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Relevant bibliographies by topics / Semisimple algebraic groups

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Academic literature on the topic 'Semisimple algebraic groups'

Author: Grafiati

Published: 21 September 2024

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Journal articles on the topic "Semisimple algebraic groups"

1

Nahlus, Nazih. "Homomorphisms of Lie Algebras of Algebraic Groups and Analytic Groups." Canadian Mathematical Bulletin 38, no. 3 (September 1, 1995): 352–59. http://dx.doi.org/10.4153/cmb-1995-051-7.

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AbstractLet be a Lie algebra homomorphism from the Lie algebra of G to the Lie algebra of H in the following cases: (i) G and H are irreducible algebraic groups over an algebraically closed field of characteristic 0, or (ii) G and H are linear complex analytic groups. In this paper, we present some equivalent conditions for ϕ to be a differential in the above two cases. That is, ϕ is the differential of a morphism of algebraic groups or analytic groups as appropriate.In the algebraic case, for example, it is shown that ϕ is a differential if and only if ϕ preserves nilpotency, semisimplicity, and integrality of elements. In the analytic case, ϕ is a differential if and only if ϕ maps every integral semisimple element of into an integral semisimple element of , where G0 and H0 are the universal algebraic subgroups of G and H. Via rational elements, we also present some equivalent conditions for ϕ to be a differential up to coverings of G in the algebraic case, and for ϕ to be a differential up to finite coverings of G in the analytic case.

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De Clercq, Charles. "Équivalence motivique des groupes algébriques semisimples." Compositio Mathematica 153, no. 10 (July 27, 2017): 2195–213. http://dx.doi.org/10.1112/s0010437x17007369.

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We prove that the standard motives of a semisimple algebraic group$G$with coefficients in a field of order$p$are determined by the upper motives of the group $G$. As a consequence of this result, we obtain a partial version of the motivic rigidity conjecture of special linear groups. The result is then used to construct the higher indexes which characterize the motivic equivalence of semisimple algebraic groups. The criteria of motivic equivalence derived from the expressions of these indexes produce a dictionary between motives, algebraic structures and the birational geometry of twisted flag varieties. This correspondence is then described for special linear groups and orthogonal groups (the criteria associated with other groups being obtained in De Clercq and Garibaldi [Tits$p$-indexes of semisimple algebraic groups, J. Lond. Math. Soc. (2)95(2017) 567–585]). The proofs rely on the Levi-type motivic decompositions of isotropic twisted flag varieties due to Chernousov, Gille and Merkurjev, and on the notion of pondered field extensions.

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De Clercq, Charles, and Skip Garibaldi. "Tits p-indexes of semisimple algebraic groups." Journal of the London Mathematical Society 95, no. 2 (January 16, 2017): 567–85. http://dx.doi.org/10.1112/jlms.12025.

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Gordeev, Nikolai, Boris Kunyavskiĭ, and Eugene Plotkin. "Word maps on perfect algebraic groups." International Journal of Algebra and Computation 28, no. 08 (December 2018): 1487–515. http://dx.doi.org/10.1142/s0218196718400052.

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We extend Borel’s theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel’s theorem to some words with constants. We also consider the surjectivity problem for particular words and groups, give a brief survey of recent results, present some generalizations and variations and discuss various approaches, with emphasis on new ideas, constructions and connections.

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Cassidy, Phyllis Joan. "The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras." Journal of Algebra 121, no. 1 (February 1989): 169–238. http://dx.doi.org/10.1016/0021-8693(89)90092-6.

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Avdeev, R. S. "On solvable spherical subgroups of semisimple algebraic groups." Transactions of the Moscow Mathematical Society 72 (2011): 1–44. http://dx.doi.org/10.1090/s0077-1554-2012-00192-7.

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Procesi, Claudio. "Book Review: Conjugacy classes in semisimple algebraic groups." Bulletin of the American Mathematical Society 34, no. 01 (January 1, 1997): 55–57. http://dx.doi.org/10.1090/s0273-0979-97-00689-7.

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Voskresenskii, V. E. "Maximal tori without effect in semisimple algebraic groups." Mathematical Notes of the Academy of Sciences of the USSR 44, no. 3 (September 1988): 651–55. http://dx.doi.org/10.1007/bf01159125.

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Mohrdieck, S. "Conjugacy classes of non-connected semisimple algebraic groups." Transformation Groups 8, no. 4 (December 2003): 377–95. http://dx.doi.org/10.1007/s00031-003-0429-3.

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Breuillard, Emmanuel, Ben Green, Robert Guralnick, and Terence Tao. "Strongly dense free subgroups of semisimple algebraic groups." Israel Journal of Mathematics 192, no. 1 (March 15, 2012): 347–79. http://dx.doi.org/10.1007/s11856-012-0030-3.

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Dissertations / Theses on the topic "Semisimple algebraic groups"

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Mohrdieck, Stephan. "Conjugacy classes of non-connected semisimple algebraic groups." [S.l. : s.n.], 2000. http://www.sub.uni-hamburg.de/disse/172/diss.pdf.

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Hazi, Amit. "Semisimple filtrations of tilting modules for algebraic groups." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/271774.

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Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. The indecomposable tilting modules $\{T(\lambda)\}$ for $G$, which are labeled by highest weight, form an important class of self-dual representations over $k$. In this thesis we investigate semisimple filtrations of minimal length (Loewy series) of tilting modules. We first demonstrate a criterion for determining when tilting modules for arbitrary quasi-hereditary algebras are rigid, i.e. have a unique Loewy series. Our criterion involves checking that $T(\lambda)$ does not have certain subquotients whose composition factors extend more than one layer in the radical or socle series. We apply this criterion to show that the restricted tilting modules for $SL_4$ are rigid when $p \geq 5$, something beyond the scope of previous work on this topic by Andersen and Kaneda. Even when $T(\lambda)$ is not rigid, in many cases it has a particularly structured Loewy series which we call a balanced semisimple filtration, whose semisimple subquotients or "layers" are symmetric about some middle layer. Balanced semisimple filtrations also suggest a remarkably straightforward algorithm for calculating tilting characters from the irreducible characters. Applying Lusztig's character formula for the simple modules, we show that the algorithm agrees with Soergel's character formula for the regular indecomposable tilting modules for quantum groups at roots of unity. We then show that these filtrations really do exist for these tilting modules. In the modular case, high weight tilting modules exhibit self-similarity in their characters at $p$-power scales. This is due to what we call higher-order linkage, an old character-theoretic result relating modular tilting characters and quantum tilting characters at $p$-power roots of unity. To better understand this behavior we describe an explicit categorification of higher-order linkage using the language of Soergel bimodules. Along the way we also develop the algebra and combinatorics of higher-order linkage at the de-categorified level. We hope that this will provide a foundation for a tilting character formula valid for all weights in the modular case when $p$ is sufficiently large.

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Kenneally, Darren John. "On eigenvectors for semisimple elements in actions of algebraic groups." Thesis, University of Cambridge, 2010. https://www.repository.cam.ac.uk/handle/1810/224782.

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Let G be a simple simply connected algebraic group defined over an algebraically closed field K and V an irreducible module defined over K on which G acts. Let E denote the set of vectors in V which are eigenvectors for some non-central semisimple element of G and some eigenvalue in K*. We prove, with a short list of possible exceptions, that the dimension of Ē is strictly less than the dimension of V provided dim V > dim G + 2 and that there is equality otherwise. In particular, by considering only the eigenvalue 1, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of V provided dim V > dim G + 2, with a short list of possible exceptions. In the majority of cases we consider modules for which dim V > dim G + 2 where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds dim G. In more difficult cases, when dim V is only slightly larger than dim G + 2, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying dim V ≤ dim G + 2, an immediate observation yields the result for dim V < dim B where B is a Borel subgroup of G, while in other cases we argue directly.

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Gandhi, Raj. "Oriented Cohomology Rings of the Semisimple Linear Algebraic Groups of Ranks 1 and 2." Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/42566.

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In this thesis, we compute minimal presentations in terms of generators and relations for the oriented cohomology rings of several semisimple linear algebraic groups of ranks 1 and 2 over algebraically closed fields of characteristic 0. The main tools we use in this thesis are the combinatorics of Coxeter groups and formal group laws, and recent results of Calm\`es, Gille, Petrov, Zainoulline, and Zhong, which relate the oriented cohomology rings of flag varieties and semisimple linear algebraic groups to the dual of the formal affine Demazure algebra.

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Maccan, Matilde. "Sous-schémas en groupes paraboliques et variétés homogènes en petites caractéristiques." Electronic Thesis or Diss., Université de Rennes (2023-....), 2024. https://ged.univ-rennes1.fr/nuxeo/site/esupversions/2e27fe72-c9e0-4d56-8e49-14fc84686d6c.

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Cette thèse achève la classification des sous-schémas en groupes paraboliques des groupes algébriques semi-simples sur un corps algébriquement clos, en particulier de caractéristique deux et trois. Dans un premier temps, nous présentons la classification en supposant que la partie réduite de ces sous-groupes soit maximale, avant de passer au cas général. Nous parvenons à une description quasiment uniforme : à l'exception d'un groupe de type G₂ en caractéristique deux, chaque sous-schémas en groupes parabolique est obtenu en multipliant des paraboliques réduits par des noyaux d'isogénies purement inséparables, puis en prenant l'intersection. En conclusion, nous discutons quelques implications géométriques de cette classification
This thesis brings to an end the classification of parabolic subgroup schemes of semisimple groups over an algebraically closed field, focusing on characteristic two and three. First, we present the classification under the assumption that the reduced part of these subgroups is maximal; then we proceed to the general case. We arrive at an almost uniform description: with the exception of a group of type G₂ in characteristic two, any parabolic subgroup scheme is obtained by multiplying reduced parabolic subgroups by kernels of purely inseparable isogenies, then taking the intersection. In conclusion, we discuss some geometric implications of this classification

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Oriente, Francesco. "Classifying semisimple orbits of theta-groups." Doctoral thesis, Università degli studi di Trento, 2012. https://hdl.handle.net/11572/368303.

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I consider the problem of classifying the semisimple orbits of a theta-group. For this purpose, once a preliminary presentation of the theoretical subjects where my problem arises from, I first give an algorithm to compute a Cartan subspace; subsequently I describe how to compute the little Weyl group.

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Oriente, Francesco. "Classifying semisimple orbits of theta-groups." Doctoral thesis, University of Trento, 2012. http://eprints-phd.biblio.unitn.it/731/1/tesi.pdf.

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I consider the problem of classifying the semisimple orbits of a theta-group. For this purpose, once a preliminary presentation of the theoretical subjects where my problem arises from, I first give an algorithm to compute a Cartan subspace; subsequently I describe how to compute the little Weyl group.

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8

Lampetti, Enrico. "Nilpotent orbits in semisimple Lie algebras." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/23595/.

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This thesis is dedicated to the introductory study of the so-called nilpotent orbits in a semisimple complex Lie algebra g, i.e., the orbits of nilpotent elements under the adjoint action of the adjoint group Gad with Lie algebra g. These orbits have an extremely rich structure and lie at the interface of Lie theory, algebraic geometry, symplectic geometry, and geometric representation theory. The Jacobson and Morozov Theorem relates the orbit of a nilpotent element X in a semisimple complex Lie algebra g with a triple {H,X,Y} that generates a subalgebra of g isomorphic to sl(2,C). There is a parabolic subalgebra associated to this triple that permits to attach a weight to each node of the Dynkin diagram of g. The resulting diagram is called a weighted Dynkin diagram associated with the nilpotent orbit of X. This is a complete invariant of the orbit that one can use in order to show that there are only _nitely many nilpotent orbits in g. The thesis is organized as follows: the first three chapters contain some preliminary material on Lie algebras (Chapter 1), on Lie groups (Chapter 3) and on the representation theory of sl(2,C) (Chapter 2). Chapter 4 and 5 are the heart of the thesis. Namely, Jacobson-Morozov, Kostant and Mal'cev Theorems are proved in Chapter 4 and Chapter 5 is dedicated to the construction of weighted Dynkin diagrams. As an example the conjugacy classes of nilpotent elements in sl(n,C) are described in detail and a formula for their dimension is given. In this case, as well as in the case of all classical Lie algebras, the description of the orbits can be done in terms of partitions and tableaux.

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9

Nishiyama, Kyo. "Representations of Weyl groups and their Hecke algebras on virtual character modules of a semisimple Lie group." 京都大学 (Kyoto University), 1986. http://hdl.handle.net/2433/86366.

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Athapattu, Mudiyanselage Chathurika Umayangani Manike Athapattu. "Chevalley Groups." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1986.

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In this thesis, we construct Chevalley groups over arbitrary fields. The construction is based on the properties of semi-simple complex Lie algebras, the existence of Chevalley bases and notion of universal enveloping algebras. Using integral lattices in universal enveloping algebras and integral properties of Chevalley bases, we present a method which produces, for any complex simple Lie group, an analogous group over an arbitrary field.

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Books on the topic "Semisimple algebraic groups"

1

Humphreys, James E. Conjugacy classes in semisimple algebraic groups. Providence, R.I: American Mathematical Society, 1995.

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2

Hiss, G. Imprimitive irreducible modules for finite quasisimple groups. Providence, Rhode Island: American Mathematical Society, 2015.

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Kapovich, Michael. The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra. Providence, R.I: American Mathematical Society, 2008.

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1959-, McGovern William M., ed. Nilpotent orbits in semisimple Lie algebras. New York: Van Nostrand Reinhold, 1993.

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5

Doran, Robert S., 1937- editor of compilation, Friedman, Greg, 1973- editor of compilation, and Nollet, Scott, 1962- editor of compilation, eds. Hodge theory, complex geometry, and representation theory: NSF-CBMS Regional Conference in Mathematics, June 18, 2012, Texas Christian University, Fort Worth, Texas. Providence, Rhode Island: American Mathematical Society, 2013.

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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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Benkart, Georgia. Stability in modules for classical lie algebras: A constructive approach. Providence, R.I., USA: American Mathematical Society, 1990.

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Strade, Helmut, Thomas Weigel, Marina Avitabile, and Jörg Feldvoss. Lie algebras and related topics: Workshop in honor of Helmut Strade's 70th birthday : lie algebras, May 22-24, 2013, Università degli studi di Milano-Bicocca, Milano, Italy. Providence, Rhode Island: American Mathematical Society, 2015.

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9

Humphreys, James E. Conjugacy Classes in Semisimple Algebraic Groups. American Mathematical Society, 1995.

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10

Gille, Philippe. Groupes algébriques semi-simples en dimension cohomologique ≤2: Semisimple algebraic groups in cohomological dimension ≤2. Springer, 2019.

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Book chapters on the topic "Semisimple algebraic groups"

1

Onishchik, Arkadij L., and Ernest B. Vinberg. "Complex Semisimple Lie Groups." In Lie Groups and Algebraic Groups, 136–220. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-74334-4_4.

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Onishchik, Arkadij L., and Ernest B. Vinberg. "Real Semisimple Lie Groups." In Lie Groups and Algebraic Groups, 221–81. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/978-3-642-74334-4_5.

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Brown, Ken A., and Ken R. Goodearl. "Primer on Semisimple Lie Algebras." In Lectures on Algebraic Quantum Groups, 39–44. Basel: Birkhäuser Basel, 2002. http://dx.doi.org/10.1007/978-3-0348-8205-7_5.

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Lakshmibai, V., and Justin Brown. "Representation Theory of Semisimple Algebraic Groups." In Texts and Readings in Mathematics, 153–63. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-1393-6_11.

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Lakshmibai, V., and Justin Brown. "Representation Theory of Semisimple Algebraic Groups." In Texts and Readings in Mathematics, 183–96. Gurgaon: Hindustan Book Agency, 2009. http://dx.doi.org/10.1007/978-93-86279-41-5_11.

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Brown, Ken A., and Ken R. Goodearl. "Generic Quantized Coordinate Rings of Semisimple Groups." In Lectures on Algebraic Quantum Groups, 59–67. Basel: Birkhäuser Basel, 2002. http://dx.doi.org/10.1007/978-3-0348-8205-7_7.

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Margulis, Gregori Aleksandrovitch. "Normal Subgroups and “Abstract” Homomorphisms of Semisimple Algebraic Groups Over Global Fields." In Discrete Subgroups of Semisimple Lie Groups, 258–87. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-51445-6_9.

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Langlands, R. "On the classification of irreducible representations of real algebraic groups." In Representation Theory and Harmonic Analysis on Semisimple Lie Groups, 101–70. Providence, Rhode Island: American Mathematical Society, 1989. http://dx.doi.org/10.1090/surv/031/03.

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Guivarc’h, Yves, Lizhen Ji, and J. C. Taylor. "Extension to Semisimple Algebraic Groups Defined Over a Local Field." In Compactification of Symmetric Spaces, 231–36. Boston, MA: Birkhäuser Boston, 1998. http://dx.doi.org/10.1007/978-1-4612-2452-5_15.

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Alperin, J. L., and Rowen B. Bell. "Semisimple Algebras." In Groups and Representations, 107–36. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-0799-3_5.

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Conference papers on the topic "Semisimple algebraic groups"

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Gupta, Shalini, and Jasbir Kaur. "Structure of some finite semisimple group algebras." In DIDACTIC TRANSFER OF PHYSICS KNOWLEDGE THROUGH DISTANCE EDUCATION: DIDFYZ 2021. AIP Publishing, 2022. http://dx.doi.org/10.1063/5.0080606.

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