Journal articles: 'Lie Algebras Expansion'

1

Rowe, D. J., and J. Carvalho. "Boson expansion of lie algebras: The fermion pair algebra." Physics Letters B 175, no. 3 (August 1986): 243–48. http://dx.doi.org/10.1016/0370-2693(86)90848-8.

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Caroca, R., N. Merino, and P. Salgado. "S expansion of higher-order Lie algebras." Journal of Mathematical Physics 50, no. 1 (January 2009): 013503. http://dx.doi.org/10.1063/1.3036177.

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Curry, Charles, Kurusch Ebrahimi-Fard, and Brynjulf Owren. "The Magnus expansion and post-Lie algebras." Mathematics of Computation 89, no. 326 (May 26, 2020): 2785–99. http://dx.doi.org/10.1090/mcom/3541.

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4

Nieto, L. M., J. Negro, and M. Santander. "Two Dimensional Cayley-Klein Algebras Generated by Expansions." International Journal of Modern Physics A 12, no. 01 (January 10, 1997): 259–64. http://dx.doi.org/10.1142/s0217751x97000384.

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Expansion methods are applied in order to get extended two dimensional Cayley-Klein Lie algebras. The keystone of the construction will be the two dimensional extended Galilei algebra whose generators are used to define, by means of formal functions, generators closing any of the nine two-dimensional extended Cayley-Klein Lie algebras.

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Yan, Wang, and Yufeng Zhang. "Expansion of the Lie algebras and integrable couplings." Chaos, Solitons & Fractals 38, no. 2 (October 2008): 541–47. http://dx.doi.org/10.1016/j.chaos.2006.12.002.

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Andrianopoli, L., N. Merino, F. Nadal, and M. Trigiante. "General properties of the expansion methods of Lie algebras." Journal of Physics A: Mathematical and Theoretical 46, no. 36 (August 21, 2013): 365204. http://dx.doi.org/10.1088/1751-8113/46/36/365204.

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Ebrahimi-Fard, Kurusch, and Dominique Manchon. "Twisted dendriform algebras and the pre-Lie Magnus expansion." Journal of Pure and Applied Algebra 215, no. 11 (November 2011): 2615–27. http://dx.doi.org/10.1016/j.jpaa.2011.03.004.

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Mencattini, Igor, and Alexandre Quesney. "Crossed morphisms, integration of post-Lie algebras and the post-Lie Magnus expansion." Communications in Algebra 49, no. 8 (March 28, 2021): 3507–33. http://dx.doi.org/10.1080/00927872.2021.1900212.

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9

Pei, Yufeng, and Jinwei Yang. "Strongly graded vertex algebras generated by vertex Lie algebras." Communications in Contemporary Mathematics 21, no. 08 (October 20, 2019): 1850069. http://dx.doi.org/10.1142/s0219199718500694.

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We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

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Quesne, C. "Boson realisations of Lie algebras and expansion of shift operators." Journal of Physics A: Mathematical and General 20, no. 12 (August 21, 1987): L753—L758. http://dx.doi.org/10.1088/0305-4470/20/12/001.

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HANSEN, SØREN KOLD, and TOSHIE TAKATA. "RESHETIKHIN–TURAEV INVARIANTS OF SEIFERT 3-MANIFOLDS FOR CLASSICAL SIMPLE LIE ALGEBRAS." Journal of Knot Theory and Its Ramifications 13, no. 05 (August 2004): 617–68. http://dx.doi.org/10.1142/s0218216504003342.

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We derive explicit formulas for the Reshetikhin–Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra [Formula: see text] in terms of the Seifert invariants and standard data for [Formula: see text]. A main corollary is a determination of the full asymptotic expansions of these invariants for lens spaces in the limit of large quantum level. This result is in agreement with the asymptotic expansion conjecture due to Andersen [1,2].

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Khalique, Chaudry Masood, and Isaiah Elvis Mhlanga. "Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation." Applied Mathematics and Nonlinear Sciences 3, no. 1 (June 5, 2018): 241–54. http://dx.doi.org/10.21042/amns.2018.1.00018.

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AbstractIn this paper we study a (2+1)-dimensional coupling system with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. Its Lax-pair and bi-Hamiltonian formulation were obtained and presented in the literature. We utilize Lie symmetry analysis along with the (G′/G)–expansion method to obtain travelling wave solutions of this system. Furthermore, conservation laws are constructed using the multiplier method.

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James Gates, Jr, S., W. D. Linch, III, Joseph Phillips, and V. G. J. Rodgers. "Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras." Communications in Mathematical Physics 246, no. 2 (April 1, 2004): 333–58. http://dx.doi.org/10.1007/s00220-004-1048-0.

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14

EGUCHI, TOHRU, YASUHIKO YAMADA, and SUNG-KIL YANG. "ON THE GENUS EXPANSION IN THE TOPOLOGICAL STRING THEORY." Reviews in Mathematical Physics 07, no. 03 (April 1995): 279–309. http://dx.doi.org/10.1142/s0129055x95000141.

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A systematic formulation of the higher genus expansion in topological string theory is considered. We also develop a simple way of evaluating genus zero correlation functions. At higher genera we derive some interesting formulas for the free energy in the A1 and A2 models. We present some evidence that topological minimal models associated with Lie algebras other than the A-D-E type do not have a consistent higher genus expansion beyond genus one. We also present some new results on the CP1 model at higher genera.

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Truini, Piero, Alessio Marrani, Michael Rios, and Klee Irwin. "Space, Matter and Interactions in a Quantum Early Universe Part I: Kac–Moody and Borcherds Algebras." Symmetry 13, no. 12 (December 6, 2021): 2342. http://dx.doi.org/10.3390/sym13122342.

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We introduce a quantum model for the universe at its early stages, formulating a mechanism for the expansion of space and matter from a quantum initial condition, with particle interactions and creation driven by algebraic extensions of the Kac–Moody Lie algebra e9. We investigate Kac–Moody and Borcherds algebras, and we propose a generalization that meets further requirements that we regard as fundamental in quantum gravity.

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Yuasa, Wataru. "The 𝔰𝔩3 colored Jones polynomials for 2-bridge links." Journal of Knot Theory and Its Ramifications 26, no. 07 (April 17, 2017): 1750038. http://dx.doi.org/10.1142/s0218216517500389.

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Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk. We derive some formulas for [Formula: see text] and [Formula: see text] clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the [Formula: see text] and [Formula: see text] colored Jones polynomials of [Formula: see text]-bridge knots and links explicitly using twist formulas.

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CARIÑENA, JOSÉ F., KURUSCH EBRAHIMI-FARD, HÉCTOR FIGUEROA, and JOSÉ M. GRACIA-BOND. "HOPF ALGEBRAS IN DYNAMICAL SYSTEMS THEORY." International Journal of Geometric Methods in Modern Physics 04, no. 04 (June 2007): 577–646. http://dx.doi.org/10.1142/s0219887807002211.

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The theory of exact and of approximate solutions for non-autonomous linear differential equations forms a wide field with strong ties to physics and applied problems. This paper is meant as a stepping stone for an exploration of this long-established theme, through the tinted glasses of a (Hopf and Rota–Baxter) algebraic point of view. By reviewing, reformulating and strengthening known results, we give evidence for the claim that the use of Hopf algebra allows for a refined analysis of differential equations. We revisit the renowned Campbell–Baker–Hausdorff–Dynkin formula by the modern approach involving Lie idempotents. Approximate solutions to differential equations involve, on the one hand, series of iterated integrals solving the corresponding integral equations; on the other hand, exponential solutions. Equating those solutions yields identities among products of iterated Riemann integrals. Now, the Riemann integral satisfies the integration-by-parts rule with the Leibniz rule for derivations as its partner; and skewderivations generalize derivations. Thus, we seek an algebraic theory of integration, with the Rota–Baxter relation replacing the classical rule. The methods to deal with noncommutativity are especially highlighted. We find new identities, allowing for an extensive embedding of Dyson–Chen series of time- or path-ordered products (of generalized integration operators); of the corresponding Magnus expansion; and of their relations, into the unified algebraic setting of Rota–Baxter maps and their inverse skewderivations. This picture clarifies the approximate solutions to generalized integral equations corresponding to non-autonomous linear (skew) differential equations.

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Ilash, N. B. "Poincare series for the algebras of joint invariants and covariants of $n$ quadratic forms." Carpathian Mathematical Publications 9, no. 1 (June 7, 2017): 57–62. http://dx.doi.org/10.15330/cmp.9.1.57-62.

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We consider one of the fundamental problems of classical invariant theory - the research of Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincare series at the point $z = 1$ give us important information about the structure of the algebra $\mathcal{I}_{d}.$ It was derived by Hilbert for the algebra ${\mathcal{I}_{d}=\mathbb{C}[V_d]^{\,SL_2}}$ of invariants for binary $d-$form (by $V_d$ denote the vector space over $\mathbb{C}$ consisting of all binary forms homogeneous of degree $d$). Springer got this result, using explicit formula for the Poincare series of this algebra. We consider this problem for the algebra of joint invariants $\mathcal{I}_{2n}=\mathbb{C}[\underbrace{V_2 \oplus V_2 \oplus \cdots \oplus V_2}_{\text{n times}}]^{SL_2}$ and the algebra of joint covariants $\mathcal{C}_{2n}=\mathbb{C}[\underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{n times}}{\oplus}\mathbb{C}^2 ]^{SL_2}$ of $n$ quadratic forms. We express the Poincare series $\mathcal{P}(\mathcal{C}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{C}_{2n})_{j}\, z^j$ and $\mathcal{P}(\mathcal{I}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{I}_{2n})_{j}\, z^j$ of these algebras in terms of Narayana polynomials. Also, for these algebras we calculate the degrees and asymptotic behavious of the degrees, using their Poincare series.

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Maslikov, Alexander, and Guennady Volkov. "Ternary SU(3)-group symmetry and its possible applications in hadron-quark substructure. Towards a new spinor-fermion structure." EPJ Web of Conferences 204 (2019): 02007. http://dx.doi.org/10.1051/epjconf/201920402007.

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The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.

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Hidalgo, Rubén A., Irina Markina, and Alexander Vasil'ev. "Finite Dimensional Grading of the Virasoro Algebra." gmj 14, no. 3 (September 2007): 419–34. http://dx.doi.org/10.1515/gmj.2007.419.

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Abstract The Virasoro algebra is a central extension of the Witt algebra, the complexified Lie algebra of the sense preserving diffeomorphism group of the circle Diff 𝑆1. It appears in Quantum Field Theories as an infinite dimensional algebra generated by the coefficients of the Laurent expansion of the analytic component of the momentum-energy tensor, Virasoro generators. The background for the construction of the theory of unitary representations of Diff 𝑆1 is found in the study of Kirillov's manifold Diff 𝑆1=𝑆1. It possesses a natural Kählerian embedding into the universal Teichmüller space with the projection into the moduli space realized as an infinite-dimensional body of the coefficients of univalent quasiconformally extendable functions. The differential of this embedding leads to an analytic representation of the Virasoro algebra based on Kirillov's operators. In this paper we overview several interesting connections between the Virasoro algebra, Teichmüller theory, Löwner representation of univalent functions, and propose a finite-dimensional grading of the Virasoro algebra such that the grades form a hierarchy of finite dimensional algebras which, in their turn, are the first integrals of Liouville partially integrable systems for coefficients of univalent functions.

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Javier Rosales, G. "Examples of S—expansions of Lie Algebras." Journal of Physics: Conference Series 2090, no. 1 (November 1, 2021): 012067. http://dx.doi.org/10.1088/1742-6596/2090/1/012067.

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Abstract In this note, we give examples of S—expansions of Lie algebras of finite and infinite dimension. For the finite dimensional case, we expand all real three-dimensional Lie algebras. In the case of infinite dimension, we perform contractions obtaining new Lie algebras of infinite dimension.

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Nikonorova, Renata, Dilara Siraeva, and Yulia Yulmukhametova. "New Exact Solutions with a Linear Velocity Field for the Gas Dynamics Equations for Two Types of State Equations." Mathematics 10, no. 1 (January 1, 2022): 123. http://dx.doi.org/10.3390/math10010123.

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In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted by the system to 12 and 14 parameters, respectively. Invariant submodels of rank one are constructed from two three-dimensional subalgebras of the corresponding Lie algebras, and exact solutions with a linear velocity field with inhomogeneous deformation are obtained. On the one hand of the special state equation, the submodel describes an isochoric vortex motion of particles, isobaric along each world line and restricted by a moving plane. The motions of particles occur along parabolas and along rays in parallel planes. The spherical volume of particles turns into an ellipsoid at finite moments of time, and as time tends to infinity, the particles end up on an infinite strip of finite width. On the other hand of the state equation of a monatomic gas, the submodel describes vortex compaction to the origin and the subsequent expansion of gas particles in half-spaces. The motion of any allocated volume of gas retains a spherical shape. It is shown that for any positive moment of time, it is possible to choose the radius of a spherical volume such that the characteristic conoid beginning from its center never reaches particles outside this volume. As a result of the generalization of the solutions with a linear velocity field, exact solutions of a wider class are obtained without conditions of invariance of density and pressure with respect to the selected three-dimensional subalgebras.

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de Azcárraga, J. A., and J. M. Izquierdo. "-Poincaré supergravities from Lie algebra expansions." Nuclear Physics B 854, no. 1 (January 2012): 276–91. http://dx.doi.org/10.1016/j.nuclphysb.2011.08.020.

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Yong, Yang, and Zhao Yan. "Expansion of Lie Algebra and Its Application." Communications in Theoretical Physics 47, no. 1 (January 2007): 19–21. http://dx.doi.org/10.1088/0253-6102/47/1/004.

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25

Malham, Simon J. A., and Anke Wiese. "Stochastic expansions and Hopf algebras." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2112 (September 21, 2009): 3729–49. http://dx.doi.org/10.1098/rspa.2009.0203.

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We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell–Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that, at low orders, this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However, it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here, we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof uses the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies.

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26

Guo, Fukui, and Yufeng Zhang. "Expansion of the Lie algebra and its applications." Chaos, Solitons & Fractals 27, no. 4 (February 2006): 1048–55. http://dx.doi.org/10.1016/j.chaos.2005.04.073.

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27

TODOROV, IVAN. "INFINITE DIMENSIONAL LIE ALGEBRAS IN 4D CONFORMAL FIELD THEORY." International Journal of Geometric Methods in Modern Physics 05, no. 08 (December 2008): 1361–71. http://dx.doi.org/10.1142/s0219887808003387.

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It is known that there are no scalar Lie fields in more than two space-time dimensions [4]. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. Recent work, [2, 3], is reviewed, in which we classify such algebras and their unitary positive energy representations in a theory of a system of scalar fields of dimension two. The results are linked to the Doplicher–Haag–Roberts theory of superselection sectors governed by a (global) compact gauge group.

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Todorov, Ivan. "Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models." Advances in Mathematical Physics 2010 (2010): 1–12. http://dx.doi.org/10.1155/2010/509538.

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It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.

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Azcárraga, J. A. de, J. M. Izquierdo, M. Picón, and O. Varela. "Extensions, expansions, Lie algebra cohomology and enlarged superspaces." Classical and Quantum Gravity 21, no. 10 (April 17, 2004): S1375—S1384. http://dx.doi.org/10.1088/0264-9381/21/10/010.

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Artebani, M., R. Caroca, M. C. Ipinza, D. M. Peñafiel, and P. Salgado. "Geometrical aspects of the Lie algebra S-expansion procedure." Journal of Mathematical Physics 57, no. 2 (February 2016): 023516. http://dx.doi.org/10.1063/1.4941135.

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Rowe, D. J., R. Le Blanc, and J. Repka. "A rotor expansion of the su(3) Lie algebra." Journal of Physics A: Mathematical and General 22, no. 8 (April 21, 1989): L309—L316. http://dx.doi.org/10.1088/0305-4470/22/8/001.

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Izaurieta, F., A. Perez, E. Rodriguez, and P. Salgado. "Dual formulation of the Lie algebra S-expansion procedure." Journal of Mathematical Physics 50, no. 7 (July 2009): 073511. http://dx.doi.org/10.1063/1.3171923.

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ROWE, D. J. "BOSON AND ROTOR EXPANSIONS OF LIE ALGEBRAS IN VECTOR COHERENT STATE THEORY." International Journal of Modern Physics E 02, supp01 (January 1993): 119–35. http://dx.doi.org/10.1142/s0218301393000510.

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A brief overview is given of some of the ways VCS theory can be used to generate boson and rotor expansions of Lie groups. It is demonstrated by examples that such representations are a powerful aid in computing the explicit matrices of the irreducible representations needed in the application of Lie groups and Lie algebras in physics. It is shown that VCS theory is a theory of induced representations and that it has some advantages over other inducing constructions. Boson and rotor expansions are applied to the microscopic theory of nuclear rotations and it is shown that, in addition to providing algorithms for the calculation of the representation matrices needed, these expansions also provide new perspectives on the theory which enable it to be extended to include intrinsic nucleon spin degrees of freedom and the adiabatic mixing of representations.

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WANG, XIAO-GUANG. "COHERENT STATES, DISPLACED NUMBER STATES AND LAGUERRE POLYNOMIAL STATES FOR su(1, 1) LIE ALGEBRA." International Journal of Modern Physics B 14, no. 10 (April 20, 2000): 1093–103. http://dx.doi.org/10.1142/s0217979200001084.

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The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1, 1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discussed.

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GE, MO-LIN, YU-QUAN LI, and KANG XUE. "EXTENDED STATE EXPANSIONS AND THE UNIVERSALITY OF WITTEN’S VERSION OF LINK POLYNOMIAL THEORY." International Journal of Modern Physics A 05, no. 10 (May 20, 1990): 1975–2003. http://dx.doi.org/10.1142/s0217751x90000921.

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The Witten’s version for constructing the skein relations of link polynomials based on (2+1) Chern-Simons Lagrangian is shown to be universal for Lie algebras. The extended state calculations are developed to give the explicit representations of braid group and new understanding of framing factor from the point of view of generalized Markov trace.

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de Azcárraga, J. A., D. Gútiez, and J. M. Izquierdo. "Extended D = 3 Bargmann supergravity from a Lie algebra expansion." Nuclear Physics B 946 (September 2019): 114706. http://dx.doi.org/10.1016/j.nuclphysb.2019.114706.

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Izmest'ev, A. A., G. S. Pogosyan, A. N. Sissakian, and P. Winternitz. "Contractions of Lie algebras and the separation of variables: interbase expansions." Journal of Physics A: Mathematical and General 34, no. 3 (January 12, 2001): 521–54. http://dx.doi.org/10.1088/0305-4470/34/3/314.

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Inostroza, Carlos, Igor Kondrashuk, Nelson Merino, and Felip Nadal. "On a Java library to perform S-expansions of Lie algebras." Journal of Physics: Conference Series 1085 (September 2018): 052010. http://dx.doi.org/10.1088/1742-6596/1085/5/052010.

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Gobet, Thomas. "Noncrossing partitions, fully commutative elements and bases of the Temperley–Lieb algebra." Journal of Knot Theory and Its Ramifications 25, no. 06 (May 2016): 1650035. http://dx.doi.org/10.1142/s0218216516500358.

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We introduce a new basis of the Temperley–Lieb algebra. It is defined using a bijection between noncrossing partitions and fully commutative elements together with a basis introduced by Zinno, which is obtained by mapping the simple elements of the Birman–Ko–Lee braid monoid to the Temperley–Lieb algebra. The combinatorics of the new basis involve the Bruhat order restricted to noncrossing partitions. As an application we can derive properties of the coefficients of the base change matrix between Zinno’s basis and the well-known diagram or Kazhdan–Lusztig basis of the Temperley–Lieb algebra. In particular, we give closed formulas for some of the coefficients of the expansion of an element of the diagram basis in the Zinno basis.

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BRINGMANN, KATHRIN, and KARL MAHLBURG. "Asymptotic formulas for coefficients of Kac–Wakimoto Characters." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (February 22, 2013): 51–72. http://dx.doi.org/10.1017/s0305004112000680.

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AbstractWe study the coefficients of Kac and Wakimoto's character formulas for the affine Lie superalgebrassℓ(n+1|1)∧. The coefficients of these characters are the weight multiplicities of irreducible modules of the Lie superalgebras, and their asymptotic study begins with Kac and Peterson's earlier use of modular forms to understand the coefficients of characters for affine Lie algebras. In the affine Lie superalgebra setting, the characters are products of weakly holomorphic modular forms and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. Using our previously developed extension of the Circle Method for products of mock modular forms along with the Saddle Point Method, we find asymptotic series expansions for the coefficients of the characters with polynomial error.

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Shojaei-Fard, Ali. "Application of Deformed Lie Algebras to Non-Perturbative Quantum Field Theory." Journal of the Indian Mathematical Society 84, no. 1-2 (January 2, 2017): 109. http://dx.doi.org/10.18311/jims/2017/5839.

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The manuscript implements Connes-Kreimer Hopf algebraic renormalization of Feynman diagrams and Dubois-Violette type noncommutative differential geometry to discover a new class of differential calculi with respect to infinite formal expansions of Feynman diagrams which are generated by Dyson-Schwinger equations.

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Romano, Luca. "Non-relativistic four dimensional p-brane supersymmetric theories and Lie algebra expansion." Classical and Quantum Gravity 37, no. 14 (July 6, 2020): 145016. http://dx.doi.org/10.1088/1361-6382/ab8bbc.

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Edelstein, José D., Mokhtar Hassaïne, Ricardo Troncoso, and Jorge Zanelli. "Lie-algebra expansions, Chern–Simons theories and the Einstein–Hilbert Lagrangian." Physics Letters B 640, no. 5-6 (September 2006): 278–84. http://dx.doi.org/10.1016/j.physletb.2006.07.058.

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Halverson, Tom, Manuela Mazzocco, and Arun Ram. "Commuting Families in Hecke and Temperley-Lieb Algebras." Nagoya Mathematical Journal 195 (2009): 125–52. http://dx.doi.org/10.1017/s0027763000009740.

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AbstractWe define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group . We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.

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Grebenev, V. N., A. N. Grishkov, and M. Oberlack. "The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence." Advances in Mathematical Physics 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/469654.

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The extended symmetry of the functional of length determined in an affine spaceK3of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variablet) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metricsdl2(t)(Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation spaceK3and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic formdlD22(t)generated bydl2(t)which is similar to the Lie algebra constructed by Grebenev et al. (2012). Then, using the properties of this Lie algebra, we show that there exists a nontrivial central extension wherein the central charge is defined by the same bilinear skew-symmetric formcas for the Witt algebra which measures the number of internal degrees of freedom of the system. For the applications in turbulence, as the main result, we establish the asymptotic expansion of the transversal correlation function for large correlation distances in the frame ofdlD22(t).

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Hudson, RL, and S. Pulmannová. "Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus." Proceedings of the London Mathematical Society 77, no. 2 (September 1998): 462–80. http://dx.doi.org/10.1112/s0024611598000537.

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Halpern, M. B., and N. A. Obers. "New Semiclassical Non-Abelian Vertex Operators for Chiral and Nonchiral WZW Theory." International Journal of Modern Physics A 12, no. 24 (September 30, 1997): 4317–55. http://dx.doi.org/10.1142/s0217751x97002358.

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We supplement the discussion of Moore and Reshetikhin and others by finding new semiclassical non-Abelian vertex operators for the chiral, antichiral and nonchiral primary fields of WZW theory. These new non-Abelian vertex operators are the natural generalization of the familiar Abelian vertex operators: they involve only the representation matrices of Lie g, the currents of affine (g × g) and certain chiral and antichiral zero modes, and they reduce to the Abelian vertex operators in the limit of Abelian algebras. Using the new constructions, we also discuss semiclassical operator product expansions, braid relations and relations to the known form of the semiclassical affine-Sugawara conformal blocks.

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Ikeda, Takeshi, Hiroshi Mizukawa, Tatsuhiro Nakajima, and Hiro-Fumi Yamada. "Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra A1(1)." Advances in Applied Mathematics 40, no. 4 (May 2008): 514–35. http://dx.doi.org/10.1016/j.aam.2007.05.003.

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Fiedler, Bernold. "Reversibility, Continued Fractions, and Infinite Meander Permutations of Planar Homoclinic Orbits in Linear Hyperbolic Anosov Maps." International Journal of Bifurcation and Chaos 24, no. 08 (August 2014): 1440008. http://dx.doi.org/10.1142/s0218127414400082.

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Meander permutations have been encountered in the context of Gauss words, singularity theory, Sturm global attractors, plane Cartesian billiards, and Temperley–Lieb algebras, among others. In this spirit, we attempt to investigate the difference of orderings of homoclinic orbits on the stable and unstable manifolds of a planar saddle. As an example, we consider reversible linear Anosov maps on the 2-torus, and their relation to continued fraction expansions.

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OZAVSAR, MUTTALIP, and GURSEL YESILOT. "DIFFERENTIAL CALCULUS ON THE LOGARITHMIC EXTENSION OF THE QUANTUM 3D SPACE AND WEYL ALGEBRA." International Journal of Geometric Methods in Modern Physics 08, no. 08 (December 2011): 1667–78. http://dx.doi.org/10.1142/s0219887811005877.

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Noncommutative derivative operators acting on the quantum 3D space in the sense of Manin are introduced. Furthermore, the quantum 3D space is extended by the series expansion of the logarithm of the grouplike generator in the quantum 3D space. We give its differential calculus and the corresponding Weyl algebra. We also obtain algebra of Cartan–Maurer forms on this extension and the corresponding Lie algebra of vector fields. All noncommutative results are found to reduce to those of the standard commutative algebra when the deformation parameter of the quantum 3D space is set to one.

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Journal articles on the topic 'Lie Algebras Expansion'

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