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Journal articles on the topic 'Lie algebras][Jordan algebras'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

Fernández López, Antonio, Esther García, and Miguel Gómez Lozano. "The Jordan algebras of a Lie algebra." Journal of Algebra 308, no. 1 (February 2007): 164–77. http://dx.doi.org/10.1016/j.jalgebra.2006.02.035.

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2

Han, Gang, Yucheng Liu, and Kang Lu. "Multiplicity-Free Gradings on Semisimple Lie and Jordan Algebras and Skew Root Systems." Algebra Colloquium 26, no. 01 (March 2019): 123–38. http://dx.doi.org/10.1142/s1005386719000129.

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A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.

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3

Popov, A. V. "Lie Type Jordan Algebras." Siberian Advances in Mathematics 29, no. 4 (October 2019): 274–307. http://dx.doi.org/10.3103/s1055134419040035.

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4

Guo, Wei, and Liangyun Chen. "Algebras of Quotients of Jordan–Lie Algebras." Communications in Algebra 44, no. 9 (May 19, 2016): 3788–95. http://dx.doi.org/10.1080/00927872.2015.1087009.

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5

Ait Ben Haddou, Malika, Saïd Benayadi, and Said Boulmane. "Malcev–Poisson–Jordan algebras." Journal of Algebra and Its Applications 15, no. 09 (August 22, 2016): 1650159. http://dx.doi.org/10.1142/s0219498816501590.

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Malcev–Poisson–Jordan algebra (MPJ-algebra) is defined to be a vector space endowed with a Malcev bracket and a Jordan structure which are satisfying the Leibniz rule. We describe such algebras in terms of a single bilinear operation, this class strictly contains alternative algebras. For a given Malcev algebra [Formula: see text], it is interesting to classify the Jordan structure ∘ on the underlying vector space of [Formula: see text] such that [Formula: see text] is an MPJ-algebra (∘ is called an MPJ-structure on Malcev algebra [Formula: see text]. In this paper we explicitly give all MPJ-structures on some interesting classes of Malcev algebras. Further, we introduce the concept of pseudo-Euclidean MPJ-algebras (PEMPJ-algebras) and we show how one can construct new interesting quadratic Lie algebras and pseudo-Euclidean Malcev (non-Lie) algebras from PEMPJ-algebras. Finally, we give inductive descriptions of nilpotent PEMPJ-algebras.

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6

Hou, Dongping, Xiang Ni, and Chengming Bai. "Pre-jordan Algebras." MATHEMATICA SCANDINAVICA 112, no. 1 (March 1, 2013): 19. http://dx.doi.org/10.7146/math.scand.a-15231.

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The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

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7

Grishkov, A. N., and Ivan P. Shestakov. "Speciality of Lie–Jordan Algebras." Journal of Algebra 237, no. 2 (March 2001): 621–36. http://dx.doi.org/10.1006/jabr.2000.8612.

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8

Gordji, M. Eshaghi, and G. H. Kim. "Approximate -Lie Homomorphisms and Jordan -Lie Homomorphisms on -Lie Algebras." Abstract and Applied Analysis 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/279632.

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9

Zalar, Borut. "Continuity of derivations on Mal'cev H*-algebras." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 3 (November 1991): 455–59. http://dx.doi.org/10.1017/s0305004100070523.

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A long time ago the concept of H*-algebra was introduced by Ambrose in [1] where the structure of complex associative H*-algebras was given. Since then this theory was extended to such classical types of non-associative algebras as alternative algebras (in [6]), Jordan algebras (in [5, 13, 14]), non-commutative Jordan algebras (in [5]), Lie algebras (in [3, 9, 10]) and Mal'cev algebras (in [2]).

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10

Kaneyuki, Soji, and Hiroshi Asano. "Graded Lie algebras and generalized Jordan triple systems." Nagoya Mathematical Journal 112 (December 1988): 81–115. http://dx.doi.org/10.1017/s002776300000115x.

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One frequently encounters (real) semisimple graded Lie algebras in various branches of differential geometry (e.g. [16], [9], [14], [18]). It is therefore desirable to study semisimple graded Lie algebras, including those which have been studied individually, in a unified way. One of our concerns is to classify (finite-dimensional) semisimple graded Lie algebras in a way that enables us to construct them.

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11

Guangyu, Shen. "On lie algebras associated with nodal noncommutative Jordan algebras." Acta Mathematica Sinica 2, no. 1 (March 1986): 14–24. http://dx.doi.org/10.1007/bf02568520.

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12

Johnson, B. E. "Symmetric amenability and the nonexistence of Lie and Jordan derivations." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 3 (October 1996): 455–73. http://dx.doi.org/10.1017/s0305004100075010.

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A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

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13

Ammar, Gregory, Christian Mehl, and Volker Mehrmann. "Schur-like forms for matrix Lie groups, Lie algebras and Jordan algebras." Linear Algebra and its Applications 287, no. 1-3 (January 1999): 11–39. http://dx.doi.org/10.1016/s0024-3795(98)10133-7.

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14

Nichita, Florin. "Unification Theories: Examples and Applications." Axioms 7, no. 4 (November 16, 2018): 85. http://dx.doi.org/10.3390/axioms7040085.

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We consider several unification problems in mathematics. We refer to transcendental numbers. Furthermore, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras.

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15

Elduque, Alberto, and Susumu Okubo. "Lie algebras with S3- or S4-action and generalized Malcev algebras." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 2 (March 25, 2009): 321–57. http://dx.doi.org/10.1017/s0308210508000164.

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Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In the case of S3-symmetry, the Lie algebras are coordinatized by some non-associative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary product, and include both the Malcev algebras and the Jordan triple systems.

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16

Cao, Peng, and Shanli Sun. "Lie algebras generated by Jordan operators." Studia Mathematica 186, no. 3 (2008): 267–74. http://dx.doi.org/10.4064/sm186-3-5.

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17

Cagliero, Leandro, and Fernando Szechtman. "Jordan–Chevalley Decomposition in Lie Algebras." Canadian Mathematical Bulletin 62, no. 02 (February 28, 2019): 349–54. http://dx.doi.org/10.4153/cmb-2018-023-7.

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AbstractWe prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0 and $A\in \mathfrak{s}$ , then the semisimple and nilpotent summands of the Jordan–Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in \mathfrak{s}$ , $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$ ) such that $A=S+N$ .

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18

Martínez, Consuelo. "Infinite Dimensional Lie and Jordan Algebras." Mediterranean Journal of Mathematics 3, no. 2 (July 2006): 273–82. http://dx.doi.org/10.1007/s00009-006-0077-7.

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19

Panyushev, Dmitri I. "Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras." Algebra & Number Theory 7, no. 6 (September 19, 2013): 1505–34. http://dx.doi.org/10.2140/ant.2013.7.1505.

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20

García, Esther, Miguel Gómez Lozano, and Rubén Muñoz Alcázar. "On the speciality of Jordan algebras and subquotients of Lie algebras." Journal of Algebra 563 (December 2020): 426–41. http://dx.doi.org/10.1016/j.jalgebra.2020.07.013.

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21

KAVALOV, AN R., and R. L. MKRTCHYAN. "ON A FERMIONIC REALIZATIONS OF W-TYPE SYMMETRIES." Modern Physics Letters A 08, no. 39 (December 21, 1993): 3735–40. http://dx.doi.org/10.1142/s0217732393003469.

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The simplest W-type algebra is considered, which includes spin-3/2 and 1 currents, with the aim of finding all its realizations in the free fermion theory through the currents of the type γi1…i2sψi1 … ψi2s. The solution of this problem appears to be related to some problem in the theory of Lie algebras, and we give a classification of the solutions for γ tensors, which turn out to be connected with structure constants of Lie algebras. This is in parallel with previously known similar bosonic construction, connected with symmetric counterpart of the Lie algebras — the Jordan algebras.

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22

Bremner, Murray, and Sara Madariaga. "Lie and Jordan Products in Interchange Algebras." Communications in Algebra 44, no. 8 (April 29, 2016): 3485–508. http://dx.doi.org/10.1080/00927872.2015.1085545.

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23

Jabeen, Aisha. "Lie (Jordan) centralizers on generalized matrix algebras." Communications in Algebra 49, no. 1 (August 26, 2020): 278–91. http://dx.doi.org/10.1080/00927872.2020.1797759.

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24

Elduque, Alberto. "Jordan gradings on exceptional simple Lie algebras." Proceedings of the American Mathematical Society 137, no. 12 (July 30, 2009): 4007–17. http://dx.doi.org/10.1090/s0002-9939-09-09994-8.

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25

Wang, Yu. "Lie (Jordan) derivations of arbitrary triangular algebras." Aequationes mathematicae 93, no. 6 (January 3, 2019): 1221–29. http://dx.doi.org/10.1007/s00010-018-0634-8.

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26

Lu, Fangyan, and Xiuping Yu. "Lie and Jordan Ideals in Reflexive Algebras." Integral Equations and Operator Theory 59, no. 2 (June 27, 2007): 189–206. http://dx.doi.org/10.1007/s00020-007-1516-y.

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27

Fernández López, Antonio, Esther García, and Miguel Gómez Lozano. "The Jordan socle and finitary Lie algebras." Journal of Algebra 280, no. 2 (October 2004): 635–54. http://dx.doi.org/10.1016/j.jalgebra.2004.06.013.

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28

Bahturin, Y. A., I. P. Shestakov, and M. V. Zaicev. "Gradings on simple Jordan and Lie algebras." Journal of Algebra 283, no. 2 (January 2005): 849–68. http://dx.doi.org/10.1016/j.jalgebra.2004.10.007.

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29

BREMNER, MURRAY R., and IRVIN R. HENTZEL. "IDENTITIES RELATING THE JORDAN PRODUCT AND THE ASSOCIATOR IN THE FREE NONASSOCIATIVE ALGEBRA." Journal of Algebra and Its Applications 05, no. 01 (February 2006): 77–88. http://dx.doi.org/10.1142/s0219498806001594.

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We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every nonassociative algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary algebras which generalizes the variety of Jordan algebras in the same way that Akivis algebras generalize Lie algebras.

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30

GÜNAYDIN, MURAT, and SEUNGJOON HYUN. "TERNARY ALGEBRAIC CONSTRUCTION OF EXTENDED SUPERCONFORMAL ALGEBRAS." Modern Physics Letters A 06, no. 19 (June 21, 1991): 1733–43. http://dx.doi.org/10.1142/s0217732391001871.

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We give a construction of extended (N = 2 and N = 4) superconformal algebras over a very general class of ternary algebras (triple systems). For N = 2 this construction leads to superconformal algebras corresponding to certain coset spaces of Lie groups with non-vanishing torsion and generalizes a previous construction over Jordan triple systems which are associated with Hermitian symmetric spaces. In general, a given Lie group admits more than one coset space of this type. We give examples for all simple Lie groups. In particular, the division algebras and their tensor products lead to N = 2 superconformal algebras associated with the groups of the Magic Square. For a very special class of ternary algebras, namely the Freudenthal triple (FT) systems, the N = 2 superconformal algebras can be extended to N = 4 superconformal algebras with the gauge group SU (2) × SU (2) × U (1). We give a complete list of the FT systems and the corresponding N = 4 models. They are associated with the unique quaternionic symmetric spaces of Lie groups.

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31

GORDJI, M. ESHAGHI, and N. GHOBADIPOUR. "STABILITY OF (α, β, γ)-DERIVATIONS ON LIE C*-ALGEBRAS." International Journal of Geometric Methods in Modern Physics 07, no. 07 (November 2010): 1093–102. http://dx.doi.org/10.1142/s0219887810004737.

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Petr Novotný and Jiřĺ Hrivnák [14] investigated and generalized the concept of Lie derivations via certain complex parameters and obtained various Lie and Jordan operator algebras as well as two one-parametric sets of linear operators. Moreover, they established the structure and properties of (α, β, γ)-derivations of Lie algebras. We say a functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to true solution of (ξ). In the present paper, we investigate the stability of (α, β, γ)-derivations on Lie C*-algebras associated with the following functional equation [Formula: see text]

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32

GORDJI, M. ESHAGHI, R. KHODABAKHSH, and H. KHODAEI. "ON APPROXIMATE n-ARY DERIVATIONS." International Journal of Geometric Methods in Modern Physics 08, no. 03 (May 2011): 485–500. http://dx.doi.org/10.1142/s0219887811005245.

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C. Park et al. proved the stability of homomorphisms and derivations in Banach algebras, Banach ternary algebras, C*-algebras, Lie C*-algebras and C*-ternary algebras. In this paper, we improve and generalize some results concerning derivations. We first introduce the following generalized Jensen functional equation [Formula: see text] and using fixed point methods, we prove the stability of n-ary derivations and n-ary Jordan derivations in n-ary Banach algebras. Secondly, we study this functional equation with *-n-ary derivations in C*-n-ary algebras.

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33

Hopkins, Nora C. "Noncommutative matrix jordon algebras from lie algebras." Communications in Algebra 19, no. 3 (January 1991): 767–75. http://dx.doi.org/10.1080/00927879108824168.

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34

Brešar, M., M. Cabrera, M. Fošner, and A. R. Villena. "Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan–Banach algebras." Studia Mathematica 169, no. 3 (2005): 207–28. http://dx.doi.org/10.4064/sm169-3-1.

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35

MAN’KO, V. I., G. MARMO, P. VITALE, and F. ZACCARIA. "A GENERALIZATION OF THE JORDAN-SCHWINGER MAP: THE CLASSICAL VERSION AND ITS q DEFORMATION." International Journal of Modern Physics A 09, no. 31 (December 20, 1994): 5541–61. http://dx.doi.org/10.1142/s0217751x94002260.

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For all three-dimensional Lie algebras the construction of generators in terms of functions on four-dimensional real phase space is given with a realization of the Lie product in terms of Poisson brackets. This is the classical Jordan-Schwinger map, which is also given for the deformed algebras [Formula: see text], ℰq(2) and ℋq(1). The algebra [Formula: see text] is discussed in the same context.

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36

Caveny, D. M., and O. N. Smirnov. "Categories of Jordan Structures and Graded Lie Algebras." Communications in Algebra 42, no. 1 (October 18, 2013): 186–202. http://dx.doi.org/10.1080/00927872.2012.709565.

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37

Zhao, Jun, Liangyun Chen, and Lili Ma. "Representations andT*-Extensions of Hom–Jordan–Lie Algebras." Communications in Algebra 44, no. 7 (June 2016): 2786–812. http://dx.doi.org/10.1080/00927872.2015.1065843.

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38

López, Antonio Feranández, Esther García, and Miguel Gómez Lozano. "3-Graded Lie Algebras with Jordan Finiteness Conditions." Communications in Algebra 32, no. 10 (December 31, 2004): 3807–24. http://dx.doi.org/10.1081/agb-200027748.

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39

Cagliero, Leandro, and Fernando Szechtman. "Jordan-Chevalley decomposition in finite dimensional Lie algebras." Proceedings of the American Mathematical Society 139, no. 11 (November 1, 2011): 3909–13. http://dx.doi.org/10.1090/s0002-9939-2011-10827-x.

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40

Anquela, José A., Teresa Cortés, Esther García, and Miguel Gómez Lozano. "Prime Quotients of Jordan Systems and Lie Algebras." Mediterranean Journal of Mathematics 13, no. 1 (July 2, 2015): 29–52. http://dx.doi.org/10.1007/s00009-014-0488-9.

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41

BOLSINOV, A. V., and P. ZHANG. "JORDAN–KRONECKER INVARIANTS OF FINITE-DIMENSIONAL LIE ALGEBRAS." Transformation Groups 21, no. 1 (December 1, 2015): 51–86. http://dx.doi.org/10.1007/s00031-015-9353-6.

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42

Li, Y., and F. Wei. "Jordan Derivations and Lie Derivations on Path Algebras." Bulletin of the Iranian Mathematical Society 44, no. 1 (February 2018): 79–92. http://dx.doi.org/10.1007/s41980-018-0006-0.

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43

Giambruno, Antonio, and Mikhail Zaicev. "Lie, Jordan and proper codimensions of associative algebras." Rendiconti del Circolo Matematico di Palermo 57, no. 2 (August 2008): 161–71. http://dx.doi.org/10.1007/s12215-008-0010-y.

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44

Truini, Piero. "Exceptional Lie algebras, SU(3), and Jordan pairs." Pacific Journal of Mathematics 260, no. 1 (October 11, 2012): 227–43. http://dx.doi.org/10.2140/pjm.2012.260.227.

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45

Brešar, Matej, and Peter Šemrl. "ELEMENTARY OPERATORS AS LIE HOMOMORPHISMS OR COMMUTATIVITY PRESERVERS." Proceedings of the Edinburgh Mathematical Society 48, no. 1 (February 2005): 37–49. http://dx.doi.org/10.1017/s0013091504000094.

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AbstractWe consider elementary operators on centrally closed prime algebras that are Lie (or Jordan) homomorphisms or commutativity preservers.AMS 2000 Mathematics subject classification: Primary 16N60. Secondary 16R50; 47B47

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46

Xie, Wenjuan, and Wende Liu. "Hom-structures on simple graded Lie algebras of finite growth." Journal of Algebra and Its Applications 16, no. 08 (August 17, 2016): 1750154. http://dx.doi.org/10.1142/s0219498817501547.

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A Hom-structure on a Lie algebra [Formula: see text] is a linear map [Formula: see text] satisfying the Hom–Jacobi identity: [Formula: see text] for all [Formula: see text]. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. In this paper, using a classification theorem due to Mathieu, we determine explicitly all the Hom-structures on the simple graded Lie algebras of finite growth. As a direct consequence, all the Hom-structures on any simple graded Lie algebras of finite growth constitute a Jordan algebra in the usual way.

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47

Lee, Sang Youl, Yongdo Lim, and Chan-Young Park. "Symmetric geodesics on conformal compactifications of Euclidean Jordan algebras." Bulletin of the Australian Mathematical Society 59, no. 2 (April 1999): 187–201. http://dx.doi.org/10.1017/s0004972700032810.

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In this article we define symmetric geodesies on conformal compactifications of Euclidean Jordan algebras and classify symmetric geodesics for the Euclidean Jordan algebra of all n × n symmetric real matrices. Furthermore, we show that the closed geodesics for the Euclidean Jordan algebra of all 2 × 2 symmetric real matrices are realised as the torus knots in the Shilov boundary of a Lie ball.

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48

Liu, Chein-Shan. "The g-Based Jordan Algebra and Lie Algebra Formulations of the Maxwell Equations." Journal of Mechanics 20, no. 4 (December 2004): 285–96. http://dx.doi.org/10.1017/s1727719100003518.

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AbstractWhen it is usually using a bigger algebra system to formulate the Maxwell equations, in this paper we consider a real four-dimensional algebra to express the Maxwell equations without appealing to the imaginary number and higher dimensional algebras. In terms of g-based Jordan algebra formulation the Lorentz gauge condition is found to be a necessary and sufficient condition to render the second pair of Maxwell equations, while the first pair of Maxwell equations is proved to be an intrinsic algebraic property. Then, we transform the g-based Jordan algebra to a Lie algebra of the dilation proper orthochronous Lorentz group, which gives us an incentive to consider a linear matrix operator of the Lie type, rendering more easy to derive the Maxwell equations and the wave equations. The new formulations fully match the requirements for the classical electrodynamic equations and the Lorentz gauge condition. The mathematical advantage of our formulations is that they are irreducible in the sense that, when compared to the formulations which using other bigger algebras (e.g., biquaternions and Clifford algebras), the number of explicit components and operations is minimal. From this aspect, the g-based Jordan algebra and Lie algebra are the most suitable algebraic systems to implement the Maxwell equations into a more compact form.

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49

Mira, Jose Antonio Cuenca, and Angel Rodriguez Palacios. "Isomorphisms of H*-algebras." Mathematical Proceedings of the Cambridge Philosophical Society 97, no. 1 (January 1985): 93–99. http://dx.doi.org/10.1017/s0305004100062629.

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H*-algebras were introduced and studied by Ambrose [1] in the associative case, and the theory has been extended to such particular classes of non-associative algebras as Lie [18, 19], Jordan[20, 21, 7], alternative [11] and non-commutative Jordan [6] algebras. In all these cases the core of the matter is showing that every H*-algebra (in the given class) with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals (each of which is a topologically simple H*-algebra), and then listing all the topologically simple H*-algebras in the class. In fact every nonassociative H*-algebra with zero annihilator is the closure of the orthogonal sum of its minimal closed ideals [6, theorem 2·7], so the problem of the classification of topologically simple non-associative H*-algebras becomes interesting. In relation with this problem the question arises whether, once an algebra A has been structured as a topologically simple H*-algebra, every H*-algebra structure on A is (up to a positive multiple of the inner product) totally isomorphic to the given one (see [3] and [11, section 4]). As a consequence of the results in this paper we give a general affirmative answer to this question.

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50

Arzikulov, Farhodjon, Jamilahon Aliyeva, and Olimjon Nuriddinov. "2-Local multipliers on associative, lie and jordan algebras." ACADEMICIA: An International Multidisciplinary Research Journal 10, no. 6 (2020): 649. http://dx.doi.org/10.5958/2249-7137.2020.00621.7.

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