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Journal articles on the topic 'Lie groups and Lie algebras'

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Published: 6 July 2024

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1

Wüstner, Michael. "Splittable Lie Groups and Lie Algebras." Journal of Algebra 226, no. 1 (April 2000): 202–15. http://dx.doi.org/10.1006/jabr.1999.8162.

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2

Lord, Nick, and N. Bourbaki. "Lie Groups and Lie Algebras (Chapters 1-3)." Mathematical Gazette 74, no. 468 (June 1990): 199. http://dx.doi.org/10.2307/3619408.

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3

Mikami, Kentaro, and Fumio Narita. "Dual Lie algebras of Heisenberg Poisson Lie groups." Tsukuba Journal of Mathematics 17, no. 2 (December 1993): 429–41. http://dx.doi.org/10.21099/tkbjm/1496162270.

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4

Hilgert, Joachim, and Karl H. Hofmann. "Semigroups in Lie groups, semialgebras in Lie algebras." Transactions of the American Mathematical Society 288, no. 2 (February 1, 1985): 481. http://dx.doi.org/10.1090/s0002-9947-1985-0776389-7.

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5

Berenstein, Arkady, and Vladimir Retakh. "Lie algebras and Lie groups over noncommutative rings." Advances in Mathematics 218, no. 6 (August 2008): 1723–58. http://dx.doi.org/10.1016/j.aim.2008.03.003.

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6

HOFMANN, K. H., and K. H. NEEB. "Pro-Lie groups which are infinite-dimensional Lie groups." Mathematical Proceedings of the Cambridge Philosophical Society 146, no. 2 (March 2009): 351–78. http://dx.doi.org/10.1017/s030500410800128x.

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AbstractA pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.
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7

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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8

Nahlus, Nazih. "Lie Algebras of Pro-Affine Algebraic Groups." Canadian Journal of Mathematics 54, no. 3 (June 1, 2002): 595–607. http://dx.doi.org/10.4153/cjm-2002-021-9.

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AbstractWe extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed fieldKof characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra ℒ(G) of the pro-affine algebraic groupGoverK, which is discrete in the finite-dimensional case and linearly compact in general. As an example, ifLis any sub Lie algebra of ℒ(G), we show that the closure of [L,L] in ℒ(G) is algebraic in ℒ(G).We also discuss the Hopf algebra of representative functions H(L) of a residually finite dimensional Lie algebraL. As an example, we show that ifLis a sub Lie algebra of ℒ(G) andGis connected, then the canonical Hopf algebra morphism fromK[G] intoH(L) is injective if and only ifLis algebraically dense in ℒ(G).
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9

Noohi, Behrang. "Integrating morphisms of Lie 2-algebras." Compositio Mathematica 149, no. 2 (February 2013): 264–94. http://dx.doi.org/10.1112/s0010437x1200067x.

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AbstractGiven two Lie 2-groups, we study the problem of integrating a weak morphism between the corresponding Lie 2-algebras to a weak morphism between the Lie 2-groups. To do so, we develop a theory of butterflies for 2-term L∞-algebras. In particular, we obtain a new description of the bicategory of 2-term L∞-algebras. An interesting observation here is that the role played by 1-connected Lie groups in Lie theory is now played by 2-connected Lie 2-groups. Using butterflies, we also give a functorial construction of 2-connected covers of Lie 2 -groups. Based on our results, we expect that a similar pattern generalizes to Lie n-groups and Lie n-algebras.
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10

Lauret, Jorge. "Degenerations of Lie algebras and geometry of Lie groups." Differential Geometry and its Applications 18, no. 2 (March 2003): 177–94. http://dx.doi.org/10.1016/s0926-2245(02)00146-8.

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11

Adian, S. I., and A. A. Razborov. "Periodic groups and Lie algebras." Russian Mathematical Surveys 42, no. 2 (April 30, 1987): 1–81. http://dx.doi.org/10.1070/rm1987v042n02abeh001307.

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12

González-Sánchez, Jon, and Alejandro P. Nicolas. "Uniform groups and Lie algebras." Journal of Algebra 334, no. 1 (May 2011): 54–73. http://dx.doi.org/10.1016/j.jalgebra.2011.03.003.

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13

Sukilovic, Tijana. "Classification of left invariant metrics on 4-dimensional solvable Lie groups." Theoretical and Applied Mechanics, no. 00 (2020): 14. http://dx.doi.org/10.2298/tam200826014s.

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In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.
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14

Deré, Jonas, and Marcos Origlia. "Simply transitive NIL-affine actions of solvable Lie groups." Forum Mathematicum 33, no. 5 (July 17, 2021): 1349–67. http://dx.doi.org/10.1515/forum-2020-0114.

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Abstract Every simply connected and connected solvable Lie group 𝐺 admits a simply transitive action on a nilpotent Lie group 𝐻 via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups 𝐺 can act simply transitively on which Lie groups 𝐻. So far, the focus was mainly on the case where 𝐺 is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action ρ : G → Aff ⁡ ( H ) \rho\colon G\to\operatorname{Aff}(H) is simply transitive by looking only at the induced morphism φ : g → aff ⁡ ( h ) \varphi\colon\mathfrak{g}\to\operatorname{aff}(\mathfrak{h}) between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group 𝐺 acts simply transitively on a given nilpotent Lie group 𝐻, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension 4.
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15

Howard, Eric. "Theory of groups and symmetries: Finite groups, Lie groups and Lie algebras." Contemporary Physics 60, no. 3 (July 3, 2019): 275. http://dx.doi.org/10.1080/00107514.2019.1663933.

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16

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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17

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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18

Chen, P. B., and T. S. Wu. "Hopf Algebras, Lie Algebras, and Analytic Groups." Journal of Algebra 181, no. 1 (April 1996): 1–15. http://dx.doi.org/10.1006/jabr.1996.0106.

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19

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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20

ANDO, Hiroshi, and Yasumichi MATSUZAWA. "Lie group-Lie algebra correspondences of unitary groups in finite von Neumann algebras." Hokkaido Mathematical Journal 41, no. 1 (February 2012): 31–99. http://dx.doi.org/10.14492/hokmj/1330351338.

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21

Jibladze, Mamuka, and Teimuraz Pirashvili. "Lie theory for symmetric Leibniz algebras." Journal of Homotopy and Related Structures 15, no. 1 (October 5, 2019): 167–83. http://dx.doi.org/10.1007/s40062-019-00248-x.

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Abstract Lie algebras and groups equipped with a multiplication $$\mu $$μ satisfying some compatibility properties are studied. These structures are called symmetric Lie $$\mu $$μ-algebras and symmetric $$\mu $$μ-groups respectively. An equivalence of categories between symmetric Lie $$\mu $$μ-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of the paper is an equivalence of categories between simply connected symmetric Lie $$\mu $$μ-groups and finite dimensional symmetric Leibniz algebras.
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22

Shaqaqha, Shadi, and Nadeen Kdaisat. "Extraction Algorithm of HOM–LIE Algebras Based on Solvable and Nilpotent Groups." International Journal of Mathematics and Mathematical Sciences 2023 (December 1, 2023): 1–9. http://dx.doi.org/10.1155/2023/6633715.

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Hom–Lie algebras are generalizations of Lie algebras that arise naturally in the study of nonassociative algebraic structures. In this paper, the concepts of solvable and nilpotent Hom–Lie algebras are studied further. In the theory of groups, investigations of the properties of the solvable and nilpotent groups are well-developed. We establish a theory of the solvable and nilpotent Hom–Lie algebras analogous to that of the solvable and nilpotent groups. We also provide examples to illustrate our results and discuss possible directions for further research.Dedicated to Al Farouk School & Kinder garten-Irbid-Jordan
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23

Wockel, Christoph, and Chenchang Zhu. "Integrating central extensions of Lie algebras via Lie 2-groups." Journal of the European Mathematical Society 18, no. 6 (2016): 1273–320. http://dx.doi.org/10.4171/jems/613.

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24

King, Ronald C. "S-functions and characters of Lie algebras and Lie groups." Banach Center Publications 26, no. 2 (1990): 327–44. http://dx.doi.org/10.4064/-26-2-327-344.

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25

Pestov, Vladimir. "Free Banach-Lie algebras, couniversal Banach-Lie groups, and more." Pacific Journal of Mathematics 157, no. 1 (January 1, 1993): 137–44. http://dx.doi.org/10.2140/pjm.1993.157.137.

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26

Hirsch, Morris W. "Smooth actions of Lie groups and Lie algebras on manifolds." Journal of Fixed Point Theory and Applications 10, no. 2 (November 12, 2011): 219–32. http://dx.doi.org/10.1007/s11784-011-0069-5.

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27

Majid, Shahn, and Wen-Qing Tao. "Noncommutative differentials on Poisson–Lie groups and pre-Lie algebras." Pacific Journal of Mathematics 284, no. 1 (July 10, 2016): 213–56. http://dx.doi.org/10.2140/pjm.2016.284.213.

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28

Fokas, A. S., and I. M. Gelfand. "Surfaces on Lie groups, on Lie algebras, and their integrability." Communications in Mathematical Physics 177, no. 1 (March 1996): 203–20. http://dx.doi.org/10.1007/bf02102436.

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29

Siciliano, S., and Th Weigel. "On powerful and p-central restricted Lie algebras." Bulletin of the Australian Mathematical Society 75, no. 1 (February 2007): 27–44. http://dx.doi.org/10.1017/s000497270003896x.

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In this note we analyse the analogy between m-potent and p-central restricted Lie algebras and p-groups. For restricted Lie algebras the notion of m-potency has stronger implications than for p-groups (Theorem A). Every finite-dimensional restricted Lie algebra  is isomorphic to for some finite-dimensional p-central restricted Lie algebra (Proposition B). In particular, for restricted Lie algebras there does not hold an analogue of J.Buckley's theorem. For p odd one can characterise powerful restricted Lie algebras in terms of the cup product map in the same way as for finite p-groups (Theorem C). Moreover, the p-centrality of the finite-dimensional restricted Lie algebra  has a strong implication on the structure of the cohomology ring H•(,) (Theorem D).
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30

XU, MAOSEN, and ZHIXIANG WU. "HOMOLOGY THEORY OF MULTIPLICATIVE HOM-LIE ALGEBRAS." Mathematical Reports 25(75), no. 2 (2023): 331–47. http://dx.doi.org/10.59277/mrar.2023.25.75.2.331.

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In this article, we establish Serre-Hochschild spectral sequences of Hom-Lie algebras. By using these spectral sequences, we describe homology groups of finite dimensional multiplicative Hom-Lie algebras in terms of homology groups of Lie algebras and abe
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31

Miah, Md Shapan, Khondokar M. Ahmed, and Salma Nasrin. "Characteristics of General Linear Group of Order 2 as Lie Group and Lie Algebra." Dhaka University Journal of Science 71, no. 1 (May 29, 2023): 82–86. http://dx.doi.org/10.3329/dujs.v71i1.65277.

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The main target of this article is to study about Lie Groups, Lie Algebras. This article will enrich our knowledge about Algebraic properties of manifolds, how Lie Groups and Lie Algebras are working with their properties. Finally, we have discussed an example by showing all the properties of Lie Algebra,Lie Groups for a special Group and a Theorem has established. Dhaka Univ. J. Sci. 71(1): 82-86, 2023 (Jan)
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32

Haghighatdoost, Ghorbanali, Zohreh Ravanpak, and Adel Rezaei-Aghdam. "Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups." International Journal of Geometric Methods in Modern Physics 16, no. 07 (July 2019): 1950097. http://dx.doi.org/10.1142/s021988781950097x.

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We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra [Formula: see text]. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all [Formula: see text]-[Formula: see text] structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between [Formula: see text]-[Formula: see text] structures and the generalized complex structures on the Lie algebras [Formula: see text] and also the solutions of modified Yang–Baxter equation (MYBE) on the double of Lie bialgebra [Formula: see text]. The results are applied to some relevant examples.
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33

Bokut, L. A., Yuqun Chen, and Abdukadir Obul. "Some new results on Gröbner–Shirshov bases for Lie algebras and around." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1403–23. http://dx.doi.org/10.1142/s0218196718400027.

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We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for [Formula: see text]-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.
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34

Segal, Dan. "On the automorphism groups of certain Lie algebras." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 1 (July 1989): 67–76. http://dx.doi.org/10.1017/s0305004100067980.

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We fix a ground field k and a finite separable extension K of k. To a Lie algebra L over k is associated the Lie algebra KL = K ⊗kL over K. If we forget the action of K, we can think of KL as a larger Lie algebra over k; in particular we can ask what is the automorphism group Autk KL of KL as a k-algebra. There does not seem to be any simple answer to this question in general; the purpose of this note is to give a simple condition on L which makes Autk KL quite easy to determine. Examples of algebras which satisfy this condition include the free nilpotent Lie algebras and the algebras of all n × n triangular nilpotent matrices.
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35

LICHTMAN, A. I. "RESTRICTED LIE ALGEBRAS OF POLYCYCLIC GROUPS." Journal of Algebra and Its Applications 05, no. 05 (October 2006): 571–627. http://dx.doi.org/10.1142/s0219498806001892.

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We consider some classes of polycyclic groups which have a p-series such that the restricted graded Lie algebra associated to this p-series is free abelian. We also study p-series and restricted Lie algebras associated to them in arbitrary polycyclic groups.
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36

SUDO, TAKAHIRO, and HIROSHI TAKAI. "STABLE RANK OF THE C*-ALGEBRAS OF NILPOTENT LIE GROUPS." International Journal of Mathematics 06, no. 03 (June 1995): 439–46. http://dx.doi.org/10.1142/s0129167x95000158.

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The stable rank of the C*-algebras of simply connected nilpotent Lie groups is determined by the dimension of the fixed point subspaces of the real adjoint spaces of the Lie algebras under the coadjoint actions, which is a generalization of the result obtained by A. J-L. Sheu. This equality is no longer affirmative in the case of non-nilpotent Lie groups.
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37

Suciu, Alexander I., and He Wang. "Formality properties of finitely generated groups and Lie algebras." Forum Mathematicum 31, no. 4 (July 1, 2019): 867–905. http://dx.doi.org/10.1515/forum-2018-0098.

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Abstract We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.
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38

Poroshenko, E. N., and E. I. Timoshenko. "Partially commutative groups and Lie algebras." Sibirskie Elektronnye Matematicheskie Izvestiya 18, no. 1 (June 4, 2021): 668–93. http://dx.doi.org/10.33048/semi.2021.18.048.

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39

Bryant, Roger M. "Automorphism Groups of Nilpotent Lie Algebras." Journal of the London Mathematical Society s2-36, no. 2 (October 1987): 257–74. http://dx.doi.org/10.1112/jlms/s2-36.2.257.

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40

Lombardo, S., and A. V. Mikhailov. "Reduction Groups and Automorphic Lie Algebras." Communications in Mathematical Physics 258, no. 1 (March 30, 2005): 179–202. http://dx.doi.org/10.1007/s00220-005-1334-5.

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41

Ryan, John G. "Lie algebras and pro-algebraic groups." Communications in Algebra 16, no. 2 (January 1988): 239–48. http://dx.doi.org/10.1080/00927878808823571.

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42

Pestov, Vladimir G. "Enlargeable Banach-Lie algebras and free topological groups." Bulletin of the Australian Mathematical Society 48, no. 1 (August 1993): 13–22. http://dx.doi.org/10.1017/s0004972700015409.

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43

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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44

Rezaei-Aghdam, A., and M. Sephid. "Classification of real low-dimensional Jacobi (generalized)–Lie bialgebras." International Journal of Geometric Methods in Modern Physics 14, no. 01 (December 20, 2016): 1750007. http://dx.doi.org/10.1142/s0219887817500074.

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We describe the definition of Jacobi (generalized)–Lie bialgebras [Formula: see text] in terms of structure constants of the Lie algebras [Formula: see text] and [Formula: see text] and components of their 1-cocycles [Formula: see text] and [Formula: see text] in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low-dimensional Jacobi–Lie bialgebras. In this way, we obtain and classify real two- and three-dimensional Jacobi–Lie bialgebras.
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45

Shao, Bing, En Tao Yuan, and Zhong Hai Yu. "The Real-Time Control of Space Robot by Computed Torque Control Law." Advanced Materials Research 225-226 (April 2011): 978–81. http://dx.doi.org/10.4028/www.scientific.net/amr.225-226.978.

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Lie groups and Lie algebras are used to research the dynamics and computed torque law control of free flying space robot systems. First the adjoint transformations and adjoint operators of Lie groups and Lie algebras are discussed. Then the free flying base systems are transformed to fixed base systems. The inverse dynamics and forward dynamics are described with Lie groups and Lie algebras. The computed torque control law is used to simulate with the results of dynamics. The simulation results show that with the method the dynamical simulation problems of space robot can be solved quickly and efficiently. This built the foundation of real-time control based on dynamics. The computed torque control law has good performance.
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46

Nahlus, Nazih. "Basic groups of Lie algebras and Hopf algebras." Pacific Journal of Mathematics 180, no. 1 (September 1, 1997): 135–51. http://dx.doi.org/10.2140/pjm.1997.180.135.

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47

Marin, Ivan. "Group Algebras of Finite Groups as Lie Algebras." Communications in Algebra 38, no. 7 (June 21, 2010): 2572–84. http://dx.doi.org/10.1080/00927870903417638.

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48

Gorbatsevich, V. V. "On decompositions and transitive actions of nilpotent Lie groups." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 4 (April 22, 2024): 3–14. http://dx.doi.org/10.26907/0021-3446-2024-4-3-14.

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The article considers decompositions of nilpotent Lie algebras and nilpotent Lie groups, and connections between them. Also, descriptions of irreducible transitive actions of nilpotent Lie groups on the plane and on three-dimensional space are given.
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49

Yu, Wen Jie, and Zhen Kuan Pan. "Dynamical Equations of Flexible Multibody Systems under Lie Group Framework." Applied Mechanics and Materials 723 (January 2015): 215–23. http://dx.doi.org/10.4028/www.scientific.net/amm.723.215.

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A new type of dynamic equations of flexible multibody systems is derived via virtual work principle together with floating frame approach. The absolute Cartesian coordinates are used to describe positions of floating frames’ origins of deformable bodies; The orientation transform matrices as special orthogonal groups are used to describe rotation motions; The modal coordinates are used to express small deflections of deformable bodies with respect to the corresponding floating frames. The resulting equations are mixed classic Euler-Lagrange equations of translational motion of bodies and deformation and Euler-Poinaré equations of rotation in Lie groups and Lie algebras, the Lie-Poisson equations as reconstruction equations obtaining Lie Groups from Lie algebras, constraint equations Lie groups and Lie algebras. In order to simplify the implementation, we separate the constant coefficient matrices in generalized mass matrices and generalized forces in detail. The results can be easily used to design geometric integrators of Lie group structure preserving.
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Hanusch, Maximilian. "The regularity problem for Lie groups with asymptotic estimate Lie algebras." Indagationes Mathematicae 31, no. 1 (January 2020): 152–76. http://dx.doi.org/10.1016/j.indag.2019.12.001.

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