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Relevant bibliographies by topics / Polycyclic groups][Algebraic groups

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Academic literature on the topic 'Polycyclic groups][Algebraic groups'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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Journal articles on the topic "Polycyclic groups][Algebraic groups"

1

Sautoy, Marcus Du. "Polycyclic Groups, Analytic Groups and Algebraic Groups." Proceedings of the London Mathematical Society 85, no. 1 (July 2002): 62–92. http://dx.doi.org/10.1112/plms/85.1.62.

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MEYEROVITCH, TOM. "Pseudo-orbit tracing and algebraic actions of countable amenable groups." Ergodic Theory and Dynamical Systems 39, no. 9 (January 24, 2018): 2570–91. http://dx.doi.org/10.1017/etds.2017.126.

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Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

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

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

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Assmann, Björn, and Bettina Eick. "Computing polycyclic presentations for polycyclic rational matrix groups." Journal of Symbolic Computation 40, no. 6 (December 2005): 1269–84. http://dx.doi.org/10.1016/j.jsc.2005.05.003.

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Sinanan, S. K., and D. F. Holt. "Algorithms for polycyclic-by-finite groups." Journal of Symbolic Computation 79 (March 2017): 269–84. http://dx.doi.org/10.1016/j.jsc.2016.02.008.

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Nikolaev, Andrey, and Alexander Ushakov. "Subset sum problem in polycyclic groups." Journal of Symbolic Computation 84 (January 2018): 84–94. http://dx.doi.org/10.1016/j.jsc.2017.03.007.

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Franciosi, Silvana, Francesco Giovanni, and Martin L. Newell. "Groups with Polycyclic Non-Normal Subgroups." Algebra Colloquium 7, no. 1 (March 2000): 33–42. http://dx.doi.org/10.1007/s10011-000-0033-1.

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Ostheimer, Gretchen. "Practical Algorithms for Polycyclic Matrix Groups." Journal of Symbolic Computation 28, no. 3 (September 1999): 361–79. http://dx.doi.org/10.1006/jsco.1999.0287.

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Asri, M. S. M., K. B. Wong, and P. C. Wong. "Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups." Algebra Colloquium 28, no. 01 (January 20, 2021): 119–30. http://dx.doi.org/10.1142/s1005386721000110.

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We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form [Formula: see text], where [Formula: see text] has infinite order and [Formula: see text] is finite.

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Garreta, Albert, Alexei Miasnikov, and Denis Ovchinnikov. "Diophantine problems in solvable groups." Bulletin of Mathematical Sciences 10, no. 01 (February 21, 2020): 2050005. http://dx.doi.org/10.1142/s1664360720500058.

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We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups [Formula: see text]. Then, we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups [Formula: see text].

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Dissertations / Theses on the topic "Polycyclic groups][Algebraic groups"

1

Du, Sautoy M. P. F. "Discrete groups, analytic groups and Poincare series." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236109.

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Assmann, Björn. "Applications of Lie methods to computations with polycyclic groups." Thesis, St Andrews, 2007. http://hdl.handle.net/10023/435.

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Silva, Jefferson dos Santos e. "Uma apresentação policíclica para o multiplicador de Schur e o quadrado tensorial não abeliano de um grupo policíclico." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/4539.

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Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq
In this work, based on [9], describes an effective method for computing a consistent polycyclic presentation for the nonanbeian tensor square G G of a group G given by a consistent polycyclic presentation.
Este trabalho, baseado em [9], determina um efetivo método para calcular uma apresentação policíclica consistente para o quadrado tensorial não abeliano G G de um grupo G dado por uma apresentação policíclica consistente.

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Garibaldi, Skip. "Trialitarian algebraic groups /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1998. http://wwwlib.umi.com/cr/ucsd/fullcit?p9906492.

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Meyer, Aurel Nathan. "Essential dimension of algebraic groups." Thesis, University of British Columbia, 2010. http://hdl.handle.net/2429/27091.

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We study the essential dimension of linear algebraic groups. For a group G, essential dimension is a measure for the complexity of G-torsors or, more generally, the complexity of any algebraic or geometric structure with automorphism group G. This makes essential dimension a powerful invariant with many interesting and surprising connections to problems in algebra and geometry. We show that for various classes of groups, including finite (algebraic) groups and algebraic tori, the essential dimension is related to minimal faithful representations. In many cases this renders the exact value of the essential dimension computable and we explore several of its consequences. An important open problem is the essential dimension of the projective linear group PGLn. This topic is closely related to the structure theory of central simple algebras, which may be viewed as twisted forms of the algebra of n x n matrices. We study central simple algebras with additional structure such as a distinguished Galois subfield. We prove new bounds on the essential dimension of these algebras and, as a corollary, of the group PGLn.

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Brundan, Jonathan Walter. "Double cosets in algebraic groups." Thesis, Imperial College London, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.244137.

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Craven, David Andrew. "Algebraic modules for finite groups." Thesis, University of Oxford, 2007. http://ora.ox.ac.uk/objects/uuid:7f641b33-d301-4445-8269-a5a33f4b7e5e.

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The main focus of this thesis is algebraic modules---modules that satisfy a polynomial equation with integer co-efficients in the Green ring---in various finite groups, as well as their general theory. In particular, we ask the question `when are all the simple modules for a finite group G algebraic?' We call this the (p-)SMA property. The first chapter introduces the topic and deals with preliminary results, together with the trivial first results. The second chapter provides the general theory of algebraic modules, with particular attention to the relationship between algebraic modules and the composition factors of a group, and between algebraic modules and the Heller operator and Auslander--Reiten quiver. The third chapter concerns itself with indecomposable modules for dihedral and elementary abelian groups. The study of such groups is both interesting in its own right, and can be applied to studying simple modules for simple groups, such as the sporadic groups in the final chapter. The fourth chapter analyzes the groups PSL(2,q); here we determine, in characteristic 2, which simple modules for PSL(2,q) are algebraic, for any odd q. The fifth chapter generalizes this analysis to many groups of Lie type, although most results here are in defining characteristic only. Notable exceptions include the small Ree groups, which have the 2-SMA property for all q. The sixth and final chapter focuses on the sporadic groups: for most groups we provide results on some simple modules, and some of the groups are completely analyzed in all characteristics. This is normally carried out by restricting to the Sylow p-subgroup. This thesis develops the current state of knowledge concerning algebraic modules for finite groups, and particularly for which simple groups, and for which primes, all simple modules are algebraic.

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Clarke, Matthew Charles. "Unipotent elements in algebraic groups." Thesis, University of Cambridge, 2012. https://www.repository.cam.ac.uk/handle/1810/241660.

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This thesis is concerned with three distinct, but closely related, research topics focusing on the unipotent elements of a connected reductive algebraic group G, over an algebraically closed field k, and nilpotent elements in the Lie algebra g = LieG. The first topic is a determination of canonical forms for unipotent classes and nilpotent orbits of G. Using an original approach, we begin by obtaining a new canonical form for nilpotent matrices, up to similarity, which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi;j) -> (xn+1-j;n+1-i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group, we thus obtain a unified approach to computing representatives for nilpotent orbits for all classical groups G. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary, we obtain a complete set of generic canonical representatives for the unipotent classes of the finite general unitary groups GUn(Fq) for all prime powers q. Our second topic is concerned with unipotent pieces, defined by G. Lusztig in [Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449-487]. We give a case-free proof of the conjectures of Lusztig from that paper. This presents a uniform picture of the unipotent elements of G, which can be viewed as an extension of the Dynkin-Kostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra g and the coadjoint action of G on g*. We also obtain several general results about the Hesselink stratification and Fq-rational structures on G-modules. Our third topic is concerned with generalised Gelfand-Graev representations of finite groups of Lie type. Let u be a unipotent element in such a group GF and let Γu be the associated generalised Gelfand-Graev representation of GF . Under the assumption that G has a connected centre, we show that the dimension of the endomorphism algebra of Γu is a polynomial in q (the order of the associated finite field), with degree given by dimCG(u). When the centre of G is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of q, unless one adopts a convention of considering separately various congruence classes of q. Subject to such a convention, we extend our result. We also present computational data related to the main theoretical results. In particular, tables of our canonical forms are given in the appendices, as well as tables of dimension polynomials for endomorphism algebras of generalised Gelfand-Graev representations, together with the relevant GAP source code.

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9

Dos, Santos João Pedro Pinto. "Fundamental groups in algebraic geometry." Thesis, University of Cambridge, 2006. https://www.repository.cam.ac.uk/handle/1810/252015.

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Sinanan, Shavak. "Algorithms for polycyclic-by-finite groups." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/49186/.

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A set of fundamental algorithms for computing with polycyclic-by-finite groups is presented here. Polycyclic-by-finite groups arise naturally in a number of contexts; for example, as automorphism groups of large finite soluble groups, as quotients of finitely presented groups, and as extensions of modules by groups. No existing mode of representation is suitable for these groups, since they will typically not have a convenient faithful permutation representation. A mixed mode is used to represent elements of such a group; utilising a polycyclic presentation or a power-conjugate presentation for the elements of the normal subgroup, and a permutation representation for the elements of the quotient.

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Books on the topic "Polycyclic groups][Algebraic groups"

1

Group and ring theoretic properties of polycyclic groups. London: Springer, 2009.

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Tschinkel, Yuri, ed. Algebraic groups. Göttingen: Göttingen University Press, 2007. http://dx.doi.org/10.17875/gup2007-57.

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Onishchik, Arkadij L. Lie Groups and Algebraic Groups. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990.

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B, Vinberg Ė. Lie groups and algebraic groups. Berlin: Springer Verlag, 1990.

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Onishchik, Arkadij L., and Ernest B. Vinberg. Lie Groups and Algebraic Groups. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-74334-4.

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Ariki, Susumu, Hiraku Nakajima, Yoshihisa Saito, Ken-ichi Shinoda, Toshiaki Shoji, and Toshiyuki Tanisaki, eds. Algebraic Groups and Quantum Groups. Providence, Rhode Island: American Mathematical Society, 2012. http://dx.doi.org/10.1090/conm/565.

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7

Linear algebraic groups. 2nd ed. Boston: Birkhäuser, 2009.

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8

Borel, Armand. Linear algebraic groups. 2nd ed. New York: Springer-Verlag, 1991.

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9

Differential algebraic groups. Orlando: Academic Press, 1985.

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Humphreys, James E. Linear algebraic groups. 5th ed. New York: Springer, 1998.

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Book chapters on the topic "Polycyclic groups][Algebraic groups"

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Lawson, Mark V. "The Polycyclic Inverse Monoids and the Thompson Groups Revisited." In Semigroups, Categories, and Partial Algebras, 179–214. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-33-4842-4_12.

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Onishchik, Arkadij L., and Ernest B. Vinberg. "Algebraic Varieties." In Lie Groups and Algebraic Groups, 59–97. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-74334-4_2.

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Onishchik, Arkadij L., and Ernest B. Vinberg. "Algebraic Groups." In Lie Groups and Algebraic Groups, 98–135. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-74334-4_3.

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Taylor, Joseph. "Algebraic groups." In Graduate Studies in Mathematics, 419–58. Providence, Rhode Island: American Mathematical Society, 2002. http://dx.doi.org/10.1090/gsm/046/15.

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Seitz, G. M. "Algebraic Groups." In Finite and Locally Finite Groups, 45–70. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-011-0329-9_2.

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Bahturin, Yuri. "Algebraic Groups." In Basic Structures of Modern Algebra, 289–334. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-017-0839-5_7.

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Harris, Joe. "Algebraic Groups." In Algebraic Geometry, 114–29. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-2189-8_10.

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Droste, Manfred, and Rüdiger Göbel. "The Automorphism Groups of Hahn Groups." In Ordered Algebraic Structures, 183–215. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-011-5640-0_8.

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Wallach, Nolan R. "Lie Groups and Algebraic Groups." In Universitext, 31–47. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-65907-7_2.

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Goodman, Roe, and Nolan R. Wallach. "Lie Groups and Algebraic Groups." In Graduate Texts in Mathematics, 1–68. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-79852-3_1.

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Conference papers on the topic "Polycyclic groups][Algebraic groups"

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MALGRANGE, B. "DIFFERENTIAL ALGEBRAIC GROUPS." In Algebraic Approach to Differential Equations. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814273244_0007.

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Wang, Lifang, and Yanming Wang. "On CN–Groups and CT–Groups." In The International Conference on Algebra 2010 - Advances in Algebraic Structures. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814366311_0051.

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KERZ, MORITZ. "ON NEGATIVE ALGEBRAIC K-GROUPS." In International Congress of Mathematicians 2018. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789813272880_0049.

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Nebe, Gabriele. "Computing with Arithmetic Groups." In ISSAC '17: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2017. http://dx.doi.org/10.1145/3087604.3087661.

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Golasiński, Marek, Daciberg L. Gonçalves, and Peter N. Wong. "A note on generalized equivariant homotopy groups." In Algebraic Topology - Old and New. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2009. http://dx.doi.org/10.4064/bc85-0-12.

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DVUREČENSKIJ, ANATOLIJ. "ON THE ROLE OF ℓ-GROUPS AND PO-GROUPS FOR ALGEBRAIC AND QUANTUM STRUCTURES." In Proceedings of the QL&SC 2012. WORLD SCIENTIFIC, 2012. http://dx.doi.org/10.1142/9789814401531_0002.

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Sharygin, G. I. "A new construction of characteristic classes for noncommutative algebraic principal bundles." In Noncommutative Geometry and Quantum Groups. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc61-0-15.

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Bialostocki, A., and T. Shaska. "Galois groups of prime degree polynomials with nonreal roots." In Computational Aspects of Algebraic Curves. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812701640_0015.

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Xu, Chuanyu. "New Properties of Algebraic Structure of Vague Groups." In 2007 IEEE International Conference on Control and Automation. IEEE, 2007. http://dx.doi.org/10.1109/icca.2007.4376487.

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Symonds, Peter. "On the construction of permutation complexes for profinite groups." In School and Conference in Algebraic Topology. Mathematical Sciences Publishers, 2007. http://dx.doi.org/10.2140/gtm.2007.11.369.

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