Relevant bibliographies by topics / Automorphic Lie Algebras

Academic literature on the topic 'Automorphic Lie Algebras'

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Published: 10 December 2022
Last updated: 28 January 2023

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Journal articles on the topic "Automorphic Lie Algebras"

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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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Knibbeler, Vincent, Sara Lombardo, and Jan A. Sanders. "Higher-Dimensional Automorphic Lie Algebras." Foundations of Computational Mathematics 17, no. 4 (April 11, 2016): 987–1035. http://dx.doi.org/10.1007/s10208-016-9312-1.

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Borcherds, Richard. "Automorphic forms and Lie Algebras." Current Developments in Mathematics 1996, no. 1 (1996): 1–36. http://dx.doi.org/10.4310/cdm.1996.v1996.n1.a1.

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Karabanov, A. "Automorphic algebras of dynamical systems and generalised In¨on¨u-Wigner contractions." Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences, no. 5 (December 20, 2022): 5–14. http://dx.doi.org/10.19110/1994-5655-2022-5-5-14.

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Lie algebras a with a complex underlying vector space V are studied that are automorphic with respect to a given linear dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂ Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a Lie algebraic structure to the vector space of trajectories of the group Gt. The basics of the general structure of automorphic algebras a are described in terms of the eigenspace decomposition of the operatorM ∈ der(a) that determines the dynamics. Symmetries encoded by the presence of nonabelian automorphic algebras are pointed out connected to conservation laws, spectral relations and root systems. It is shown that, for a given dynamics Gt, automorphic algebras can be found via a limit transition in the space of Lie algebras on V along the trajectories of the group Gt itself. This procedure generalises the well-known Inönü-Wigner contraction and links adjoint representations of automorphic algebras to isospectral Lax representations on gl(V ). These results can be applied to physically important symmetry groups and their representations, including classical and relativistic mechanics, open quantum dynamics and nonlinear evolution equations. Simple examples are given.
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Mikhalev, Alexander A., and Jie-Tai Yu. "Test Elements, Retracts and Automorphic Orbits of Free Algebras." International Journal of Algebra and Computation 08, no. 03 (June 1998): 295–310. http://dx.doi.org/10.1142/s0218196798000144.

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A nonzero element a of an algebra A is called a test element if for any endomorphism φ of A it follows from φ(a)=a that φ is an automorphism of the algebra A. A subalgebra B of A is a retract if there is an ideal I of A such that A=B ⊕ I. We consider the main types of free algebras with the Nielsen–Schreier property: free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. For any free algebra F of finite rank of such type we prove that an element u is a test element if and only if it does not belong to any proper retract of F. Test elements for monomorphisms of F are exactly elements that are not contained in proper free factors of F. These results give analogs of Turner's results on test elements of free groups. We also characterize retracts of the algebra F. We prove that if some endomorphism φ preserve the automorphic orbit of some nonzero element of F, then φ is a monomorphism. For free Lie algebras and superalgebras over a field of characteristic zero and for free Lie p-(super)algebras over a field of prime characteristic p we show that in this situation φ is an automorphism. We discuss some related topics and formulate open problems.
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Knibbeler, V., S. Lombardo, and J. A Sanders. "Automorphic Lie algebras with dihedral symmetry." Journal of Physics A: Mathematical and Theoretical 47, no. 36 (August 21, 2014): 365201. http://dx.doi.org/10.1088/1751-8113/47/36/365201.

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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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GRITSENKO, VALERI A., and VIACHESLAV V. NIKULIN. "AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART I." International Journal of Mathematics 09, no. 02 (March 1998): 153–99. http://dx.doi.org/10.1142/s0129167x98000105.

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Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac–Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac–Moody algebras and formulate finiteness Conjecture for these forms. Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II.
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Lombardo, Sara, and Jan A. Sanders. "On the Classification of Automorphic Lie Algebras." Communications in Mathematical Physics 299, no. 3 (July 24, 2010): 793–824. http://dx.doi.org/10.1007/s00220-010-1092-x.

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Bury, Rhys T., and Alexander V. Mikhailov. "Automorphic Lie algebras and corresponding integrable systems." Differential Geometry and its Applications 74 (February 2021): 101710. http://dx.doi.org/10.1016/j.difgeo.2020.101710.

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Dissertations / Theses on the topic "Automorphic Lie Algebras"

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Knibbeler, Vincent. "Invariants of automorphic Lie algebras." Thesis, Northumbria University, 2015. http://nrl.northumbria.ac.uk/23590/.

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Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, denied by invariance under the action of a finite group, the reduction group. Since their introduction in 2005 [29, 31], mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic [4, 30]. In this thesis we set out to find the origin of these observations by searching for properties that are independent of the reduction group, called invariants of Automorphic Lie Algebras. The uniformity of Automorphic Lie Algebras with nonisomorphic reduction groups starts at the Riemann sphere containing the spectral parameter, restricting the finite groups to the polyhedral groups. Through the use of classical invariant theory and the properties of this class of groups it is shown that Automorphic Lie Algebras are freely generated modules over the polynomial ring in one variable. Moreover, the number of generators equals the dimension of the base Lie algebra, yielding an invariant. This allows the definition of the determinant of invariant vectors which will turn out to be another invariant. A surprisingly simple formula is given expressing this determinant as a monomial in ground forms. All invariants are used to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras signicantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity. In addition, the structure theory advances the classification project. For example, it clarifies the effect of a change in pole orbit resulting in various new Cartan-Weyl normal form generators for Automorphic Lie Algebras. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring.
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Chopp, Mikaël. "Lie-admissible structures on Witt type algebras and automorphic algebras." Thesis, Metz, 2011. http://www.theses.fr/2011METZ020S/document.

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L’algèbre de Witt a été intensivement étudiée. Elle est présente dans de nombreux domaines des Mathématiques. Cette thèse est l’étude de deux généralisations de l’algèbre de Witt: les algèbres de type Witt et les algèbres de Krichever-Novikov. Dans une première partie on s’intéresse aux structures Lie-admissibles sur les algèbres de type Witt. On donne toutes les structures troisième-puissance associatives et flexibles Lie-admissibles sur ces algèbres. De plus, on étudie les formes symplectiques qui induisent un produit symétrique gauche. Dans une seconde partie on étudie les algèbres automorphes. Partant d’une surface de Riemann compacte quelconque, on considère l’action d’un sous-groupe fini du groupe des automorphismes de la surface sur des algèbres d’origines géométriques comme les algèbres de Krichever-Novikov. Plus précisément nous faisons le lien entre la sous-algèbre des éléments invariants sur la surface et l’algèbre sur la surface quotient. La structure presque-gradue des algèbres de Krichever-Novikov induit une presque-graduation sur ces sous-algèbres de certaines algèbres de Krichever- Novikov
The Witt algebra has been intensively studied and arise in many research fields in Mathematics. We are interested in two generalizations of the Witt algebra: the Witt type algebras and the Krichever-Novikov algebras. In a first part we study the problem of finding Lie-admissible structures on Witt type algebras. We give all third-power associative Lie-admissible structures and flexible Lie-admissible structures on these algebras. Moreover we study the symplectic forms which induce a graded left-symmetric product. In the second part of the thesis we study the automorphic algebras. Starting from arbitrary compact Riemann surfaces we consider the action of finite subgroups of the automorphism group of the surface on certain geometrically defined Lie algebras as the Krichever-Novikov type algebras. More precisely, we relate for G a finite subgroup of automorphism acting on the Riemann surface, the invariance subalgebras living on the surface to the algebras on the quotient surface under the group action. The almost-graded Krichever-Novikov algebras structure on the quotient gives in this way a subalgebra of a certain Krichever-Novikov algebra (with almost-grading) on the original Riemann surface
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Lombardo, Sara. "Reductions of integrable equations and automorphic Lie algebras." Thesis, University of Leeds, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.417747.

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Bury, Rhys Thomas. "Automorphic lie algebras, corresponding integrable systems and their soliton solutions." Thesis, University of Leeds, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.539708.

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Persson, Daniel. "Arithmetic and hyperbolic structures in string theory." Doctoral thesis, Universite Libre de Bruxelles, 2009. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210323.

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This thesis consists of an introductory text followed by two separate parts which may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of spacelike singularities (the BKL-limit). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of gravity. We investigate the modification of the billiard dynamics when the original gravitational theory is formulated on a compact spatial manifold of arbitrary topology, revealing fascinating mathematical structures known as galleries. We further use the conjectured hyperbolic symmetry E10 to generate and classify certain cosmological (S-brane) solutions in eleven-dimensional supergravity. Finally, we show in detail that eleven-dimensional supergravity and massive type IIA supergravity are dynamically unified within the framework of a geodesic sigma model for a particle moving on the infinite-dimensional coset space E10/K(E10).

Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are typically given by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by moduli-dependent functions which are manifestly invariant under G(Z), known as automorphic forms. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on a special class of functions known as (non-holomorphic) Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also discuss the possibility that certain generalized Eisenstein series, which are covariant under the maximal compact subgroup K(G), could play a role in determining the exact effective action for toroidally compactified higher derivative corrections. Finally, we propose that in the case of rigid Calabi-Yau compactifications in type IIA string theory, the exact universal hypermultiplet moduli space exhibits a quantum duality group given by the emph{Picard modular group} SU(2,1;Z[i]). To verify this proposal we construct an SU(2,1;Z[i])-invariant Eisenstein series, and we present preliminary results for its Fourier expansion which reveals the expected contributions from D2-brane and NS5-brane instantons.

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Résumé francais:

Cette thèse est composée d'une introduction suivie de deux parties qui peuvent être lues indépendemment. Dans la première partie, nous analysons des structures hyperboliques apparaissant dans l'étude de la gravité au voisinage d'une singularité de type espace (la limite BKL). Dans cette limite, les points spatiaux se découplent et la dynamique suit un comportement ultralocal qui peut être reformulé en termes d'un billiard hyperbolique (qui peut être chaotique). Dans toutes les supergravités qui sont des limites de basse énergie de théories de cordes ou de la théorie M, la dynamique du billiard prend place à l'intérieur des chambres de Weyl fondamentales de certaines algèbres de Kac-Moody hyperboliques, ce qui suggère que ces algèbres correspondent à des symétries cachées de dimension infinie de la gravité. Nous examinons comment la dynamique du billard est modifiée quand la théorie de gravité originale est formulée sur une variété spatiale compacte de topologie arbitraire, révélant ainsi de fascinantes structures mathématiques appelées galleries. De plus, dans le cadre de la supergravité à onze dimensions, nous utilisons la symétrie hyperbolique conjecturée E10 pour engendrer et classifier certaines solutions cosmologiques (S-branes). Finalement, nous montrons en détail que la supergravité à onze dimensions et la supergravité de type IIA massive sont dynamiquement unifiées dans le contexte d'un modèle sigma géodesique pour une particule se déplaçant sur l'espace quotient de dimension infinie E10/K(E10).

La deuxième partie de cette thèse est consacrée à étudier comment les dualités U en théorie des cordes fournissent des contraintes puissantes sur les corrections quantiques perturbatives et non perturbatives. Ces dualités sont typiquement données par des groupes arithmétiques G(Z) dont il est conjecturé qu'ils préservent l'action effective. Les couplages exacts sont donnés par des fonctions des moduli qui sont manifestement invariantes sous G(Z), et qu'on appelle des formes automorphiques. Nous discutons en détail différentes méthodes de construction de ces formes automorphiques, en insistant particulièrement sur une classe spéciale de fonctions appelées séries d'Eisenstein (non holomorphiques). Nous présentons comme exemples les cas de SL(2,Z) et SL(3,Z), qui sont physiquement pertinents. Nous construisons les séries d'Eisenstein correspondantes et leurs expansions de Fourier (non abéliennes). Nous discutons également la possibilité que certaines séries d'Eisenstein généralisées, qui sont covariantes sous le sous-groupe compact maximal, pourraient jouer un rôle dans la détermination des actions effectives exactes pour les théories incluant des corrections de dérivées supérieures compactifiées sur des tores.


Doctorat en Sciences
info:eu-repo/semantics/nonPublished

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Yu, Jun. "Symmetric subgroups of automorphism groups of compact simple Lie algebras /." View abstract or full-text, 2009. http://library.ust.hk/cgi/db/thesis.pl?MATH%202009%20YU.

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Hindeleh, Firas Y. "Tangent and Cotangent Bundles, Automorphism Groups and Representations of Lie Groups." University of Toledo / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1153933389.

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Gray, Jonathan Nathan. "On the homology of automorphism groups of free groups." 2011. http://trace.tennessee.edu/utk_graddiss/974.

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Following the work of Conant and Vogtmann on determining the homology of the group of outer automorphisms of a free group, a new nontrivial class in the rational homology of Outer space is established for the free group of rank eight. The methods started in [8] are heavily exploited and used to create a new graph complex called the space of good chord diagrams. This complex carries with it significant computational advantages in determining possible nontrivial homology classes.Next, we create a basepointed version of the Lie operad and explore some of it proper- ties. In particular, we prove a Kontsevich-type theorem that relates the Lie homology of a particular space to the cohomology of the group of automorphisms of the free group.
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Roeseler, Karsten. "Oktaven und Reduktionstheorie." Doctoral thesis, 2011. http://hdl.handle.net/11858/00-1735-0000-0006-B3F4-C.

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Books on the topic "Automorphic Lie Algebras"

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1938-, Griffiths Phillip, and Kerr Matthew D. 1975-, eds. Hodge theory, complex geometry, and representation theory. Providence, Rhode Island: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 2013.

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Flicker, Yuval Z. Drinfeld Moduli Schemes and Automorphic Forms: The Theory of Elliptic Modules with Applications. New York, NY: Springer New York, 2013.

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Alladi, Krishnaswami, Frank Garvan, and Ae Ja Yee. Ramanujan 125: International conference to commemorate the 125th anniversary of Ramanujan's birth, Ramanujan 125, November 5--7, 2012, University of Florida, Gainesville, Florida. Providence, Rhode Island: American Mathematical Society, 2014.

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1943-, Labesse J. P., and Schwermer Joachim, eds. Cohomology of arithmetic groups and automorphic forms: Proceedings of a conference held in Luminy/Marseille, France, May 22-27, 1989. Berlin: Springer-Verlag, 1990.

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Trends in number theory: Fifth Spanish meeting on number theory, July 8-12, 2013, Universidad de Sevilla, Sevilla, Spain. Providence, Rhode Island: American Mathematical Society, 2015.

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Shahidi, Freydoon, Dihua Jiang, David Soudry, and James W. Cogdell. Advances in the theory of automorphic forms and their L-functions: Workshop in honor of James Cogdell's 60th birthday, October 16-25, 2013, Erwin Schrodinger Institute, University of Vienna, Vienna, Austria. Providence, Rhode Island: American Mathematical Society, 2016.

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Cogdell, James W., 1953- editor, Shahidi Freydoon editor, and Soudry David 1956 editor, eds. Automorphic forms and related geometry: Assessing the legecy of I.I. Piatetski-Shapiro : April 23-27, 2012, Yale University, New Haven, CT. Providence, Rhode Island: American Mathematical Society, 2014.

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D, Goldfeld, ed. Collected works of Hervé Jacquet. Providence, R.I: American Mathematical Society, 2011.

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On certain L-functions: Conference in honor of Freydoon Shahidi on certain L-functions, Purdue Univrsity, West Lafayette, Indiana, July 23-27, 2007. Providence, R.I: American Mathematical Society, 2011.

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1937-, Doran Robert S., Sally Paul, and Spice Loren 1981-, eds. Harmonic analysis on reductive, p-adic groups: AMS Special Session on Harmonic Analysis and Representations of Reductive, p-adic Groups, January 16, 2010, San Francisco, CA. Providence, R.I: American Mathematical Society, 2011.

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Book chapters on the topic "Automorphic Lie Algebras"

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Scheithauer, Nils R. "Lie Algebras, Vertex Algebras, and Automorphic Forms." In Developments and Trends in Infinite-Dimensional Lie Theory, 151–68. Boston: Birkhäuser Boston, 2010. http://dx.doi.org/10.1007/978-0-8176-4741-4_5.

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Komori, Yasushi, Kohji Matsumoto, and Hirofumi Tsumura. "On Witten Multiple Zeta-Functions Associated with Semisimple Lie Algebras III." In Multiple Dirichlet Series, L-functions and Automorphic Forms, 223–86. Boston, MA: Birkhäuser Boston, 2012. http://dx.doi.org/10.1007/978-0-8176-8334-4_11.

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Tanaka, Makiko Sumi, and Hiroyuki Tasaki. "Maximal Antipodal Subgroups of the Automorphism Groups of Compact Lie Algebras." In Hermitian–Grassmannian Submanifolds, 39–47. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-5556-0_4.

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Naito, Satoshi, and Daisuke Sagaki. "Crystal Base Elements of an ExtremalWeight Module Fixed by a Diagram Automorphism II: Case of Affine Lie Algebras." In Representation Theory of Algebraic Groups and Quantum Groups, 225–55. Boston: Birkhäuser Boston, 2010. http://dx.doi.org/10.1007/978-0-8176-4697-4_9.

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WILLENBRING, JEB F., and GREGG J. ZUCKERMAN. "SMALL SEMISIMPLE SUBALGEBRAS OF SEMISIMPLE LIE ALGEBRAS." In Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory, 403–34. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812770790_0011.

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OSHIMA, TOSHIO. "ANNIHILATORS OF GENERALIZED VERMA MODULES OF THE SCALAR TYPE FOR CLASSICAL LIE ALGEBRAS." In Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory, 277–319. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812770790_0009.

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"Lie groups and linear algebraic groups I. Complex and real groups." In Lie Groups and Automorphic Forms, 1–49. Providence, Rhode Island: American Mathematical Society, 2006. http://dx.doi.org/10.1090/amsip/037/01.

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"Invariant lattices in Lie algebras and their automorphism groups." In Group Theory, 171–82. De Gruyter, 1989. http://dx.doi.org/10.1515/9783110848397-012.

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Griess,, Robert L. "A vertex operator algebra related to E8 with automorphism group O+( 10,2)." In The Monster and Lie Algebras, edited by Joseph Ferrar and Koichiro Harada. Berlin, New York: DE GRUYTER, 1998. http://dx.doi.org/10.1515/9783110801897.43.

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Conference papers on the topic "Automorphic Lie Algebras"

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LOMBARDO, S., and A. V. MIKHAILOV. "REDUCTIONS OF INTEGRABLE EQUATIONS AND AUTOMORPHIC LIE ALGEBRAS." In Proceedings of the International Conference on SPT 2004. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702142_0022.

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Choi, Seul Hee, Xueqing Chen, and Ki-Bong Nam. "Automorphism Groups of Some Stable Lie Algebras with Exponential Functions I." In The International Conference on Algebra 2010 - Advances in Algebraic Structures. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814366311_0008.

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Demin Yu and Yinghui Zhang. "The automorphism and simple quality of the generalized Virasoro-like Lie algebra." In 2011 International Conference on Multimedia Technology (ICMT). IEEE, 2011. http://dx.doi.org/10.1109/icmt.2011.6002408.

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