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Journal articles on the topic 'Abelian structures'

Author: Grafiati

Published: 28 September 2024

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1

Lu, Jianwei, and Liguo He. "On the Structures of Abelianπ-Regular Rings." International Journal of Mathematics and Mathematical Sciences 2014 (2014): 1–4. http://dx.doi.org/10.1155/2014/842313.

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Assume thatRis an Abelian ring. In this paper, we characterize the structure ofRwheneverRisπ-regular. It is also proved that an Abelianπ-regular ring is isomorphic to the subdirect sum of some metadivision rings.

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2

Clarke, Francis. "Counting abelian group structures." Proceedings of the American Mathematical Society 134, no. 10 (April 10, 2006): 2795–99. http://dx.doi.org/10.1090/s0002-9939-06-08396-1.

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3

CONSOLE, S., A. FINO, and Y. S. POON. "STABILITY OF ABELIAN COMPLEX STRUCTURES." International Journal of Mathematics 17, no. 04 (April 2006): 401–16. http://dx.doi.org/10.1142/s0129167x06003576.

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Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

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4

Poon, Yat Sun. "Abelian Complex Structures and Generalizations." Complex Manifolds 8, no. 1 (January 1, 2021): 247–66. http://dx.doi.org/10.1515/coma-2020-0117.

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Abstract After a review on the development of deformation theory of abelian complex structures from both the classical and generalized sense, we propose the concept of semi-abelian generalized complex structure. We present some observations on such structure and illustrate this new concept with a variety of examples.

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5

Tang, Guoliang. "Abelian model structures on comma categories." Ukrains’kyi Matematychnyi Zhurnal 76, no. 3 (March 25, 2024): 373–81. http://dx.doi.org/10.3842/umzh.v76i3.7289.

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UDC 512.64 Let A and B be bicomplete Abelian categories, which both have enough projectives and injectives and let T : A → B be a right exact functor. Under some mild conditions, we show that hereditary Abelian model structures on A and B can be amalgamated into a global hereditary Abelian model structure on the comma category ( T ↓ B ) . As an application of this result, we give an explicit description of a subcategory that consists of all trivial objects of the Gorenstein flat model structure on the category of modules over a triangular matrix ring.

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6

Goswami, Amartya. "Salamander lemma for non-abelian group-like structures." Journal of Algebra and Its Applications 19, no. 02 (March 15, 2019): 2050022. http://dx.doi.org/10.1142/s021949882050022x.

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It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper, we establish such a generalization of the “salamander lemma” due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Márki and W. Tholen and exact categories in the sense of M. Grandis.

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7

Yu, Chia-Fu. "Abelian varieties over finite fields as basic abelian varieties." Forum Mathematicum 29, no. 2 (March 1, 2017): 489–500. http://dx.doi.org/10.1515/forum-2014-0141.

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AbstractIn this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic ${p>0}$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.

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8

Fokina, E., J. F. Knight, A. Melnikov, S. M. Quinn, and C. Safranski. "Classes of Ulm type and coding rank-homogeneous trees in other structures." Journal of Symbolic Logic 76, no. 3 (September 2011): 846–69. http://dx.doi.org/10.2178/jsl/1309952523.

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AbstractThe first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelianp-groups, the class of Abelian torsion groups, and the special class of “rank-homogeneous” trees. We consider these conditions as a possible definition of what it means for a class of structures to have “Ulm type”. The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply this result to show that there is no Turing computable embedding of the class of graphs into the class of “rank-homogeneous” trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a “rank-preserving” Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank.

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9

Remm, Elisabeth, and Michel Goze. "Affine structures on abelian Lie groups." Linear Algebra and its Applications 360 (February 2003): 215–30. http://dx.doi.org/10.1016/s0024-3795(02)00452-4.

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10

Yu, Chia-Fu. "Lifting abelian varieties with additional structures." Mathematische Zeitschrift 242, no. 3 (April 1, 2002): 427–41. http://dx.doi.org/10.1007/s002090100350.

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11

Brasch, Hans-J�rgen. "Lifting levelD-structures of abelian varieties." Archiv der Mathematik 60, no. 6 (June 1993): 553–62. http://dx.doi.org/10.1007/bf01236082.

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12

Bojowald, Martin. "Abelianized Structures in Spherically Symmetric Hypersurface Deformations." Universe 8, no. 3 (March 15, 2022): 184. http://dx.doi.org/10.3390/universe8030184.

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In canonical gravity, general covariance is implemented by hypersurface-deformation symmetries on thephase space. The different versions of hypersurface deformations required for full covariance have complicated interplays with one another, governed by non-Abelian brackets with structure functions. For spherically symmetric space-times, it is possible to identify a certain Abelian substructure within general hypersurface deformations, which suggests a simplified realization as a Lie algebra. The generators of this substructure can be quantized more easily than full hypersurface deformations, but the symmetries they generate do not directly correspond to hypersurface deformations. The availability of consistent quantizations therefore does not guarantee general covariance or a meaningful quantum notion thereof. In addition to placing the Abelian substructure within the full context of spherically symmetric hypersurface deformation, this paper points out several subtleties relevant for attempted applications in quantized space-time structures. In particular, it follows that recent constructions by Gambini, Olmedo, and Pullin in an Abelianized setting fail to address the covariance crisis of loop quantum gravity.

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13

Hrushovski, Ehud, and James Loveys. "Strongly and co-strongly minimal abelian structures." Journal of Symbolic Logic 75, no. 2 (June 2010): 442–58. http://dx.doi.org/10.2178/jsl/1268917489.

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AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

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14

Dalezios, Georgios. "Abelian model structures on categories of quiver representations." Journal of Algebra and Its Applications 19, no. 10 (October 29, 2019): 2050195. http://dx.doi.org/10.1142/s0219498820501959.

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Let [Formula: see text] be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in [Formula: see text]. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding–Chen rings is given.

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15

Paradiso, Fabio. "Locally conformally balanced metrics on almost abelian Lie algebras." Complex Manifolds 8, no. 1 (January 1, 2021): 196–207. http://dx.doi.org/10.1515/coma-2020-0111.

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Abstract We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.

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16

Benoist, Franck, and Françoise Delon. "Questions de corps de définition pour les variétés abéliennes en caractéristique positive." Journal of the Institute of Mathematics of Jussieu 7, no. 4 (October 2008): 623–39. http://dx.doi.org/10.1017/s1474748008000145.

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AbstractDichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.

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17

GIAVARINI, G., and E. ONOFRI. "VECTOR COHERENT STATES AND NON-ABELIAN GAUGE STRUCTURES IN QUANTUM MECHANICS." International Journal of Modern Physics A 05, no. 22 (November 20, 1990): 4311–31. http://dx.doi.org/10.1142/s0217751x9000180x.

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We set the general formalism for calculating Berry's phase in quantum systems with Hamiltonian belonging to the algebra of a semisimple Lie group of any rank in the framework of generalized coherent states. Within this approach the geometric properties of Berry's connection are also studied, both in the Abelian and non-Abelian cases. In particular we call attention to the non-Abelian case where we make use of a vectorial generalization of coherent states. In this respect a thorough and self-contained exposition of the formalism of vector coherent states is given. The specific examples of the groups SU(3) and Sp(2) are worked out in detail.

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18

WANG, JUNFU, LI HUANHUAN, and ZHAOYONG HUANG. "Applications of exact structures in abelian categories." Publicationes Mathematicae Debrecen 88, no. 3-4 (April 1, 2016): 269–86. http://dx.doi.org/10.5486/pmd.2016.7220.

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19

Bajo, Ignacio, and Saïd Benayadi. "Abelian para-Kähler structures on Lie algebras." Differential Geometry and its Applications 29, no. 2 (March 2011): 160–73. http://dx.doi.org/10.1016/j.difgeo.2011.02.003.

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20

Garbagnati, Alice, and Yulieth Prieto-Montañez. "Generalized Shioda–Inose structures of order 3." Advances in Geometry 24, no. 2 (April 1, 2024): 183–207. http://dx.doi.org/10.1515/advgeom-2024-0005.

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Abstract A Shioda–Inose structure is a geometric construction which associates to an Abelian surface a projective K3 surface in such a way that their transcendental lattices are isometric. This geometric construction was described by Morrison by considering special symplectic involutions on the K3 surfaces. After Morrison several authors provided explicit examples. The aim of this paper is to generalize Morrison’s results and some of the known examples to an analogous geometric construction involving not involutions, but order 3 automorphisms. Therefore, we define generalized Shioda–Inose structures of order 3, we identify the K3 surfaces and the Abelian surfaces which appear in these structures and we provide explicit examples.

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21

Mikaelian, Vahagn H. "Subvariety structures in certain product varieties of groups." Journal of Group Theory 21, no. 5 (September 1, 2018): 865–84. http://dx.doi.org/10.1515/jgth-2018-0017.

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Abstract We classify certain cases when the wreath products of distinct pairs of groups generate the same variety. This allows us to investigate the subvarieties of some nilpotent-by-abelian product varieties {{\mathfrak{U}}{\mathfrak{V}}} with the help of wreath products of groups. In particular, using wreath products, we find such subvarieties in nilpotent-by-abelian {{\mathfrak{U}}{\mathfrak{V}}} , which have the same nilpotency class, the same length of solubility, and the same exponent, but which still are distinct subvarieties. The classification we obtain strengthens our recent work on varieties generated by wreath products.

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22

Simon, Pierre. "On dp-minimal ordered structures." Journal of Symbolic Logic 76, no. 2 (June 2011): 448–60. http://dx.doi.org/10.2178/jsl/1305810758.

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AbstractWe show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.

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23

Huczynska, Sophie, Christopher Jefferson, and Silvia Nepšinská. "Strong external difference families in abelian and non-abelian groups." Cryptography and Communications 13, no. 2 (February 8, 2021): 331–41. http://dx.doi.org/10.1007/s12095-021-00473-3.

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AbstractStrong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right. We extend the definition of SEDF from abelian groups to all finite groups, and introduce the concept of equivalence. We prove new recursive constructions for SEDFs and generalized SEDFs (GSEDFs) in cyclic groups, and present the first family of non-abelian SEDFs. We prove there exist at least two non-equivalent (k2 + 1,2,k,1)-SEDFs for every k > 2, and begin the task of enumerating SEDFs, via a computational approach which yields complete results for all groups up to order 24.

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24

SAYGILI, K. "TOPOLOGICALLY MASSIVE ABELIAN GAUGE THEORY." International Journal of Modern Physics A 23, no. 13 (May 20, 2008): 2015–35. http://dx.doi.org/10.1142/s0217751x08039840.

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We discuss three mathematical structures which arise in topologically massive Abelian gauge theory. First, the Euclidean topologically massive Abelian gauge theory defines a contact structure on a manifold. We briefly discuss three solutions and the related contact structures on the flat 3-torus, the AdS space, the 3-sphere which respectively correspond to Bianchi type I, VIII, IX spaces. We also present solutions on Bianchi type II, VI and VII spaces. Secondly, we discuss a family of complex (anti-)self-dual solutions of the Euclidean theory in Cartesian coordinates on [Formula: see text] which are given by (anti)holomorpic functions. The orthogonality relation of contact structures which are determined by the real parts of these complex solutions separates them into two classes: the self-dual and the anti-self-dual solutions. Thirdly, we apply the curl transformation to this theory. An arbitrary solution is given by a vector tangent to a sphere whose radius is determined by the topological mass in transform space. Meanwhile a gauge transformation corresponds to a vector normal to this sphere. We discuss the quantization of topological mass in an example.

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25

Abdulali, Salman. "Hodge Structures on Abelian Varieties of Type III." Annals of Mathematics 155, no. 3 (May 2002): 915. http://dx.doi.org/10.2307/3062136.

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26

Bajo, Ignacio, and Esperanza Sanmartín. "Pseudo-Kähler Lie algebras with Abelian complex structures." Journal of Physics A: Mathematical and Theoretical 45, no. 46 (October 30, 2012): 465205. http://dx.doi.org/10.1088/1751-8113/45/46/465205.

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27

Abramovich, Dan, and Anthony Várilly-Alvarado. "Level structures on abelian varieties and Vojta’s conjecture." Compositio Mathematica 153, no. 2 (February 2017): 373–94. http://dx.doi.org/10.1112/s0010437x16008253.

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Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

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28

ANDRADA, ADRIÁN, and RAQUEL VILLACAMPA. "ABELIAN BALANCED HERMITIAN STRUCTURES ON UNIMODULAR LIE ALGEBRAS." Transformation Groups 21, no. 4 (November 25, 2015): 903–27. http://dx.doi.org/10.1007/s00031-015-9352-7.

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29

Abdulali, Salman. "Hodge structures on abelian varieties of type IV." Mathematische Zeitschrift 246, no. 1-2 (January 1, 2004): 203–12. http://dx.doi.org/10.1007/s00209-003-0595-y.

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30

Gavrilovich, Misha. "Covers of Abelian varieties as analytic Zariski structures." Annals of Pure and Applied Logic 163, no. 11 (November 2012): 1524–48. http://dx.doi.org/10.1016/j.apal.2011.12.007.

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31

Moreno, Andrés J. "Harmonic G2-structures on almost Abelian Lie groups." Differential Geometry and its Applications 91 (December 2023): 102060. http://dx.doi.org/10.1016/j.difgeo.2023.102060.

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32

Zhang, Yakun, Guoping Tang, and Hong Chen. "On the Structures of K2(ℤ[G]), G a Finite Abelian p-Group." Algebra Colloquium 26, no. 01 (March 2019): 105–12. http://dx.doi.org/10.1142/s1005386719000105.

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Let G be a finite abelian p-group, Γ the maximal ℤ-order of ℤ[G]. We prove that the 2-primary torsion subgroups of K2(ℤ[G]) and K2(Γ) are isomorphic when p ≡ 3, 5, 7 (mod 8), and [Formula: see text] is isomorphic to [Formula: see text] when p ≡ 2, 3, 5, 7. As an application, we give the structure of K2(ℤ[G]) for G a cyclic p-group or an elementary abelian p-group.

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33

DITO, GIUSEPPE. "ON GENERALIZED ABELIAN DEFORMATIONS." Reviews in Mathematical Physics 11, no. 06 (July 1999): 711–25. http://dx.doi.org/10.1142/s0129055x99000246.

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We study sun-products on ℝn, i.e. generalized Abelian deformations associated with star-products for general Poisson structures on ℝn. We show that their cochains are given by differential operators. As a consequence, the weak triviality of sun-products is established and we show that strong equivalence classes are quite small. When the Poisson structure is linear (i.e. on the dual of a Lie algebra), we show that the differentiability of sun-products implies that covariant starproducts on the dual of any Lie algebra are equivalent each other.

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34

Watase, Yasushige. "Embedding Principle for Rings and Abelian Groups." Formalized Mathematics 31, no. 1 (September 1, 2023): 143–50. http://dx.doi.org/10.2478/forma-2023-0013.

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Summary The article concerns about formalizing a certain lemma on embedding of algebraic structures in the Mizar system, claiming that if a ring A is embedded in a ring B then there exists a ring C which is isomorphic to B and includes A as a subring. This construction applies to algebraic structures such as Abelian groups and rings.

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35

Di Nola, Antonio, Giacomo Lenzi, Gaetano Vitale, and Roberto Giuntini. "Expanding Lattice Ordered Abelian Groups to Riesz Spaces." Mathematica Slovaca 72, no. 1 (February 1, 2022): 1–10. http://dx.doi.org/10.1515/ms-2022-0001.

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Abstract First we give a necessary and sufficient condition for an abelian lattice ordered group to admit an expansion to a Riesz space (or vector lattice). Then we construct a totally ordered abelian group with two non-isomorphic Riesz space structures, thus improving a previous paper where the example was a non-totally ordered lattice ordered abelian group. This answers a question raised by Conrad in 1975. We give also a partial solution to another problem considered in the same paper. Finally, we apply our results to MV-algebras and Riesz MV-algebras.

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36

Lenzi, Giacomo. "On the Riesz structures of a lattice ordered abelian group." Mathematica Slovaca 69, no. 6 (December 18, 2019): 1237–44. http://dx.doi.org/10.1515/ms-2017-0304.

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Abstract A Riesz structure on a lattice ordered abelian group G is a real vector space structure where the product of a positive element of G and a positive real is positive. In this paper we show that for every cardinal k there is a totally ordered abelian group with at least k Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same l-group with strong unit. This gives a solution to a problem posed by Conrad in 1975. Finally we apply the main result to MV-algebras and Riesz MV-algebras.

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37

Ceballos, Manuel, Juan Núñez, and Ángel F. Tenorio. "Abelian subalgebras on Lie algebras." Communications in Contemporary Mathematics 17, no. 04 (June 22, 2015): 1550050. http://dx.doi.org/10.1142/s0219199715500509.

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Abelian subalgebras play an important role in the study of Lie algebras and their properties and structures. In this paper, the historical evolution of this concept is shown, analyzing the current status for the research on this topic. So, the main results obtained from previous years are indicated and commented here. Additionally, a list of some related open problems is also given.

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38

Koch, Alan. "Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures." Proceedings of the American Mathematical Society, Series B 8, no. 16 (June 9, 2021): 189–203. http://dx.doi.org/10.1090/bproc/87.

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Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

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39

Feigelstock, Shalom. "Groups Admitting only Finitely Many Nilpotent Ring Structures." Canadian Mathematical Bulletin 29, no. 2 (June 1, 1986): 197–203. http://dx.doi.org/10.4153/cmb-1986-032-2.

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AbstractThe abelian groups which are the additive groups of only finitely many non-isomorphic (associative) nilpotent rings are studied. Progress is made toward a complete classification of these groups. In the torsion free case, the actual number of non-isomorphic nilpotent rings these groups support is obtained.

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40

Rump, Wolfgang. "Quasi-linear Cycle Sets and the Retraction Problem for Set-theoretic Solutions of the Quantum Yang-Baxter Equation." Algebra Colloquium 23, no. 01 (January 6, 2016): 149–66. http://dx.doi.org/10.1142/s1005386716000183.

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Cycle sets were introduced to reduce non-degenerate unitary Yang-Baxter maps to an algebraic system with a single binary operation. Every finite cycle set extends uniquely to a finite cycle set with a compatible abelian group structure. Etingof et al. introduced affine Yang-Baxter maps. These are equivalent to cycle sets with a specific abelian group structure. Abelian group structures have also been essential to get partial results for the still unsolved retraction problem. We introduce two new classes of cycle sets with an underlying abelian group structure and show that they can be transformed into each other while keeping the group structure fixed. This leads to a proper extension of the retractibility conjecture and new evidence for its truth.

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41

BAI, CHENGMING. "A FURTHER STUDY ON NON-ABELIAN PHASE SPACES: LEFT-SYMMETRIC ALGEBRAIC APPROACH AND RELATED GEOMETRY." Reviews in Mathematical Physics 18, no. 05 (June 2006): 545–64. http://dx.doi.org/10.1142/s0129055x06002711.

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The notion of non-abelian phase space of a Lie algebra was first formulated and then discussed by Kuperschmidt. In this paper, we further study the non-abelian phase spaces in terms of left-symmetric algebras. We interpret the natural appearance of left-symmetric algebras from the intrinsic algebraic properties and the close relations with the classical Yang–Baxter equation. Furthermore, using the theory of left-symmetric algebras, we study some interesting geometric structures related to phase spaces. Moreover, we also discuss the generalized phase spaces with certain non-trivial algebraic structures on the dual spaces.

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42

ARNAUDON, D., L. FRAPPAT, J. AVAN, and M. ROSSI. "DEFORMED DOUBLE YANGIAN STRUCTURES." Reviews in Mathematical Physics 12, no. 07 (July 2000): 945–63. http://dx.doi.org/10.1142/s0129055x00000290.

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Scaling limits at q → 1 of the elliptic vertex algebras [Formula: see text] are defined for any N, extending the previously known case of N = 2. They realise deformed, centrally extended double Yangian structures [Formula: see text]. As in the quantum affine algebras [Formula: see text], and quantum elliptic affine algebras [Formula: see text], these algebras contain subalgebras at critical values of the central charge c = -N -Mr (M integer, 2r = ln p/ ln q), which become Abelian when c = -N or 2r = Nh for h integer. Poisson structures and quantum exchange relations are derived for their abstract generators.

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43

Baguis, P., and T. Stavracou. "Normal Lie subsupergroups and non-abelian supercircles." International Journal of Mathematics and Mathematical Sciences 30, no. 10 (2002): 581–91. http://dx.doi.org/10.1155/s0161171202012395.

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We propose and study an appropriate analog of normal Lie subgroups in the supergeometrical context. We prove that the ringed space obtained taking the quotient of a Lie supergroup by a normal Lie subsupergroup, is still a Lie supergroup. We show how one can construct Lie supergroup structures over topologically nontrivial Lie groups and how the previous property of normal Lie subsupergroups can be used, in order to explicitly obtain the coproduct, counit, and antipode of these structures. We illustrate the general theory by carrying out the previous constructions over the circle, which leads to non-abelian super generalizations of the circle.

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44

Gao, Su. "On automorphism groups of countable structures." Journal of Symbolic Logic 63, no. 3 (September 1998): 891–96. http://dx.doi.org/10.2307/2586718.

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AbstractStrengthening a theorem of D. W. Kueker, this paper completely charaterizes which countable structures do not admit uncountable Lω1ω-elementarily equivalent models. In particular, it is shown that if the automorphism group of a countable structure M is abelian, or even just solvable, then there is no uncountable model of the Scott sentence of M. These results arise as part of a study of Polish groups with compatible left-invariant complete metrics.

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45

Zhou, Kun, and Gongxiang Liu. "On the quasitriangular structures of abelian extensions of ℤ2." Communications in Algebra 49, no. 11 (June 10, 2021): 4755–62. http://dx.doi.org/10.1080/00927872.2021.1929274.

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46

Baro, Elías, and Alessandro Berarducci. "Topology of definable abelian groups in o-minimal structures." Bulletin of the London Mathematical Society 44, no. 3 (November 17, 2011): 473–79. http://dx.doi.org/10.1112/blms/bdr108.

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Pérez, Marco A. "Homological dimensions and Abelian model structures on chain complexes." Rocky Mountain Journal of Mathematics 46, no. 3 (June 2016): 951–1010. http://dx.doi.org/10.1216/rmj-2016-46-3-951.

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Gerasimov, Anton A., and Samson L. Shatashvili. "On non-abelian structures in open string field theory." Journal of High Energy Physics 2001, no. 06 (June 27, 2001): 066. http://dx.doi.org/10.1088/1126-6708/2001/06/066.

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Cvetič, Mirjam, and Paul Langacker. "Implications of Abelian extended gauge structures from string models." Physical Review D 54, no. 5 (September 1, 1996): 3570–79. http://dx.doi.org/10.1103/physrevd.54.3570.

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MACLAUGHLIN, C., H. PEDERSEN, Y. S. POON, and S. SALAMON. "DEFORMATION OF 2-STEP NILMANIFOLDS WITH ABELIAN COMPLEX STRUCTURES." Journal of the London Mathematical Society 73, no. 01 (February 2006): 173–93. http://dx.doi.org/10.1112/s0024610705022519.

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