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Journal articles on the topic 'Double groupoids'

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Author: Grafiati

Published: 9 March 2023

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1

MACKENZIE, K. C. H. "ON SYMPLECTIC DOUBLE GROUPOIDS AND THE DUALITY OF POISSON GROUPOIDS." International Journal of Mathematics 10, no. 04 (June 1999): 435–56. http://dx.doi.org/10.1142/s0129167x99000185.

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We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic double groupoid are Poisson groupoids in duality. Further, we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids (and thus of Lie bialgebroids) in duality. To handle the structures involved effectively we extend to this context the dualities and canonical isomorphisms for tangent and cotangent structures of the author and Ping Xu.

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2

Epstein, Marcelo. "Material Geometry of Binary Composites." Symmetry 13, no. 5 (May 18, 2021): 892. http://dx.doi.org/10.3390/sym13050892.

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The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable mathematical framework for a unified formulation of this and similar kinds of problems in continuum mechanics.

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3

Temel, Sedat, Tunşar Şahan, and Osman Mucuk. "Crossed modules, double group-groupoids and crossed squares." Filomat 34, no. 6 (2020): 1755–69. http://dx.doi.org/10.2298/fil2006755t.

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The purpose of this paper is to obtain the notion of crossed module over group-groupoids considering split extensions and prove a categorical equivalence between these types of crossed modules and double group-groupoids. This equivalence enables us to produce various examples of double groupoids. We also prove that crossed modules over group-groupoids are equivalent to crossed squares.

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4

Mehta, Rajan Amit, and Xiang Tang. "From double Lie groupoids to local Lie 2-groupoids." Bulletin of the Brazilian Mathematical Society, New Series 42, no. 4 (December 2011): 651–81. http://dx.doi.org/10.1007/s00574-011-0033-4.

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5

Andruskiewitsch, Nicolás, and Juan Martín Mombelli. "Examples of Weak Hopf Algebras Arising from Vacant Double Groupoids." Nagoya Mathematical Journal 181 (March 2006): 1–27. http://dx.doi.org/10.1017/s0027763000025642.

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AbstractWe construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi [H]) via vacant double groupoids as explained in [AN]. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.

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6

Andruskiewitsch, Nicolás, and Sonia Natale. "The structure of double groupoids." Journal of Pure and Applied Algebra 213, no. 6 (June 2009): 1031–45. http://dx.doi.org/10.1016/j.jpaa.2008.11.003.

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7

Andruskiewitsch, Nicolas, Jesus Ochoa Arango, and Alejandro Tiraboschi. "On slim double Lie groupoids." Pacific Journal of Mathematics 256, no. 1 (May 6, 2012): 1–17. http://dx.doi.org/10.2140/pjm.2012.256.1.

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8

MUCUK, Osman, and Serap DEMİR. "Normality and quotient in crossed modules over groupoids and double groupoids." TURKISH JOURNAL OF MATHEMATICS 42, no. 5 (September 9, 2018): 2336–47. http://dx.doi.org/10.3906/mat-1606-126.

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9

LU, JIANG-HUA, and VICTOR MOUQUIN. "DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS." Transformation Groups 23, no. 3 (August 8, 2017): 765–800. http://dx.doi.org/10.1007/s00031-017-9437-6.

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10

Cañez, Santiago. "Double groupoids and the symplectic category." Journal of Geometric Mechanics 10, no. 2 (2018): 217–50. http://dx.doi.org/10.3934/jgm.2018009.

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11

Mombelli, Juan Martín, and Sonia Natale. "Tensor categories and Vacant Double Groupoids." Mathematical Research Letters 14, no. 1 (2007): 1–18. http://dx.doi.org/10.4310/mrl.2007.v14.n1.a1.

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12

Cegarra, Antonio Martínez, Benjamín A. Heredia, and Josué Remedios. "Double Groupoids and Homotopy 2-types." Applied Categorical Structures 20, no. 4 (December 3, 2010): 323–78. http://dx.doi.org/10.1007/s10485-010-9240-1.

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13

Andruskiewitsch, Nicolás, and Sonia Natale. "Tensor categories attached to double groupoids." Advances in Mathematics 200, no. 2 (March 2006): 539–83. http://dx.doi.org/10.1016/j.aim.2005.02.008.

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14

Dupont, Benjamin, and Philippe Malbos. "Coherent confluence modulo relations and double groupoids." Journal of Pure and Applied Algebra 226, no. 10 (October 2022): 107037. http://dx.doi.org/10.1016/j.jpaa.2022.107037.

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15

Mikami, Kentaro. "Symplectic Double Groupoids Over Poisson (ax + b)-Groups." Transactions of the American Mathematical Society 324, no. 1 (March 1991): 447. http://dx.doi.org/10.2307/2001517.

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16

Mikami, Kentaro. "Symplectic double groupoids over Poisson $(ax+b)$-groups." Transactions of the American Mathematical Society 324, no. 1 (January 1, 1991): 447–63. http://dx.doi.org/10.1090/s0002-9947-1991-1025757-x.

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17

Szlachányi, Kornél. "The double algebraic view of finite quantum groupoids." Journal of Algebra 280, no. 1 (October 2004): 249–94. http://dx.doi.org/10.1016/j.jalgebra.2004.06.005.

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18

Mokri, Tahar. "Matched pairs of Lie algebroids." Glasgow Mathematical Journal 39, no. 2 (May 1997): 167–81. http://dx.doi.org/10.1017/s0017089500032055.

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AbstractWe extend to Lie algebroids the notion variously known as a double Lie algebra (Lu and Weinstein), matched pair of Lie algebras (Majid), or twilled extension of Lie algebras (Kosmann-Schwarzbach and Magri). It is proved that a matched pair of Lie groupoids induces a matched pair of Lie algebroids. Conversely, we show that under certain conditions a matched pair of Lie algebroids integrates to a matched pair of Lie groupoids. The importance of matched pairs of Lie algebroids has been recently demonstrated by Lu.

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19

GHORBANI, SHOKOOFEH. "Pseudo Commutative Double Basic Algebras." Kragujevac Journal of Mathematics 45, no. 6 (December 2021): 977–94. http://dx.doi.org/10.46793/kgjmat2106.977g.

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In this paper, we study the concept of pseudo commutative double basic algebras and investigate some related results. We prove that there are relations among pseudo commutative double basic algebras and other logical algebras such as pseudo hoops, pseudo BCK-algebras and double MV-algebras. We obtain a close relation between pseudo commutative double basic algebras and pseudo residuted l-groupoids. Then we investigate the properties of the boolean center of pseudo commutative double basic algebras and we use the boolean elements to deﬁne and study algebras on subintervals of pseudo commutative double basic algebras.

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20

Izhar, Muhammad, Asghar Khan, and Tariq Mahmood. "(M, N)-Double framed soft ideals of Abel Grassmann’s groupoids." Journal of Intelligent & Fuzzy Systems 35, no. 6 (December 24, 2018): 6313–27. http://dx.doi.org/10.3233/jifs-181119.

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21

Martins, João Faria, and Roger Picken. "Surface holonomy for non-abelian 2-bundles via double groupoids." Advances in Mathematics 226, no. 4 (March 2011): 3309–66. http://dx.doi.org/10.1016/j.aim.2010.10.017.

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22

Hila, Kostaq, Muhammad Izhar, Muhammad Farooq, and Asghar Khan. "$(M,N)$-double-framed soft $bi$-ideals of Abel Grassmann's groupoids." Discussiones Mathematicae - General Algebra and Applications 42, no. 2 (2022): 425. http://dx.doi.org/10.7151/dmgaa.1400.

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23

Izhar, Muhammad, Asghar Khan, and Kostaq Hila. "Double-framed soft generalized bi-ideals of intra-regular AG-groupoids." Journal of Intelligent & Fuzzy Systems 35, no. 4 (October 27, 2018): 4701–15. http://dx.doi.org/10.3233/jifs-181188.

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24

Mouquin, Victor. "Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells." Journal of Symplectic Geometry 19, no. 4 (2021): 993–1045. http://dx.doi.org/10.4310/jsg.2021.v19.n4.a4.

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25

Willerton, Simon. "The twisted Drinfeld double of a finite group via gerbes and finite groupoids." Algebraic & Geometric Topology 8, no. 3 (September 3, 2008): 1419–57. http://dx.doi.org/10.2140/agt.2008.8.1419.

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26

Qiao, Yu. "Double layer potentials on three-dimensional wedges and pseudodifferential operators on Lie groupoids." Journal of Mathematical Analysis and Applications 462, no. 1 (June 2018): 428–47. http://dx.doi.org/10.1016/j.jmaa.2018.01.077.

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27

Ciaglia, F. M., A. Ibort, and G. Marmo. "Schwinger’s picture of quantum mechanics III: The statistical interpretation." International Journal of Geometric Methods in Modern Physics 16, no. 11 (November 2019): 1950165. http://dx.doi.org/10.1142/s0219887819501652.

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Schwinger’s algebra of selective measurements has a natural interpretation in the formalism of groupoids. Its kinematical foundations, as well as the structure of the algebra of observables of the theory, were presented in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, Int. J. Geom. Meth. Mod. Phys. (2019), arXiv: 1905.12274 [math-ph]. https://doi.org/10.1142/S0219887819501196 . F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics II: Algebras and observables, Int. J. Geom. Meth. Mod. Phys. (2019). https://doi.org/10.1142/S0219887819501366 ]. In this paper, a closer look to the statistical interpretation of the theory is taken and it is found that an interpretation in terms of Sorkin’s quantum measure emerges naturally. It is proven that a suitable class of states of the algebra of virtual transitions of the theory allows to define quantum measures by means of the corresponding decoherence functionals. Quantum measures satisfying a reproducing property are described and a class of states, called factorizable states, possessing the Dirac–Feynman “exponential of the action” form are characterized. Finally, Schwinger’s transformation functions are interpreted similarly as transition amplitudes defined by suitable states. The simple examples of the qubit and the double slit experiment are described in detail, illustrating the main aspects of the theory.

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28

Liu, Ling, and Shuan-hong Wang. "The Generalized C.M.Z.-Theorem and a Drinfel'd Double Construction for WT-Coalgebras and Graded Quantum Groupoids." Communications in Algebra 36, no. 9 (September 17, 2008): 3393–417. http://dx.doi.org/10.1080/00927870802107611.

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29

LU, JIANG-HUA. "HOPF ALGEBROIDS AND QUANTUM GROUPOIDS." International Journal of Mathematics 07, no. 01 (February 1996): 47–70. http://dx.doi.org/10.1142/s0129167x96000050.

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We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras of the Drinfeld doubles of Hopf algebras when the R-matrices act properly. When this construction is applied to quantum groups, we get examples of quantum groupoids, which have Poisson groupoids as their semi-classical limits. The example of quantum sl(2) is worked out in details.

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30

Grabowska, Katarzyna, and Janusz Grabowski. "n-Tuple principal bundles." International Journal of Geometric Methods in Modern Physics 15, no. 12 (December 2018): 1850211. http://dx.doi.org/10.1142/s0219887818502110.

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31

Brown, Ronald, and Kirill C. H. Mackenzie(xe). "Determination of a double Lie groupoid by its core diagram." Journal of Pure and Applied Algebra 80, no. 3 (July 1992): 237–72. http://dx.doi.org/10.1016/0022-4049(92)90145-6.

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32

Brown, Ronald, and George Janelidze. "Galois Theory and a New Homotopy Double Groupoid of a Map of Spaces." Applied Categorical Structures 12, no. 1 (February 2004): 63–80. http://dx.doi.org/10.1023/b:apcs.0000013811.15727.1a.

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33

COQUEREAUX, R., and A. O. GARCÍA. "ON BIALGEBRAS ASSOCIATED WITH PATHS AND ESSENTIAL PATHS ON ADE GRAPHS." International Journal of Geometric Methods in Modern Physics 02, no. 03 (June 2005): 441–66. http://dx.doi.org/10.1142/s0219887805000582.

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We define a graded multiplication on the vector space of essential paths on a graph G (a tree) and show that it is associative. In most interesting applications, this tree is an ADE Dynkin diagram. The vector space of length-preserving endomorphisms of essential paths has a grading obtained from the length of paths and possesses several interesting bialgebra structures. One of these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct gives a filtrated convolution product on the dual vector space. Another bialgebra structure is obtained by replacing this filtered convolution product by a graded associative product. It can be obtained from the former by projection on a subspace of maximal grade, but it is interesting to define it directly, without using the DTA. What is obtained is a weak bialgebra, not a weak Hopf algebra.

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34

Lu, Jiang-Hua, Victor Mouquin, and Shizhuo Yu. "Configuration Poisson Groupoids of Flags." International Mathematics Research Notices, November 26, 2022. http://dx.doi.org/10.1093/imrn/rnac321.

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Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

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35

Lean, Madeleine Jotz, and Kirill C. H. Mackenzie. "Transitive double Lie algebroids via core diagrams." Journal of Geometric Mechanics, 2021, 0. http://dx.doi.org/10.3934/jgm.2021023.

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<p style='text-indent:20px;'>The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called <i>transitive</i>. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram.</p><p style='text-indent:20px;'>The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram.</p><p style='text-indent:20px;'>Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.</p>

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36

Cabrera, Alejandro, and Miquel Cueca. "Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids." Letters in Mathematical Physics 112, no. 5 (October 2022). http://dx.doi.org/10.1007/s11005-022-01596-1.

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AbstractWe show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated with the double of the underlying Lie bialgebroid. This field-theoretic result follows from a Lie-theoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of Crainic–Marcut to a particular gauge fixing of the 3d theory.

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37

Zhang, Xiaohong. "BCC-algebras and residuated partially-ordered groupoid." Mathematica Slovaca 63, no. 3 (January 1, 2013). http://dx.doi.org/10.2478/s12175-013-0104-7.

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AbstractThe aim of the paper is to investigate the relationship between BCC-algebras and residuated partially-ordered groupoids. We prove that an integral residuated partially-ordered groupoid is an integral residuated pomonoid if and only if it is a double BCC-algebra. Moreover, we introduce the notion of weakly integral residuated pomonoid, and give a characterization by the notion of pseudo-BCI algebra. Finally, we give a method to construct a weakly integral residuated pomonoid (pseudo-BCI algebra) from any bounded pseudo-BCK algebra with pseudo product and any group.

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38

Bursztyn, Henrique, Alejandro Cabrera, and Matias del Hoyo. "Poisson double structures." Journal of Geometric Mechanics, 2021, 0. http://dx.doi.org/10.3934/jgm.2021029.

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<p style='text-indent:20px;'>We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are related by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.</p>

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39

DEMİR, Serap, and Osman MUCUK. "Actions of Double Group-groupoids and Covering Morphism." GAZI UNIVERSITY JOURNAL OF SCIENCE, June 2, 2020, 1. http://dx.doi.org/10.35378/gujs.604849.

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40

Esen, Ogul, and Serkan Sutlu. "Discrete dynamical systems over double cross product Lie groupoids." International Journal of Geometric Methods in Modern Physics, December 31, 2020. http://dx.doi.org/10.1142/s0219887821500572.

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41

Qiao, Yu, and Hengguang Li. "Double Layer Potentials on Polygons and Pseudodifferential Operators on Lie Groupoids." Integral Equations and Operator Theory 90, no. 2 (March 7, 2018). http://dx.doi.org/10.1007/s00020-018-2441-y.

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42

Ahmad, Nisar, Syed Aleem Shah, Wali Khan Mashwani, and Nasim Ullah. "Corrections and Extensions in Left and Right Almost Semigroups." Punjab University Journal of Mathematics, July 27, 2021, 475–96. http://dx.doi.org/10.52280/pujm.2021.530703.

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In this paper we elaborated the concept that on what conditions left almost semigroup (LA-Semigroup), right almost semigroup (RA-Semigroup) and a groupoid become commutative and further extended these results on medial, LA-Group and RA-Group. We proved that the relation of LA-Semigroup with left double displacement semigroup (LDD-semigroup), RA-Semigroup with left double displacement semigroup (RDD-semigroup) is only commutative property. We highlighted the errors in the recently developed results on LA-Semigroup and semigroup [17, 1, 18] and proved that example discussed in [18] is semigroup with left identity but not paramedial. We extended results on locally associative LA-Semigroup explained in [20, 21] towards LA-Semigroup and RA-Semigroup with left zero and right zero respectively. We also discussed results on n-dimensional LA-Semigroup, n-dimensional RASemigroup, non commutative finite medials with three or more than three left or right identities and finite as well as infinite commutative idempotent medials not studied in literature.

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