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Journal articles on the topic 'Lie quadratic'

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Published: 4 June 2021
Last updated: 1 February 2022

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1

Ardizzoni, Alessandro, and Fabio Stumbo. "Quadratic Lie Algebras." Communications in Algebra 39, no. 8 (August 2011): 2723–51. http://dx.doi.org/10.1080/00927872.2010.489917.

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2

Albuquerque, Helena, Elisabete Barreiro, and Saïd Benayadi. "Odd-quadratic Lie superalgebras." Journal of Geometry and Physics 60, no. 2 (February 2010): 230–50. http://dx.doi.org/10.1016/j.geomphys.2009.09.013.

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3

Zhu, Linsheng. "Solvable quadratic Lie algebras." Science in China Series A 49, no. 4 (March 26, 2006): 477–93. http://dx.doi.org/10.1007/s11425-006-0477-y.

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4

Duong, Minh Thanh, and Rosane Ushirobira. "Singular quadratic Lie superalgebras." Journal of Algebra 407 (June 2014): 372–412. http://dx.doi.org/10.1016/j.jalgebra.2014.02.034.

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5

AYADI, IMEN, HEDI BENAMOR, and SAÏD BENAYADI. "LIE SUPERALGEBRAS WITH SOME HOMOGENEOUS STRUCTURES." Journal of Algebra and Its Applications 11, no. 05 (September 26, 2012): 1250095. http://dx.doi.org/10.1142/s0219498812500958.

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We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [A. Aubert, Structures affines et pseudo-métriques invariantes à gauche sur des groupes de Lie, Thèse, Université Montpellier II (1996)]. We use this concept to give an inductive description of nilpotent homogeneous-symplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than even-quadratic even-symplectic considered in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608]. We study the structures of even (respectively, odd)-quadratic odd (respectively, even)-symplectic Lie superalgebras and odd-quadratic odd-symplectic Lie superalgebras and we give its inductive descriptions in terms of quadratic generalized double extensions and odd quadratic generalized double extensions. This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superalgebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [E. Barreiro and S. Benayadi, Quadratic symplectic Lie superalgebras and Lie bi-superalgebras, J. Algebra 321(2) (2009) 582–608].
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6

Barreiro, Elisabete, and Saïd Benayadi. "Quadratic symplectic Lie superalgebras and Lie bi-superalgebras." Journal of Algebra 321, no. 2 (January 2009): 582–608. http://dx.doi.org/10.1016/j.jalgebra.2008.09.026.

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7

Wang, Song, and Linsheng Zhu. "Non-degenerate Invariant Bilinear Forms on Lie Color Algebras." Algebra Colloquium 17, no. 03 (September 2010): 365–74. http://dx.doi.org/10.1142/s1005386710000362.

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In this paper, we study Lie color algebras 𝔤 with a non-degenerate color-symmetric, 𝔤-invariant bilinear form B, such a (𝔤,B) is called a quadratic Lie color algebra. Our first result generalizes the notion of double extensions to quadratic Lie color algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Lie color algebra to be a quadratic Lie color algebra by double extension. At last, we generalize the notion of T*-extensions to Lie color algebras.
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8

Baklouti, Amir. "Quadratic Hom-Lie triple systems." Journal of Geometry and Physics 121 (November 2017): 166–75. http://dx.doi.org/10.1016/j.geomphys.2017.06.013.

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9

Kotov, Alexei, and Thomas Strobl. "Integration of quadratic Lie algebroids to Riemannian Cartan–Lie groupoids." Letters in Mathematical Physics 108, no. 3 (January 12, 2018): 737–56. http://dx.doi.org/10.1007/s11005-018-1048-1.

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10

Jurdjevic, Velimir. "Affine-quadratic problems on Lie groups." Mathematical Control & Related Fields 3, no. 3 (2013): 347–74. http://dx.doi.org/10.3934/mcrf.2013.3.347.

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11

Benamor, Hedi, and Saïd Benayadi. "Double extension of quadratic lie superalgebras." Communications in Algebra 27, no. 1 (January 1999): 67–88. http://dx.doi.org/10.1080/00927879908826421.

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12

Hong, Yanyong, and Zhixiang Wu. "Simplicity of quadratic Lie conformal algebras." Communications in Algebra 45, no. 1 (October 11, 2016): 141–50. http://dx.doi.org/10.1080/00927872.2016.1175466.

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13

Arnal, Didier, Wissem Bakbrahem, and Ridha Chatbouri. "Hom-Lie quadratic and Pinczon Algebras." Communications in Algebra 45, no. 12 (May 11, 2017): 5471–86. http://dx.doi.org/10.1080/00927872.2017.1327058.

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14

Lin, Qian, Zhangju Liu, and Yunhe Sheng. "Quadratic Deformations of Lie–Poisson Structures." Letters in Mathematical Physics 83, no. 3 (January 22, 2008): 217–29. http://dx.doi.org/10.1007/s11005-008-0221-3.

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15

Abdelmoula, Lobna, and Yasmine Bouaziz. "Quadratic overgroups for diamond Lie groups." Bulletin des Sciences Mathématiques 138, no. 7 (October 2014): 870–86. http://dx.doi.org/10.1016/j.bulsci.2014.04.001.

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16

Kath, I., and M. Olbrich. "Metric Lie algebras and quadratic extensions." Transformation Groups 11, no. 1 (March 2006): 87–131. http://dx.doi.org/10.1007/s00031-005-1106-5.

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17

Bajo, Ignacio, Saïd Benayadi, and Alberto Medina. "Symplectic structures on quadratic Lie algebras." Journal of Algebra 316, no. 1 (October 2007): 174–88. http://dx.doi.org/10.1016/j.jalgebra.2007.06.001.

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18

Kang, Yi Fang, and Zhi Qi Chen. "Dirac Operators on Quadratic Lie Superalgebras." Acta Mathematica Sinica, English Series 37, no. 8 (August 2021): 1229–53. http://dx.doi.org/10.1007/s10114-021-0556-6.

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19

Lang, Honglei, Yu Qiao, and Yanbin Yin. "On Lie bialgebroid crossed modules." International Journal of Mathematics 32, no. 04 (March 2021): 2150021. http://dx.doi.org/10.1142/s0129167x2150021x.

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We study Lie bialgebroid crossed modules which are pairs of Lie algebroid crossed modules in duality that canonically give rise to Lie bialgebroids. A one-one correspondence between such Lie bialgebroid crossed modules and co-quadratic Manin triples [Formula: see text] is established, where [Formula: see text] is a co-quadratic Lie algebroid and [Formula: see text] is a pair of transverse Dirac structures in [Formula: see text].
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20

Hofmann, Karl H., and Verena S. Keith. "Invariant quadratic forms on finite dimensional lie algebras." Bulletin of the Australian Mathematical Society 33, no. 1 (February 1986): 21–36. http://dx.doi.org/10.1017/s0004972700002835.

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Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [L, L] ∩ R (with the radical R) in the orthogonal L⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].
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21

García-Delgado, R., G. Salgado, and O. A. Sánchez-Valenzuela. "Invariant metrics on central extensions of quadratic Lie algebras." Journal of Algebra and Its Applications 19, no. 12 (December 16, 2019): 2050224. http://dx.doi.org/10.1142/s0219498820502242.

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A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.
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22

LIN, JIE, ZHIQI CHEN, and LIANGYUN CHEN. "Quadratic Lie triple systems admitting symplectic structures." Publicationes Mathematicae Debrecen 88, no. 3-4 (April 1, 2016): 369–80. http://dx.doi.org/10.5486/pmd.2016.7368.

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23

Duong, Minh Thanh, Georges Pinczon, and Rosane Ushirobira. "A New Invariant of Quadratic Lie Algebras." Algebras and Representation Theory 15, no. 6 (May 17, 2011): 1163–203. http://dx.doi.org/10.1007/s10468-011-9284-4.

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24

Benito, P., D. de-la-Concepción, and J. Laliena. "Free nilpotent and nilpotent quadratic Lie algebras." Linear Algebra and its Applications 519 (April 2017): 296–326. http://dx.doi.org/10.1016/j.laa.2017.01.007.

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25

Bajo, Ignacio, and Said Benayadi. "Lie algebras admitting a unique quadratic structure." Communications in Algebra 25, no. 9 (January 1997): 2795–805. http://dx.doi.org/10.1080/00927879708826023.

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26

Chen, Zhiqi. "Underlying lie algebras of quadratic Novikov algebras." Czechoslovak Mathematical Journal 61, no. 2 (June 2011): 323–28. http://dx.doi.org/10.1007/s10587-011-0077-z.

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27

Zhu, Linsheng, and Daoji Meng. "QUADRATIC LIE ALGEBRAS AND COMMUTATIVE ASSOCIATIVE ALGEBRAS." Communications in Algebra 29, no. 5 (April 30, 2001): 2249–68. http://dx.doi.org/10.1081/agb-100002182.

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28

Zhang, Erchuan, and Lyle Noakes. "Riemannian cubics in quadratic matrix Lie groups." Applied Mathematics and Computation 375 (June 2020): 125082. http://dx.doi.org/10.1016/j.amc.2020.125082.

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29

ARON, ANANIA, CORA DĂNIASĂ, and MIRCEA PUTA. "QUADRATIC AND HOMOGENEOUS HAMILTON–POISSON SYSTEM ON (so(3))." International Journal of Geometric Methods in Modern Physics 04, no. 07 (November 2007): 1173–86. http://dx.doi.org/10.1142/s0219887807002491.

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30

YILMAZ, Koray, and Elis SOYLU YILMAZ. "Baues cofibration for quadratic modules of Lie algebras." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 68, no. 2 (July 1, 2019): 1653–63. http://dx.doi.org/10.31801/cfsuasmas.468743.

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31

Benayadi, Saïd. "Socle and some invariants of quadratic Lie superalgebras." Journal of Algebra 261, no. 2 (March 2003): 245–91. http://dx.doi.org/10.1016/s0021-8693(02)00682-8.

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32

Bajo, Ignacio, and Saïd Benayadi. "Lie algebras with quadratic dimension equal to 2." Journal of Pure and Applied Algebra 209, no. 3 (June 2007): 725–37. http://dx.doi.org/10.1016/j.jpaa.2006.07.010.

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33

Albuquerque, Helena, Elisabete Barreiro, and Saïd Benayadi. "Quadratic Lie superalgebras with a reductive even part." Journal of Pure and Applied Algebra 213, no. 5 (May 2009): 724–31. http://dx.doi.org/10.1016/j.jpaa.2008.09.016.

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34

Benayadi, Saïd, and Alberto Elduque. "Classification of quadratic Lie algebras of low dimension." Journal of Mathematical Physics 55, no. 8 (August 2014): 081703. http://dx.doi.org/10.1063/1.4890646.

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35

Jarvis, P. D., G. Rudolph, and L. A. Yates. "A class of quadratic deformations of Lie superalgebras." Journal of Physics A: Mathematical and Theoretical 44, no. 23 (May 11, 2011): 235205. http://dx.doi.org/10.1088/1751-8113/44/23/235205.

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36

Tao, Yi, Zhi Qi Chen, and Yan Wang. "Quadratic Lie Superalgebras Generalized by Balinsky–Novikov Superalgebras." Acta Mathematica Sinica, English Series 35, no. 2 (November 20, 2018): 213–26. http://dx.doi.org/10.1007/s10114-018-7210-y.

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37

Accardi, L., I. Ya Aref’eva, and I. V. Volovich. "Fermionic Meixner classes, Lie algebras and quadratic Hamiltonians." Indian Journal of Pure and Applied Mathematics 46, no. 4 (August 2015): 517–38. http://dx.doi.org/10.1007/s13226-015-0150-7.

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38

Almeida, M. A., F. C. Santos, and I. C. Moreira. "Lie symmetries of quadratic two-dimensional difference equations." International Journal of Theoretical Physics 36, no. 2 (February 1997): 551–58. http://dx.doi.org/10.1007/bf02435748.

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39

Cohen, David Bruce. "Lipschitz 1-connectedness for Some Solvable Lie Groups." Canadian Journal of Mathematics 71, no. 03 (January 9, 2019): 533–55. http://dx.doi.org/10.4153/cjm-2017-038-8.

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AbstractA space X is said to be Lipschitz 1-connected if every Lipschitz loop 𝛾 : S 1 → X bounds a O (Lip(𝛾))-Lipschitz disk f : D 2 → X. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.
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40

Zhai, Guisheng, Xuping Xu, Hai Lin, and Derong Liu. "Extended Lie Algebraic Stability Analysis for Switched Systems with Continuous-Time and Discrete-Time Subsystems." International Journal of Applied Mathematics and Computer Science 17, no. 4 (December 1, 2007): 447–54. http://dx.doi.org/10.2478/v10006-007-0036-x.

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Extended Lie Algebraic Stability Analysis for Switched Systems with Continuous-Time and Discrete-Time SubsystemsWe analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.
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41

ZHANG, YUFENG, HONWAH TAM, and JIANQIN MEI. "INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION." Modern Physics Letters B 24, no. 14 (June 10, 2010): 1573–94. http://dx.doi.org/10.1142/s0217984910023360.

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Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.
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42

Zhou, Jinsen, and Yanyong Hong. "Quadratic Leibniz conformal algebras." Journal of Algebra and Its Applications 18, no. 10 (August 6, 2019): 1950195. http://dx.doi.org/10.1142/s0219498819501950.

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In this paper, we study a class of Leibniz conformal algebras called quadratic Leibniz conformal algebras. An equivalent characterization of a Leibniz conformal algebra [Formula: see text] through three algebraic operations on [Formula: see text] is given. By this characterization, several constructions of quadratic Leibniz conformal algebras are presented. Moreover, one-dimensional central extensions of quadratic Leibniz conformal algebras are considered using some bilinear forms on [Formula: see text]. In particular, we also study one-dimensional Leibniz central extensions of quadratic Lie conformal algebras.
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43

Benito, Pilar, Daniel de-la-Concepción, Jorge Roldán-López, and Iciar Sesma. "Quadratic 2-step Lie algebras: Computational algorithms and classification." Journal of Symbolic Computation 94 (September 2019): 70–89. http://dx.doi.org/10.1016/j.jsc.2018.07.001.

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44

Zhu, Fuhai, and Linsheng Zhu. "THE UNIQUENESS OF THE DECOMPOSITION OF QUADRATIC LIE ALGEBRAS." Communications in Algebra 29, no. 11 (January 1, 2001): 5145–54. http://dx.doi.org/10.1081/agb-100106807.

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45

Kriukova, G. V. "The Lie algebra associated to an integer quadratic form." Вісник Київського національного університету імені Тараса Шевченка. Серія "Фізико-математичні науки", Вип. 3 (2010): 43–46.

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46

Abdelmoula, Lobna, Didier Arnal, and Mohamed Selmi. "Weak quadratic overgroups for a class of Lie groups." Monatshefte für Mathematik 171, no. 2 (August 28, 2012): 129–56. http://dx.doi.org/10.1007/s00605-012-0438-1.

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47

Wolf, T. "Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket." General Relativity and Gravitation 38, no. 6 (March 22, 2006): 1115–27. http://dx.doi.org/10.1007/s10714-006-0293-2.

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48

Bloch, Anthony M., Peter E. Crouch, Jerrold E. Marsden, and Amit K. Sanyal. "Optimal Control and Geodesics on Quadratic Matrix Lie Groups." Foundations of Computational Mathematics 8, no. 4 (February 28, 2008): 469–500. http://dx.doi.org/10.1007/s10208-008-9025-1.

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49

Rubenthaler, Hubert. "Minimal graded Lie algebras and representations of quadratic algebras." Journal of Algebra 473 (March 2017): 29–65. http://dx.doi.org/10.1016/j.jalgebra.2016.10.011.

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50

Earnest, A. G., and Ji Young Kim. "Integral quadratic forms avoiding arithmetic progressions." International Journal of Number Theory 16, no. 10 (July 25, 2020): 2141–48. http://dx.doi.org/10.1142/s1793042120501109.

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For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.
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