiah (1772–1864) was a powerful opponent of slavery, president of Harvard, and mayor of Boston, a post also held by several of his descendants. The traditional pronunciation is “Quinzyâ€.
Surname or Lastname
English
English : from the personal name Horace, Latin Horatius, a Roman family name of unknown origin, associated chiefly with the name of the poet Quintus Horatius Flaccus (65–8 bc).
Boy/Male
Danish, Finnish, French, German, Latin, Shakespearean, Swedish
Born Fifth
QUANTUM GROUPOID
QUANTUM GROUPOID
Boy/Male
Tamil
Bajrangbali | பஜரஂகபலீÂ
With strength of diamond, Lord Hanuman
Girl/Female
American, Australian, British, Christian, English, French, Hebrew, Welsh
White Wave; Fair Phantom; Juniper Berry; Form of Geneva; White and Smooth; Soft; Race of Women; White Race
Boy/Male
Czech
Fighting far away.
Boy/Male
Hindu, Indian
Owning Knowledge; God of Education
Girl/Female
Indian
Beautiful, To consult with Allah, Diverted toward Allah
Girl/Female
Arabic, Australian, French, Muslim
Reconciling
Girl/Female
Indian
The one who knows the supreme
Boy/Male
Hindu, Indian
Lord of the Clan
Boy/Male
Indian, Sanskrit
Solid; Dense; Harsh
Boy/Male
Hindu, Indian
Lord of Light
QUANTUM GROUPOID
QUANTUM GROUPOID
QUANTUM GROUPOID
QUANTUM GROUPOID
QUANTUM GROUPOID
n.
A quantic of the eighth degree.
n.
A quantic of the fifth degree. See Quantic.
n.
A quantic of the seventh degree.
n.
A homogeneous algebraic function of two or more variables, in general containing only positive integral powers of the variables, and called quadric, cubic, quartic, etc., according as it is of the second, third, fourth, fifth, or a higher degree. These are further called binary, ternary, quaternary, etc., according as they contain two, three, four, or more variables; thus, the quantic / is a binary cubic.
n.
A quantic of the second degree. See Quantic.
n.
Quantity; amount.
pl.
of Quantum
n.
A definite portion of a manifoldness, limited by a mark or by a boundary.
n.
A function involving the coefficients and the variables of a quantic, and such that when the quantic is lineally transformed the same function of the new variables and coefficients shall be equal to the old function multiplied by a factor. An invariant is a like function involving only the coefficients of the quantic.
n.
A punting pole with a broad flange near the end to prevent it from sinking into the mud; a setting pole.
n.
A fanciful, odd, or extravagant notion; a quant fancy; an unnatural or affected conception; a witty thought or turn of expression; a fanciful device; a whim; a quip.
n.
One of the variables of a quantic as distinguished from a coefficient.
n.
Part or proportion; quota.
a.
Of, pertaining to, or in the manner of, the Roman general, Quintus Fabius Maximus Verrucosus; cautious; dilatory; avoiding a decisive contest.
n.
A quantic of the sixth degree.
n.
A quantic of the fourth degree. See Quantic.