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Journal articles on the topic 'Lie algebras][Euclidean space'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

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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2

Kalnins, Ernest G., and W. Miller. "Quadratic algebra contractions and second-order superintegrable systems." Analysis and Applications 12, no. 05 (August 28, 2014): 583–612. http://dx.doi.org/10.1142/s0219530514500377.

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Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of second-order superintegrable systems in two dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces, we show that the free quadratic algebras generated by the first- and second-order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

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3

BLOHMANN, CHRISTIAN. "PERTURBATIVE SYMMETRIES ON NONCOMMUTATIVE SPACES." International Journal of Modern Physics A 19, no. 32 (December 30, 2004): 5693–706. http://dx.doi.org/10.1142/s0217751x04021238.

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Perturbative deformations of symmetry structures on noncommutative spaces are studied in view of noncommutative quantum field theories. The rigidity of enveloping algebras of semisimple Lie algebras with respect to formal deformations is reviewed in the context of star products. It is shown that rigidity of symmetry algebras extends to rigidity of the action of the symmetry on the space. This implies that the noncommutative spaces considered can be realized as star products by particular ordering prescriptions which are compatible with the symmetry. These symmetry preserving ordering prescriptions are calculated for the quantum plane and four-dimensional quantum Euclidean space. The result can be used to construct invariant Lagrangians for quantum field theory on noncommutative spaces with a deformed symmetry.

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4

SANTANDER, MARIANO. "A PERSPECTIVE ON THE MAGIC SQUARE AND THE "SPECIAL UNITARY" REALIZATION OF REAL SIMPLE LIE ALGEBRAS." International Journal of Geometric Methods in Modern Physics 10, no. 08 (August 7, 2013): 1360002. http://dx.doi.org/10.1142/s0219887813600025.

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This paper contains the last part of the minicourse "Spaces: A Perspective View" delivered at the IFWGP2012. The series of three lectures was intended to bring the listeners from the more naive and elementary idea of space as "our physical Space" (which after all was the dominant one up to the 1820s) through the generalization of the idea of space which took place in the last third of the 19th century. That was a consequence of first the discovery and acceptance of non-Euclidean geometry and second, of the views afforded by the works of Riemann and Klein and continued since then by many others, outstandingly Lie and Cartan. Here we deal with the part of the minicourse which centers on the classification questions associated to the simple real Lie groups. We review the original introduction of the Magic Square "á la Freudenthal", putting the emphasis in the role played in this construction by the four normed division algebras ℝ, ℂ, ℍ, 𝕆. We then explore the possibility of understanding some simple real Lie algebras as "special unitary" over some algebras 𝕂 or tensor products 𝕂1 ⊗ 𝕂2, and we argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions ℂ′, ℍ′, 𝕆′ of complex, quaternions and octonions as well. This way we get a "Grand Magic Square" and we fill in all details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as "special unitary" (or only unitary when n = 2) over some tensor product of two *-algebras 𝕂1, 𝕂2, which in all cases are obtained from ℝ, ℂ, ℂ′, ℍ, ℍ′, 𝕆, 𝕆′ as sets, endowing them with a *-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.

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5

Izmest'ev, A. A., G. S. Pogosyan, A. N. Sissakian, and P. Winternitz. "Contractions of Lie Algebras and Separation of Variables." International Journal of Modern Physics A 12, no. 01 (January 10, 1997): 53–61. http://dx.doi.org/10.1142/s0217751x97000074.

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The Inönü-Wigner contraction from the Lorentz group O(2,1) to the Euclidean group E(2) is used to relate the separation of variables in the Laplace-Beltrami operators on the two corresponding homogeneous spaces. We consider the contractions on four levels: the Lie algebra, the commuting sets of second order operators in the enveloping algebra o(2,1), the coordinate systems and some eigenfunctions of the Laplace-Beltrami operators.

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6

ANDERSON, ARLEN. "SYMMETRIC SPACE TWO-MATRIX MODELS." International Journal of Modern Physics A 07, no. 23 (September 20, 1992): 5781–96. http://dx.doi.org/10.1142/s0217751x92002635.

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The radial form of the partition function of a two-matrix model is formally given in terms of a spherical function for matrices representing any Euclidean symmetric space. An explicit expression is obtained by constructing the spherical function by the method of intertwining. The reduction of two-matrix models based on Lie algebras is an elementary application. A model based on the rank one symmetric space isomorphic to RN is less trivial and is treated in detail. This model may be interpreted as an Ising model on a random branched polymer. It has the unusual feature that the maximum order of criticality is different in the planar and double-scaling limits.

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7

Cortés, V., L. Gall, and T. Mohaupt. "Four-dimensional vector multiplets in arbitrary signature (I)." International Journal of Geometric Methods in Modern Physics 17, no. 10 (August 26, 2020): 2050150. http://dx.doi.org/10.1142/s0219887820501509.

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We derive a necessary and sufficient condition for Poincaré Lie superalgebras in any dimension and signature to be isomorphic. This reduces the classification problem, up to certain discrete operations, to classifying the orbits of the Schur group on the vector space of superbrackets. We then classify four-dimensional [Formula: see text] supersymmetry algebras, which are found to be unique in Euclidean and in neutral signature, while in Lorentz signature there exist two algebras with R-symmetry groups [Formula: see text] and [Formula: see text], respectively.

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8

DOUGLAS, ANDREW, and JOE REPKA. "INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF." Bulletin of the Australian Mathematical Society 83, no. 3 (April 1, 2011): 439–49. http://dx.doi.org/10.1017/s0004972711002115.

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AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

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9

POPOV, A. D. "SELF-DUAL YANG–MILLS: SYMMETRIES AND MODULI SPACE." Reviews in Mathematical Physics 11, no. 09 (October 1999): 1091–149. http://dx.doi.org/10.1142/s0129055x99000350.

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Geometry of the solution space of the self-dual Yang–Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern–Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N=2 strings are briefly discussed.

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Cortés, V., L. Gall, and T. Mohaupt. "Four-dimensional vector multiplets in arbitrary signature (II)." International Journal of Geometric Methods in Modern Physics 17, no. 10 (August 26, 2020): 2050151. http://dx.doi.org/10.1142/s0219887820501510.

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Following the classification up to isomorphism of [Formula: see text] Poincaré Lie superalgebras in four dimensions with arbitrary signature obtained in a companion paper, we present off-shell vector multiplet representations and invariant Lagrangians realizing these algebras. By dimensional reduction of five-dimensional off-shell vector multiplets, we obtain two representations in each four-dimensional signature. In Euclidean and neutral signature, these representations can be mapped to each other by a field redefinition induced by the action of the Schur group on the space of superbrackets. In Minkowski signature, we show that the superbrackets underlying the two vector multiplet representations belong to distinct open orbits of the Schur group and are therefore inequivalent. Our formalism allows to answer questions about the possible relative signs between terms in the Lagrangian systematically by relating them to the underlying space of superbrackets.

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11

Boniver, F., and P. B. A. Lecomte. "A Remark About the Lie Algebra of Infinitesimal Conformal Transformations of the Euclidean Space." Bulletin of the London Mathematical Society 32, no. 3 (May 2000): 263–66. http://dx.doi.org/10.1112/s0024609300006986.

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12

Park, Wooram, Yan Liu, Yu Zhou, Matthew Moses, and Gregory S. Chirikjian. "Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map." Robotica 26, no. 4 (July 2008): 419–34. http://dx.doi.org/10.1017/s0263574708004475.

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SUMMARYA nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker–Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space,SO(3), and the Euclidean motion group of the plane,SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

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Szajewska, Marzena. "Faces of Platonic solids in all dimensions." Acta Crystallographica Section A Foundations and Advances 70, no. 4 (June 11, 2014): 358–63. http://dx.doi.org/10.1107/s205327331400638x.

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This paper considers Platonic solids/polytopes in the real Euclidean space {\bb R}^n of dimension 3 ≤n< ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤d≤n− 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of typesAn,Bn,Cn,F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groupsH3,H4. The method consists of recursively decorating the appropriate Coxeter–Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit,i.e.are identical, the solid is called Platonic. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.

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JURDJEVIC, V., and J. ZIMMERMAN. "Rolling sphere problems on spaces of constant curvature." Mathematical Proceedings of the Cambridge Philosophical Society 144, no. 3 (May 2008): 729–47. http://dx.doi.org/10.1017/s0305004108001084.

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AbstractThe rolling sphere problem on Euclidean space consists of determining the path of minimal length traced by the point of contact of the oriented unit sphere$\mathbb{S}^{n}$as it rolls on$\mathbb{E}^{n}$without slipping between two points of$\mathbb{E}^{n}\times SO_{n+1}(\mathbb{R})$. This problem is extended to situations in which an oriented sphere$\mathbb{S}_{\rho}^{n}$of radius ρ rolls on a stationary sphere$\mathbb{S}_{\sigma}^{n}$and to the hyperbolic analogue in which the spheres$\mathbb{S}_{\rho}^{n}$and$\mathbb{S}_{\sigma}^{n}$are replaced by the hyperboloids$\mathbb{H}_{\rho}^{n}$and$\mathbb{H}_{\sigma}^{n}$respectively. The notion of “rolling” is defined in an isometric sense: the length of the path traced by the point of contact is measured by the Riemannian metric of the stationary manifold, and the orientation of the rolling object is measured by a matrix in its isometry group. These rolling problems are formulated as left invariant optimal control problems on Lie groups whose Hamiltonian extremal equations reveal two remarkable facts: on the level of Lie algebras the extremal equations of all these rolling problems are governed by a single set of equations, and the projections onto the stationary manifold of the extremal equations havingI4=0, whereI4is an integral of motion, coincide with the elastic curves on this manifold. The paper then outlines some explicit solutions based on the use of symmetries and the corresponding integrals of motion.

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15

Francis, John. "The tangent complex and Hochschild cohomology of -rings." Compositio Mathematica 149, no. 3 (December 10, 2012): 430–80. http://dx.doi.org/10.1112/s0010437x12000140.

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AbstractIn this work, we study the deformation theory of${\mathcal {E}}_n$-rings and the${\mathcal {E}}_n$analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence$A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the${\mathcal {E}}_n$-tangent complex and${\mathcal {E}}_n$-Hochschild cohomology of an${\mathcal {E}}_n$-ring$A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups,$B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here${\mathfrak B}^n\!A$is an enriched$(\infty ,n)$-category constructed from$A$, and${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of${\mathfrak B}^n\!A$. These groups are associated to moduli problems in${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex$T_A[-n]$is a nonunital${\mathcal {E}}_{n+1}$-algebra. The${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known${\mathcal {E}}_{n+1}$-algebra structure on${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed$n$-manifolds with coefficients given by${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

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Boucetta, Mohamed, and Hicham Lebzioui. "On flat pseudo-Euclidean nilpotent Lie algebras." Journal of Algebra 537 (November 2019): 459–77. http://dx.doi.org/10.1016/j.jalgebra.2019.07.018.

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17

Kalnins, E. G., and P. Winternitz. "Maximal Abelian subalgebras of complex Euclidean Lie algebras." Canadian Journal of Physics 72, no. 7-8 (July 1, 1994): 389–404. http://dx.doi.org/10.1139/p94-055.

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Maximal Abelian subalgebras (MASAs) of the complex Euclidean Lie algebra [Formula: see text] are classified into conjugacy classes under the action of the Lie group [Formula: see text] Use is made of an earlier classification of MASAs of the orthogonal Lie algebra [Formula: see text] These are then extended to nonsplitting MASAs of [Formula: see text] in which maximal Abelian nilpotent subalgebras of [Formula: see text] are coupled with translations in a nontrivial manner. The methods presented are applicable to the classification of MASAs of any affine Lie algebra.

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18

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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KUPERSHMIDT, B. A. "MODIFIED KORTEWEG-DE VRIES EQUATIONS ON EUCLIDEAN LIE ALGEBRAS." International Journal of Modern Physics B 03, no. 06 (June 1989): 853–61. http://dx.doi.org/10.1142/s0217979289000622.

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For any finite-dimensional Euclidean Lie alegebra [Formula: see text], a commuting hierarchy of generalized modified Korteweg-de Vries equations is constructed, together with a nonabelian generalization of the classical Miura map. The classical situation is recovered for the case when [Formula: see text] is abelian one-dimensional. Localization of differential formulae yields a representation of the Virasoro algebra in terms of elements of the current Lie algebra associated to [Formula: see text].

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20

Adashev, J. Q., B. A. Omirov, and S. Uguz. "Leibniz Algebras Associated with Representations of Euclidean Lie Algebra." Algebras and Representation Theory 23, no. 2 (January 4, 2019): 285–301. http://dx.doi.org/10.1007/s10468-018-09849-1.

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21

Douglas, Andrew, Joe Repka, and Wainwright Joseph. "The Euclidean algebra in rank 2 classical Lie algebras." Journal of Mathematical Physics 55, no. 6 (June 2014): 061701. http://dx.doi.org/10.1063/1.4880195.

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Joyce, W. P., and P. H. Butler. "The geometric associative algebras of Euclidean space." Advances in Applied Clifford Algebras 12, no. 2 (December 2002): 195–233. http://dx.doi.org/10.1007/bf03161247.

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23

Henriques, André. "Integrating -algebras." Compositio Mathematica 144, no. 4 (July 2008): 1017–45. http://dx.doi.org/10.1112/s0010437x07003405.

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AbstractGiven a Lie n-algebra, we provide an explicit construction of its integrating Lie n-group. This extends work done by Getzler in the case of nilpotent $L_\infty $-algebras. When applied to an ordinary Lie algebra, our construction yields the simplicial classifying space of the corresponding simply connected Lie group. In the case of the string Lie 2-algebra of Baez and Crans, we obtain the simplicial nerve of their model of the string group.

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24

Ku, Jong-Min. "Structure of the Verma module M(−ϱ) over Euclidean Lie algebras." Journal of Algebra 124, no. 2 (August 1989): 367–87. http://dx.doi.org/10.1016/0021-8693(89)90138-5.

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Peyghan, E., and L. Nourmohammadifar. "Para-Kähler hom-Lie algebras." Journal of Algebra and Its Applications 18, no. 03 (March 2019): 1950044. http://dx.doi.org/10.1142/s0219498819500440.

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In this paper, we introduce the notions of pseudo-Riemannian, para-Hermitian and para-Kähler structures on hom-Lie algebras. In addition, we present the characterization of these structures. Also, we provide an example including these structures. We then introduce the phase space of a hom-Lie algebra and using the hom-left symmetric product, we show that a para-Kähler hom-Lie algebra gives a phase space and conversely, we can construct a para-Kähler hom-Lie algebra using a phase space.

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GORDJI, M. ESHAGHI, A. FAZELI, and CHOONKIL PARK. "3-LIE MULTIPLIERS ON BANACH 3-LIE ALGEBRAS." International Journal of Geometric Methods in Modern Physics 09, no. 07 (September 7, 2012): 1250052. http://dx.doi.org/10.1142/s0219887812500521.

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In this paper, we study the space of 3-Lie multipliers on Banach 3-Lie algebras. Moreover, we investigate a characterization of 3-Lie multipliers on commutative and without order Banach 3-Lie algebras. Finally, we establish the stability and superstability of 3-Lie multipliers.

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Knibbeler, Vincent, Sara Lombardo, and Jan A. Sanders. "Hereditary automorphic Lie algebras." Communications in Contemporary Mathematics 22, no. 08 (December 20, 2019): 1950076. http://dx.doi.org/10.1142/s0219199719500767.

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We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

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Ren, Bin, and Linsheng Zhu. "Quasi Qn-Filiform Lie Algebras." Algebra Colloquium 18, no. 01 (March 2011): 139–54. http://dx.doi.org/10.1142/s1005386711000083.

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In this paper, we explicitly determine the derivation algebra, automorphism group of quasi Qn-filiform Lie algebras, and by applying some properties of the root space decomposition, we obtain their isomorphism theorem.

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FIALOWSKI, ALICE, and MICHAEL PENKAVA. "DEFORMATIONS OF FOUR-DIMENSIONAL LIE ALGEBRAS." Communications in Contemporary Mathematics 09, no. 01 (February 2007): 41–79. http://dx.doi.org/10.1142/s0219199707002344.

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We study the moduli space of four-dimensional ordinary Lie algebras, and their versal deformations. Their classification is well known; our focus in this paper is on the deformations, which yield a picture of how the moduli space is assembled. Surprisingly, we get a nice geometric description of this moduli space essentially as an orbifold, with just a few exceptional points.

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Neeb, Karl-Hermann, and Friedrich Wagemann. "The Second Cohomology of Current Algebras of General Lie Algebras." Canadian Journal of Mathematics 60, no. 4 (August 1, 2008): 892–922. http://dx.doi.org/10.4153/cjm-2008-038-6.

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AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

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FIALOWSKI, ALICE, and MICHAEL PENKAVA. "VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ ALGEBRAS." Communications in Contemporary Mathematics 07, no. 02 (April 2005): 145–65. http://dx.doi.org/10.1142/s0219199705001702.

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We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞ algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

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TAO, JIYUAN, and M. SEETHARAMA GOWDA. "A REPRESENTATION THEOREM FOR LYAPUNOV-LIKE TRANSFORMATIONS ON EUCLIDEAN JORDAN ALGEBRAS." International Game Theory Review 15, no. 04 (November 18, 2013): 1340034. http://dx.doi.org/10.1142/s0219198913400343.

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A Lyapunov-like (linear) transformation L on a Euclidean Jordan algebra V is defined by the condition [Formula: see text]where K is the symmetric cone of V. In this paper, we give an elementary proof (avoiding Lie algebraic ideas and results) of the fact that Lyapunov-like transformations on V are of the form La + D, where a ∈ V, D is a derivation, and La(x) = a ◦ x for all x ∈ V.

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Dobrogowska, Alina, and Karolina Wojciechowicz. "Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras." Symmetry 13, no. 8 (August 9, 2021): 1455. http://dx.doi.org/10.3390/sym13081455.

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We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

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Xu, C., G. Zhu, and K. Yang. "SCENE CLASSIFICATION BASED ON THE INTRINSIC MEAN OF LIE GROUP." ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences V-3-2020 (August 3, 2020): 75–82. http://dx.doi.org/10.5194/isprs-annals-v-3-2020-75-2020.

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Abstract. Remote Sensing scene classification aims to identify semantic objects with similar characteristics from high resolution images. Even though existing methods have achieved satisfactory performance, the features used for classification modeling are still limited to some kinds of vector representation within a Euclidean space. As a result, their models are not robust to reflect the essential scene characteristics, hardly to promote classification accuracy higher. In this study, we propose a novel scene classification method based on the intrinsic mean on a Lie Group manifold. By introducing Lie Group machine learning into scene classification, the new method uses the geodesic distance on the Lie Group manifold, instead of Euclidean distance, solving the problem that non-euclidean space samples could not be calculated by Euclidean distance directly. The experiments show that our method produces satisfactory performance on two public and challenging remote sensing scene datasets, UC Merced and SIRI-WHU, respectively.

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JORGENSEN, PALLE E. T. "REPRESENTATIONS OF LIE ALGEBRAS BUILT OVER HILBERT SPACE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 14, no. 03 (September 2011): 419–42. http://dx.doi.org/10.1142/s0219025711004468.

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Starting with a complex Hilbert space, using inductive limits, we build Lie algebras, and find families of representations. They include those often studied in mathematical physics in order to model quantum statistical mechanics or quantum fields. We explore natural actions on infinite tensor algebras T(H) built with a functorial construction, starting with a fixed Hilbert space H. While our construction applies also when H is infinite-dimensional, the case with N ≔ dim H finite is of special interest as the symmetry group we consider is then a copy of the non-compact Lie group U(N, 1). We give the tensor algebra T(H) the structure of a Hilbert space, i.e. the unrestricted infinite tensor product Fock space [Formula: see text]. The tensor algebra T(H) is naturally represented as acting by bounded operators on [Formula: see text], and U (N, 1) as acting as a unitary representation. From this we built a covariant system, and we explore how the fermion, the boson, and the q on Hilbert spaces are reduced by the representations. In particular we display the decomposition into irreducible representations of the naturally defined U (N, 1) representation.

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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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del Amor, José, Ángel Giménez, and Pascual Lucas. "Null Curve Evolution in Four-Dimensional Pseudo-Euclidean Spaces." Advances in Mathematical Physics 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/5725234.

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We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinitely many local symmetries for the null localized induction equation is provided.

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38

LI, HAISHENG. "VERTEX ALGEBRAS AND VERTEX POISSON ALGEBRAS." Communications in Contemporary Mathematics 06, no. 01 (February 2004): 61–110. http://dx.doi.org/10.1142/s0219199704001264.

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This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex Poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex Poisson algebra are revisited and certain general construction theorems of vertex Poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it is proved that the associated graded vector space of a filtered vertex algebra is naturally a vertex Poisson algebra. For any vertex algebra V, a general construction and a classification of good filtrations are given. To each ℕ-graded vertex algebra V=∐n∈ℕV(n) with [Formula: see text], a canonical (good) filtration is associated and certain results about generating subspaces of certain types of V are also obtained. Furthermore, a notion of formal deformation of a vertex (Poisson) algebra is formulated and a formal deformation of vertex Poisson algebras associated with vertex Lie algebras is constructed.

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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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Wu, Weicai, Shouchuan Zhang, and Yao-Zhong Zhang. "On Nichols (braided) Lie algebras." International Journal of Mathematics 26, no. 10 (September 2015): 1550082. http://dx.doi.org/10.1142/s0129167x15500822.

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We prove (i) Nichols algebra 𝔅(V) of vector space V is finite dimensional if and only if Nichols braided Lie algebra 𝔏(V) is finite dimensional; (ii) if the rank of connected V is 2 and 𝔅(V) is an arithmetic root system, then 𝔅(V) = F ⊕ 𝔏(V); and (iii) if Δ(𝔅(V)) is an arithmetic root system and there does not exist any m-infinity element with puu ≠ 1 for any u ∈ D(V), then dim (𝔅(V)) = ∞ if and only if there exists V′, which is twisting equivalent to V, such that dim (𝔏-(V′)) = ∞. Furthermore, we give an estimation of dimensions of Nichols Lie algebras and two examples of Lie algebras which do not have maximal solvable ideals.

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Xie, Binhai, Shuling Dai, and Feng Liu. "A Lie Group-Based Iterative Algorithm Framework for Numerically Solving Forward Kinematics of Gough–Stewart Platform." Mathematics 9, no. 7 (April 1, 2021): 757. http://dx.doi.org/10.3390/math9070757.

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In this work, we began to take forward kinematics of the Gough–Stewart (G-S) platform as an unconstrained optimization problem on the Lie group-structured manifold SE(3) instead of simply relaxing its intrinsic orthogonal constraint when algorithms are updated on six-dimensional local flat Euclidean space or adding extra unit norm constraint when orientation parts are parametrized by a unit quaternion. With this thought in mind, we construct two kinds of iterative problem-solving algorithms (Gauss–Newton (G-N) and Levenberg–Marquardt (L-M)) with mathematical tools from the Lie group and Lie algebra. Finally, a case study for a general G-S platform was carried out to compare these two kinds of algorithms on SE(3) with corresponding algorithms that updated on six-dimensional flat Euclidean space or seven-dimensional quaternion-based parametrization Euclidean space. Experiment results demonstrate that those algorithms on SE(3) behave better than others in convergence performance especially when the initial guess selection is near to branch solutions.

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Bai, Ruipu, and Ying Li. "Tθ∗-Extensions of n-Lie Algebras." ISRN Algebra 2011 (September 11, 2011): 1–11. http://dx.doi.org/10.5402/2011/381875.

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The paper is mainly concerned with Tθ∗-extensions of n-Lie algebras. The Tθ∗-extension Lθ(L∗) of an n-Lie algebra L by a cocycle θ is defined, and a class of cocycles is constructed by means of linear mappings from an n-Lie algebra on to its dual space. Finally all Tθ∗-extensions of (n+1)-dimensional n-Lie algebras are classified, and the explicit multiplications are given.

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Ayala, Víctor, Heriberto Román-Flores, María Torreblanca Todco, and Erika Zapana. "Observability and Symmetries of Linear Control Systems." Symmetry 12, no. 6 (June 4, 2020): 953. http://dx.doi.org/10.3390/sym12060953.

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The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space R n and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups.

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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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Tan, Yan, and Zhixiang Wu. "Extending Structures for Lie 2-Algebras." Mathematics 7, no. 6 (June 18, 2019): 556. http://dx.doi.org/10.3390/math7060556.

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The extending structures problem for strict Lie 2-algebras is studied. To provide the theoretical answer to this problem, this paper introduces the unified product of a given strict Lie 2-algebra g and 2-vector space V. The unified product includes some interesting products such as semi-direct product, crossed product, and bicrossed product. The paper focuses on crossed and bicrossed products, which give the answer to the extension problem and factorization problem, respectively.

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46

Martin, Paul P., and David Woodcock. "Generalized Blob Algebras and Alcove Geometry." LMS Journal of Computation and Mathematics 6 (2003): 249–96. http://dx.doi.org/10.1112/s1461157000000450.

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AbstractA sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht⁄standard modules. As in the weight space formalism of algebraic Lie theory, there is an action of an affine reflection group on this weight space that fixes the set of labelling weights. A linkage principle is proved in each case. Further, it is shown that the simplest non-trivial example may essentially be identified with the blob algebra (a physically motivated quasihereditary algebra whose representation theory is very well understood by Lie-theory-like methods). An extended role is hence proposed for Soergel's tilting algorithm, away from its algebraic Lie theory underpinning, in determining the simple content of standard modules for these algebras. This role is explicitly verified in the blob algebra case. A tensor space representation of the blob algebra is constructed, as a candidate for a full tilting module (subsequently proven to be so in a paper by Martin and Ryom-Hansen), further evidencing the extended utility of Lie-theoretic methods. Possible generalisations of this representation to other elements of the sequence are discussed.

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47

ALEKSEEVSKY, D. V., and V. CORTÉS. "SPECIAL VINBERG CONES." Transformation Groups 26, no. 2 (April 6, 2021): 377–402. http://dx.doi.org/10.1007/s00031-021-09649-w.

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AbstractThe paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; 𝔤) and $$ {a}_{ji}={a}_{ij}^{\ast }=\mathfrak{g}\left({a}_{ij},\cdot \right)\in {V}_{ij}^{\ast } $$ a ji = a ij ∗ = g a ij ⋅ ∈ V ij ∗ belongs to the dual space $$ {V}_{ij}^{\ast }. $$ V ij ∗ . The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric 𝔤S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone 𝒱 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone 𝒱2 ⊂ ℋ2(V ) to the special rank 3 case.

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Gorbatsevich, Vladimir V. "Some properties of the space ofn-dimensional Lie algebras." Sbornik: Mathematics 200, no. 2 (February 28, 2009): 185–213. http://dx.doi.org/10.1070/sm2009v200n02abeh003991.

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Fialowski, Alice, and Michael Penkava. "The moduli space of complex 5-dimensional Lie algebras." Journal of Algebra 458 (July 2016): 422–44. http://dx.doi.org/10.1016/j.jalgebra.2016.03.029.

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Stachura, Piotr. "The κ-Poincaré group on a C∗-level." International Journal of Mathematics 30, no. 04 (April 2019): 1950022. http://dx.doi.org/10.1142/s0129167x19500228.

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The [Formula: see text]-algebraic [Formula: see text]-Poincaré group is constructed. The construction uses groupoid algebras of differential groupoids associated to Lie group decomposition. It turns out the underlying [Formula: see text]-algebra is the same as for “[Formula: see text]-Euclidean group” but a comultiplication is twisted by some unitary multiplier. Generators and commutation relations among them are presented.

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