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Journal articles on the topic 'Lie algebras'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 July 2024

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1

Gao, Yongcun, and Daoji Meng. "Affine Lie Algebra Modules and Complete Lie Algebras." Algebra Colloquium 13, no. 03 (September 2006): 481–86. http://dx.doi.org/10.1142/s1005386706000423.

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In this paper, we first construct some new infinite dimensional Lie algebras by using the integrable modules of affine Lie algebras. Then we prove that these new Lie algebras are complete. We also prove that the generalized Borel subalgebras and the generalized parabolic subalgebras of these Lie algebras are complete.

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2

Burde, Dietrich. "LIE ALGEBRA PREDERIVATIONS AND STRONGLY NILPOTENT LIE ALGEBRAS." Communications in Algebra 30, no. 7 (August 7, 2002): 3157–75. http://dx.doi.org/10.1081/agb-120004482.

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3

Burde, Dietrich, Christof Ender, and Wolfgang Alexander Moens. "Post-Lie algebra structures for nilpotent Lie algebras." International Journal of Algebra and Computation 28, no. 05 (August 2018): 915–33. http://dx.doi.org/10.1142/s0218196718500406.

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We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

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4

Burde, Dietrich, and Wolfgang Alexander Moens. "Commutative post-Lie algebra structures on Lie algebras." Journal of Algebra 467 (December 2016): 183–201. http://dx.doi.org/10.1016/j.jalgebra.2016.07.030.

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5

Burde, Dietrich, and Christof Ender. "Commutative post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras." Linear Algebra and its Applications 584 (January 2020): 107–26. http://dx.doi.org/10.1016/j.laa.2019.09.010.

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6

Li, Haisheng, Shaobin Tan, and Qing Wang. "Trigonometric Lie algebras, affine Lie algebras, and vertex algebras." Advances in Mathematics 363 (March 2020): 106985. http://dx.doi.org/10.1016/j.aim.2020.106985.

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7

Towers, David A., and Vicente R. Varea. "Elementary Lie algebras and Lie A-algebras." Journal of Algebra 312, no. 2 (June 2007): 891–901. http://dx.doi.org/10.1016/j.jalgebra.2006.11.034.

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8

Wu, Linli, Mengping Wang, and Yongsheng Cheng. "Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices." Advances in Mathematical Physics 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/6128102.

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Our aim is to classify the Rota-Baxter operators of weight 0 on the 3-dimensional Lie algebra whose derived algebra’s dimension is 2. We explicitly determine all Rota-Baxter operators (of weight zero) on the 3-dimensional Lie algebras g. Furthermore, we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras g ⋉ad⁎ g⁎ and the induced left-symmetry algebra structures on g.

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9

Jurisich, Elizabeth. "Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra." Journal of Pure and Applied Algebra 126, no. 1-3 (April 1998): 233–66. http://dx.doi.org/10.1016/s0022-4049(96)00142-9.

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10

Azam, Saeid. "Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori." Canadian Journal of Mathematics 58, no. 2 (April 1, 2006): 225–48. http://dx.doi.org/10.4153/cjm-2006-009-8.

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AbstractWe investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study themin this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

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11

Burde, Dietrich, and Karel Dekimpe. "Post-Lie algebra structures on pairs of Lie algebras." Journal of Algebra 464 (October 2016): 226–45. http://dx.doi.org/10.1016/j.jalgebra.2016.05.026.

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12

Jin, Quanqin, and Xiaochao Li. "Hom-Lie algebra structures on semi-simple Lie algebras." Journal of Algebra 319, no. 4 (February 2008): 1398–408. http://dx.doi.org/10.1016/j.jalgebra.2007.12.005.

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13

Zhou, Yanqiu, Yumeng Li, and Yunhe Sheng. "3-Lie∞-algebras and 3-Lie 2-algebras." Journal of Algebra and Its Applications 16, no. 09 (September 30, 2016): 1750171. http://dx.doi.org/10.1142/s0219498817501717.

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In this paper, we introduce the notions of a [Formula: see text]-[Formula: see text]-algebra and a 3-Lie 2-algebra. The former is a model for a 3-Lie algebra that satisfy the fundamental identity up to all higher homotopies, and the latter is the categorification of a 3-Lie algebra. We prove that the 2-category of 2-term [Formula: see text]-[Formula: see text]-algebras is equivalent to the 2-category of 3-Lie 2-algebras. Skeletal and strict 3-Lie 2-algebras are studied in detail. A construction of a 3-Lie 2-algebra from a symplectic 3-Lie algebra is given.

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14

Bai, Ruipu, Yong Wu, Jiaqian Li, and Heng Zhou. "Constructing (n+ 1)-Lie algebras fromn-Lie algebras." Journal of Physics A: Mathematical and Theoretical 45, no. 47 (November 8, 2012): 475206. http://dx.doi.org/10.1088/1751-8113/45/47/475206.

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15

Yau, Donald. "Enveloping algebra of Hom-Lie algebras." Journal of Generalized Lie Theory and Applications 2, no. 2 (2008): 95–108. http://dx.doi.org/10.4303/jglta/s070209.

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16

Camacho, L. M., J. R. Gómez, and R. M. Navarro. "Algebra of derivations of Lie algebras." Linear Algebra and its Applications 332-334 (August 2001): 371–80. http://dx.doi.org/10.1016/s0024-3795(01)00247-6.

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17

Park, Chun-Gil. "Lie ∗-homomorphisms between Lie C∗-algebras and Lie ∗-derivations on Lie C∗-algebras." Journal of Mathematical Analysis and Applications 293, no. 2 (May 2004): 419–34. http://dx.doi.org/10.1016/j.jmaa.2003.10.051.

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18

Figueroa-O’Farrill, José. "Lie algebraic Carroll/Galilei duality." Journal of Mathematical Physics 64, no. 1 (January 1, 2023): 013503. http://dx.doi.org/10.1063/5.0132661.

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We characterize Lie groups with bi-invariant bargmannian, galilean, or carrollian structures. Localizing at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian, or galilean structures are actually determined by the same data: a metric Lie algebra with a skew-symmetric derivation. This is the same data defining a one-dimensional double extension of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with such double extensions, containing carrollian Lie algebras as an ideal and projecting to galilean Lie algebras. This sets up a canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras. This reformulation allows us to use the structure theory of metric Lie algebras to give a list of bargmannian, carrollian, and galilean Lie algebras in the positive-semidefinite case. We also characterize Lie groups admitting a bi-invariant (ambient) leibnizian structure. Leibnizian Lie algebras extend the class of bargmannian Lie algebras and also set up a non-canonical correspondence between carrollian and galilean Lie algebras.

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19

Martinez-Villa, R. "G-algebras, Lie algebras, Hopf algebras." International Journal of Algebra 10 (2016): 141–61. http://dx.doi.org/10.12988/ija.2016.6215.

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20

Liu, Xu-Feng, and Min Qian. "Fock Representations of Lie Algebras, Loop Algebras and Affine Lie Algebras." Communications in Theoretical Physics 24, no. 3 (October 30, 1995): 311–20. http://dx.doi.org/10.1088/0253-6102/24/3/311.

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21

Beidar, K. I., and M. A. Chebotar. "On Lie-Admissible Algebras Whose Commutator Lie Algebras Are Lie Subalgebras of Prime Associative Algebras." Journal of Algebra 233, no. 2 (November 2000): 675–703. http://dx.doi.org/10.1006/jabr.2000.8449.

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22

Yu-Feng, Zhang, and Liu Jing. "Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra." Communications in Theoretical Physics 50, no. 2 (August 2008): 289–94. http://dx.doi.org/10.1088/0253-6102/50/2/01.

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23

Liu, Yonghong. "Lie Algebras with BCL Algebras." European Journal of Pure and Applied Mathematics 11, no. 2 (April 27, 2018): 444–48. http://dx.doi.org/10.29020/nybg.ejpam.v11i2.3219.

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The subject matter of this work is hoping for a new relationship between the Lie algebras and the algebra of logic, which will constitute an important part of our study of "pure'' algebra theory. $BCL$ algebras as a class of logical algebras is can be generated by a Lie algebra. The opposite is also true that when special conditions occur. The aim of this paper is to prove several theorems on Lie algebras with $BCL$ algebras. I introduce the notion of a "pseudo-association'' which I propose as the adjoint notion of $BCL$ algebra in the abelian group.

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24

Putra, Ricardo Eka, Edi Kurniadi, and Ema Carnia. "On Properties of Five-dimensional Nonstandard Filiform Lie algebra." CAUCHY: Jurnal Matematika Murni dan Aplikasi 8, no. 2 (November 15, 2023): 89–97. http://dx.doi.org/10.18860/ca.v8i2.21018.

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In this paper, we study the five-dimensional nonstandard Filiform Lie algebra and their basis elements representations. The aim of this research is to determine the basis elements of five-dimensional nonstandard Filiform Lie algebras representation in the form of real matrices. The method used in this study is by following Ceballos, Núñez, and Tenorio’s work. The results of this study are five real matrices as the realization of the basis elements of the five-dimensional nonstandard Filiform Lie algebra. We also discuss some results relate to five-dimensional nonstandard Filiform Lie algebra’s properties. The five-dimensional nonstandard Filiform Lie algebra is always nilpotent. For further research, it can be extended to five classes of Filiform Lie algebra, both standard and nonstandard with six dimensions. Moreover, it can be computed their split torus such that their direct sums are Frobenius Lie algebras.

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25

OKASSA, EUGÈNE. "ON LIE–RINEHART–JACOBI ALGEBRAS." Journal of Algebra and Its Applications 07, no. 06 (December 2008): 749–72. http://dx.doi.org/10.1142/s0219498808003107.

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We show that Jacobi algebras (Poisson algebras respectively) can be defined only as Lie–Rinehart–Jacobi algebras (as Lie–Rinehart–Poisson algebras respectively). Also we show that contact manifolds, locally conformal symplectic manifolds (symplectic manifolds respectively) can be defined only as symplectic Lie–Rinehart–Jacobi algebras (only as symplectic Lie–Rinehart–Poisson algebras respectively). We define symplectic Lie algebroids.

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26

Nam, Ki-Bong, and Moon-Ok Wang. "SIMPLE LIE ALGEBRAS WHICH GENERALIZE KPS`S LIE ALGEBRAS." Communications of the Korean Mathematical Society 17, no. 2 (April 1, 2002): 237–43. http://dx.doi.org/10.4134/ckms.2002.17.2.237.

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27

Chapoton, Frédéric. "Free Pre-Lie Algebras are Free as Lie Algebras." Canadian Mathematical Bulletin 53, no. 3 (September 1, 2010): 425–37. http://dx.doi.org/10.4153/cmb-2010-063-2.

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AbstractWe prove that the -module PreLie is a free Lie algebra in the category of -modules and can therefore be written as the composition of the -module Lie with a new -module X. This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using X. Furthermore, we define a natural filtration on the -module X. We also obtain a relationship between X and the -module coming from the anticyclic structure of the PreLie operad.

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28

Campoamor-Stursberg, R., and M. Rausch De Traubenberg. "Color Lie algebras and Lie algebras of order F." Journal of Generalized Lie Theory and Applications 3, no. 2 (2009): 113–30. http://dx.doi.org/10.4303/jglta/s090203.

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29

King, Jeremy D. "Finite Presentations of Lie Algebras And Restricted Lie Algebras." Bulletin of the London Mathematical Society 28, no. 3 (May 1996): 249–54. http://dx.doi.org/10.1112/blms/28.3.249.

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30

Benito, Pilar, Murray Bremner, and Sara Madariaga. "Symmetric matrices, orthogonal Lie algebras and Lie-Yamaguti algebras." Linear and Multilinear Algebra 63, no. 6 (June 27, 2014): 1257–81. http://dx.doi.org/10.1080/03081087.2014.930141.

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31

Abramov, Viktor. "Super 3-Lie Algebras Induced by Super Lie Algebras." Advances in Applied Clifford Algebras 27, no. 1 (October 10, 2015): 9–16. http://dx.doi.org/10.1007/s00006-015-0604-3.

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32

Shalev, Aner. "Simple Lie algebras and Lie algebras of maximal class." Archiv der Mathematik 63, no. 4 (October 1994): 297–301. http://dx.doi.org/10.1007/bf01189564.

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33

Burde, Dietrich, Karel Dekimpe, and Bert Verbeke. "Almost inner derivations of Lie algebras." Journal of Algebra and Its Applications 17, no. 11 (November 2018): 1850214. http://dx.doi.org/10.1142/s0219498818502146.

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We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for [Formula: see text]-step nilpotent Lie algebras determined by graphs, free [Formula: see text] and [Formula: see text]-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand, we also exhibit families of nilpotent Lie algebras having an arbitrary large space of non-inner almost inner derivations.

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34

GEBERT, REINHOLD W. "INTRODUCTION TO VERTEX ALGEBRAS, BORCHERDS ALGEBRAS AND THE MONSTER LIE ALGEBRA." International Journal of Modern Physics A 08, no. 31 (December 20, 1993): 5441–503. http://dx.doi.org/10.1142/s0217751x93002162.

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The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory In this context Borcherds algebras arise as certain “physical” subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction to this rapidly developing area of mathematics. Based on the machinery of formal calculus, we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analyzed from the point of view of symmetry in quantum theory and the construction of the monster Lie algebra is sketched.

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35

Hussain, Naveed, Stephen S. T. Yau, and Huaiqing Zuo. "Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities." Forum Mathematicum 34, no. 2 (January 6, 2022): 323–45. http://dx.doi.org/10.1515/forum-2021-0227.

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Abstract The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

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36

Burde, Dietrich, and Vsevolod Gubarev. "Decompositions of algebras and post-associative algebra structures." International Journal of Algebra and Computation 30, no. 03 (December 2, 2019): 451–66. http://dx.doi.org/10.1142/s0218196720500071.

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We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

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37

Burde, Dietrich. "Derivation double Lie algebras." Journal of Algebra and Its Applications 15, no. 06 (March 30, 2016): 1650114. http://dx.doi.org/10.1142/s0219498816501140.

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We study classical [Formula: see text]-matrices [Formula: see text] for Lie algebras [Formula: see text] such that [Formula: see text] is also a derivation of [Formula: see text]. This yields derivation double Lie algebras [Formula: see text]. The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

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38

Kori, Tosiaki, and Yuto Imai. "Lie algebra extensions of current algebras on S3." International Journal of Geometric Methods in Modern Physics 12, no. 09 (October 2015): 1550087. http://dx.doi.org/10.1142/s0219887815500875.

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An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

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39

Su, Yucai. "Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations." Canadian Journal of Mathematics 55, no. 4 (August 1, 2003): 856–96. http://dx.doi.org/10.4153/cjm-2003-036-7.

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AbstractXu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

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40

Fang, Yingjue, and Liangang Peng. "Generalized Heisenberg Algebras and Toroidal Lie Algebras." Algebra Colloquium 17, no. 03 (September 2010): 375–88. http://dx.doi.org/10.1142/s1005386710000374.

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In this article we provide two kinds of infinite presentations of toroidal Lie algebras. At first we define generalized Heisenberg algebras and prove that each toroidal Lie algebra is an amalgamation of a simple Lie algebra and a generalized Heisenberg algebra in the sense of Saito and Yoshii. This is one kind of presentations of toroidal Lie algebras given by the generators of generalized Heisenberg algebras and the Chevalley generators of simple Lie algebras with certain amalgamation relations. Secondly by using the generalized Chevalley generators, we give another kind of presentations. These two kinds of presentations are different from those given by Moody, Eswara Rao and Yokonuma.

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41

Bai, Ruipu, Lixin Lin, Yan Zhang, and Chuangchuang Kang. "q-Deformations of 3-Lie Algebras." Algebra Colloquium 24, no. 03 (September 2017): 519–40. http://dx.doi.org/10.1142/s1005386717000347.

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q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra [Formula: see text], where [Formula: see text] and [Formula: see text], is a vector space A over a field 𝔽 with 3-ary linear multiplications [ , , ]q and [Formula: see text] from [Formula: see text] to A, and a map [Formula: see text] satisfying the q-Jacobi identity [Formula: see text] for all [Formula: see text]. If the multiplications satisfy that [Formula: see text] and [Formula: see text] is skew-symmetry, then [Formula: see text] is called a type (I)-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type (I)-q-3-Lie algebras are constructed. Also, the non-trivial onedimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras [Formula: see text], and [Formula: see text] are obtained.

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42

AVITABILE, MARINA. "SOME LOOP ALGEBRAS OF HAMILTONIAN LIE ALGEBRAS." International Journal of Algebra and Computation 12, no. 04 (August 2002): 535–67. http://dx.doi.org/10.1142/s0218196702001097.

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We consider a class of thin Lie algebras with second diamond in weight a power of the characteristic of the underlying field. We identify these Lie algebras with loop algebras of a graded Hamiltonian algebra or loop algebras of an extension of the Hamiltonian algebra by an outer derivation. We also prove that the Lie algebras considered are not finitely presented.

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43

Martinez-Villa, R. "G-algebras, Lie algebras, Hopf algebras II." International Journal of Algebra 10 (2016): 351–72. http://dx.doi.org/10.12988/ija.2016.6748.

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44

Cox, Ben, Xiangqian Guo, Rencai Lu, and Kaiming Zhao. "Simple superelliptic Lie algebras." Communications in Contemporary Mathematics 19, no. 03 (April 5, 2017): 1650032. http://dx.doi.org/10.1142/s0219199716500322.

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Let [Formula: see text], [Formula: see text]. Then we have the algebraic curve [Formula: see text], and its coordinate algebras (the Riemann surfaces) [Formula: see text] and [Formula: see text] The Lie algebras [Formula: see text] and [Formula: see text] are called the [Formula: see text]th superelliptic Lie algebras associated to [Formula: see text]. In this paper, we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras, which will help to understand the birational equivalence of some algebraic curves of the form [Formula: see text].

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45

SALEMKAR, ALI REZA, HADI BIGDELY, and VAHID ALAMIAN. "SOME PROPERTIES ON ISOCLINISM OF LIE ALGEBRAS AND COVERS." Journal of Algebra and Its Applications 07, no. 04 (August 2008): 507–16. http://dx.doi.org/10.1142/s0219498808002965.

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In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

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46

Zhu, Fuyang, and Wen Teng. "Cohomology and Crossed Modules of Modified Rota–Baxter Pre-Lie Algebras." Mathematics 12, no. 14 (July 19, 2024): 2260. http://dx.doi.org/10.3390/math12142260.

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The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras.

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47

ISAEV, A. P., and O. OGIEVETSKY. "BRST OPERATOR FOR QUANTUM LIE ALGEBRAS: EXPLICIT FORMULA." International Journal of Modern Physics A 19, supp02 (May 2004): 240–47. http://dx.doi.org/10.1142/s0217751x04020440.

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Abstract:

We continue our study of quantum Lie algebras, an important class of quadratic algebras arising in the Woronowicz calculus on a quantum group. Quantum Lie algebras are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit "quantum" analogues. In particular, there is a BRST operator Q(Q2=0) which generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers a recurrence relation for the operator Q for quantum Lie algebras was given. Here we solve this recurrence relation and obtain an explicit formula for the BRST operator.

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48

Littelmann, Peter. "A Plactic Algebra for Semisimple Lie Algebras." Advances in Mathematics 124, no. 2 (December 1996): 312–31. http://dx.doi.org/10.1006/aima.1996.0085.

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49

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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50

Towers, David A. "Almost nilpotent Lie algebras." Glasgow Mathematical Journal 29, no. 1 (January 1987): 7–11. http://dx.doi.org/10.1017/s0017089500006625.

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Abstract:

Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.

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