Journal articles: 'Linear algebraic groups'

1

Pierce, Stephen. "Linear maps on algebraic groups." Linear Algebra and its Applications 162-164 (February 1992): 237–42. http://dx.doi.org/10.1016/0024-3795(92)90378-n.

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2

M�ller, Peter. "Sharply transitive linear algebraic groups." Geometriae Dedicata 45, no. 2 (February 1993): 203–24. http://dx.doi.org/10.1007/bf01264521.

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3

Hoffmann, Detlev, Alexander Merkurjev, and Jean-Pierre Tignol. "Quadratic Forms and Linear Algebraic Groups." Oberwolfach Reports 10, no. 2 (2013): 1819–59. http://dx.doi.org/10.4171/owr/2013/31.

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4

Brion, Michel. "Some Automorphism Groups are Linear Algebraic." Moscow Mathematical Journal 21, no. 3 (June 18, 2021): 453–66. http://dx.doi.org/10.17323/1609-4514-2021-21-3-453-466.

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5

Petrov, Viktor, Nikita Semenov, and Kirill Zainoulline. "$J$-invariant of linear algebraic groups." Annales scientifiques de l'École normale supérieure 41, no. 6 (2008): 1023–53. http://dx.doi.org/10.24033/asens.2088.

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6

Fomina, T. V. "Integral forms of linear algebraic groups." Mathematical Notes 61, no. 3 (March 1997): 346–51. http://dx.doi.org/10.1007/bf02355417.

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7

Platonov, Vladimir P., and Dragomir Ž. Doković. "Linear preserver problems and algebraic groups." Mathematische Annalen 303, no. 1 (September 1995): 165–84. http://dx.doi.org/10.1007/bf01460985.

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8

Pink, Richard. "Compact Subgroups of Linear Algebraic Groups." Journal of Algebra 206, no. 2 (August 1998): 438–504. http://dx.doi.org/10.1006/jabr.1998.7439.

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9

Kelarev, A. V., and P. V. Shumyatsky. "Soluble and linear repetitive groups." Bulletin of the Australian Mathematical Society 52, no. 2 (October 1995): 253–61. http://dx.doi.org/10.1017/s0004972700014684.

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10

Singer, Michael F. "Linear algebraic groups as parameterized Picard–Vessiot Galois groups." Journal of Algebra 373 (January 2013): 153–61. http://dx.doi.org/10.1016/j.jalgebra.2012.09.037.

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11

WATANABE, Takao. "Fundamental Hermite constants of linear algebraic groups." Journal of the Mathematical Society of Japan 55, no. 4 (October 2003): 1061–80. http://dx.doi.org/10.2969/jmsj/1191418764.

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12

Elmer, Jonathan, and Martin Kohls. "Zero-Separating Invariants for Linear Algebraic Groups." Proceedings of the Edinburgh Mathematical Society 59, no. 4 (December 22, 2015): 911–24. http://dx.doi.org/10.1017/s0013091515000322.

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AbstractAbstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.

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13

Ovchinnikov, Alexey. "Tannakian Approach to Linear Differential Algebraic Groups." Transformation Groups 13, no. 2 (June 2008): 413–46. http://dx.doi.org/10.1007/s00031-008-9010-4.

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14

Minchenko, Andrey, Alexey Ovchinnikov, and Michael F. Singer. "Unipotent differential algebraic groups as parameterized differential Galois groups." Journal of the Institute of Mathematics of Jussieu 13, no. 4 (July 18, 2013): 671–700. http://dx.doi.org/10.1017/s1474748013000200.

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AbstractWe deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a parameterized linear differential equation if its Galois group has differential type $0$.

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15

IWAMOTO, Takashi. "ALGEBRAIC GROUPS AND CO-COMPACT SUBGROUPS OF COMPLEX LINEAR GROUPS." Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics 42, no. 1 (1988): 1–7. http://dx.doi.org/10.2206/kyushumfs.42.1.

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16

Conversano, Annalisa. "Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups." Journal of Algebra 587 (December 2021): 295–309. http://dx.doi.org/10.1016/j.jalgebra.2021.08.004.

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17

Bachmayr, Annette, and Michael Wibmer. "Algebraic groups as difference Galois groups of linear differential equations." Journal of Pure and Applied Algebra 226, no. 2 (February 2022): 106854. http://dx.doi.org/10.1016/j.jpaa.2021.106854.

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18

Lourdeaux, Alexandre. "Degree 2 cohomological invariants of linear algebraic groups." Journal of Pure and Applied Algebra 226, no. 10 (October 2022): 107059. http://dx.doi.org/10.1016/j.jpaa.2022.107059.

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19

Fuchs, Clemens, and Duc Hiep Pham. "Commutative algebraic groups and p-adic linear forms." Acta Arithmetica 169, no. 2 (2015): 115–47. http://dx.doi.org/10.4064/aa169-2-2.

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20

Adams, Scot, and Alexander S. Kechris. "Linear algebraic groups and countable Borel equivalence relations." Journal of the American Mathematical Society 13, no. 4 (June 23, 2000): 909–43. http://dx.doi.org/10.1090/s0894-0347-00-00341-6.

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21

Minchenko, Andrey, and Alexey Ovchinnikov. "Zariski closures of reductive linear differential algebraic groups." Advances in Mathematics 227, no. 3 (June 2011): 1195–224. http://dx.doi.org/10.1016/j.aim.2011.03.002.

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22

Korhonen, Mikko. "Unipotent elements forcing irreducibility in linear algebraic groups." Journal of Group Theory 21, no. 3 (May 1, 2018): 365–96. http://dx.doi.org/10.1515/jgth-2018-0003.

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Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic {p>0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G. A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G. We generalize their result and show that if u has order p, then except for two known examples which occur in the case {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order {>p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

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23

Lienau, Christoph. "Analytic Dirac approximation for real linear algebraic groups." Mathematische Annalen 351, no. 2 (November 20, 2010): 403–10. http://dx.doi.org/10.1007/s00208-010-0607-2.

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24

Brownawell, W. Dale. "Minimal Extensions of Algebraic Groups and Linear Independence." Journal of Number Theory 90, no. 2 (October 2001): 239–54. http://dx.doi.org/10.1006/jnth.2001.2638.

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25

Vavilov, Nikolai А. "St Petersburg school of linear groups. I. Prehistorical period." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 10 (68), no. 3 (2023): 381–405. http://dx.doi.org/10.21638/spbu01.2023.301.

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The present survey describes the contribution of St Petersburg mathematicians to the theory of linear, classical and algebraic groups. The first part is dedicated to the prehistorical period, the historical genesis of Tartakowski and Faddeev algebra schools, and to the general outline of the works by Borewicz and Suslin of the mid-1970s that initiated systematical research in the fields of classical groups and algebraic K-theory in St Petersburg.

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26

Minchenko, Andrey, Alexey Ovchinnikov, and Michael F. Singer. "Reductive Linear Differential Algebraic Groups and the Galois Groups of Parameterized Linear Differential Equations." International Mathematics Research Notices 2015, no. 7 (January 10, 2014): 1733–93. http://dx.doi.org/10.1093/imrn/rnt344.

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27

Altinel, Tuna, and Gregory Cherlin. "On central extensions of algebraic groups." Journal of Symbolic Logic 64, no. 1 (March 1999): 68–74. http://dx.doi.org/10.2307/2586751.

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In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem 1. Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0 is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0 = 1.Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.Corollary 1. Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4], Section 27.5).An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of tame simple K*-groups of finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption of tameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results from K-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.

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28

Ovchinnikov, Alexey. "Tannakian Categories, Linear Differential Algebraic Groups, and Parametrized Linear Differential Equations." Transformation Groups 14, no. 1 (November 27, 2008): 195–223. http://dx.doi.org/10.1007/s00031-008-9042-9.

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29

BASHKIROV, E. L., and C. K. GUPTA. "LINEAR GROUPS OVER LOCALLY FINITE EXTENSIONS OF INFINITE FIELDS." International Journal of Algebra and Computation 17, no. 05n06 (August 2007): 905–22. http://dx.doi.org/10.1142/s0218196707003937.

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Let P be a field of characteristic different from 2, let K be an associative commutative P-algebra with an identity 1 and let n be an integer, n ≥ 2. Assume that K is an algebraic extension of P having, in general, zero divisors and P is an algebraic separable extension of an infinite subfield k. The paper studies subgroups X of the group GLn (K) such that X contains a root k-subgroup, i.e. a subgroup which is conjugate in GLn (K) to a group of all matrices [Formula: see text], a ∈ k.

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30

JOSHUA, ROY. "Algebraic K-theory and higher Chow groups of linear varieties." Mathematical Proceedings of the Cambridge Philosophical Society 130, no. 1 (January 2001): 37–60. http://dx.doi.org/10.1017/s030500410000476x.

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The main focus in this paper is the algebraic K-theory and higher Chow groups of linear varieties and schemes. We provide Kunneth spectral sequences for the higher algebraic K-theory of linear schemes flat over a base scheme and for the motivic cohomology of linear varieties defined over a field. The latter provides a Kunneth formula for the usual Chow groups of linear varieties originally obtained by different means by Totaro. We also obtain a general condition under which the higher cycle maps of Bloch from mod-lv higher Chow groups to mod-lv étale cohomology are isomorphisms for projective nonsingular varieties defined over an algebraically closed field of arbitrary characteristic p [ges ] 0 with l ≠ p. It is observed that the Kunneth formula for the Chow groups implies this condition for linear varieties and we compute the mod-lv motivic cohomology and mod-lv algebraic K-theory of projective nonsingular linear varieties to be free ℤ/lv-modules.

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31

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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32

Di Vizio, Lucia, Charlotte Hardouin, and Michael Wibmer. "DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS." Journal of the Institute of Mathematics of Jussieu 16, no. 1 (April 17, 2015): 59–119. http://dx.doi.org/10.1017/s1474748015000080.

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We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

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33

Roy, Damien, and Michel Waldschmidt. "Autour du Théorème du Sous-Groupe Algébrique." Canadian Mathematical Bulletin 36, no. 3 (September 1, 1993): 358–67. http://dx.doi.org/10.4153/cmb-1993-049-8.

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RésuméA known consequence of the theorem of the algebraic subgroup is a lower bound for the rank of matrices whose entries are linear combinations, with algebraic coefficients, of logarithms of algebraic numbers. We extend this kind of result to commutative algebraic groups.

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34

Bérczi, Gergely. "On the Popov–Pommerening conjecture for linear algebraic groups." Compositio Mathematica 154, no. 1 (October 9, 2017): 36–79. http://dx.doi.org/10.1112/s0010437x17007473.

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Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H\subseteq G$ an observable subgroup normalised by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 1970s that the invariant algebra $k[X]^{H}$ is finitely generated. We prove the conjecture for: (1) subgroups of $\operatorname{SL}_{n}(k)$ closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of $\operatorname{SL}_{n}(k)$.

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35

Maksimov, A. D. "Typical equivalence of linear groups and other algebraic systems." Journal of Mathematical Sciences 183, no. 3 (May 3, 2012): 397–406. http://dx.doi.org/10.1007/s10958-012-0820-5.

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36

Putcha, Mohan S. "A Semigroup Approach to Linear Algebraic Groups III. Buildings." Canadian Journal of Mathematics 38, no. 3 (June 1, 1986): 751–68. http://dx.doi.org/10.4153/cjm-1986-039-1.

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Introduction. Let K be an algebraically closed field, G = SL(3, K) the group of 3 × 3 matrices over K of determinant 1. Let denote the monoid of all 3 × 3 matrices over K. If e is an idempotent in , thenare opposite parabolic subgroups of G in the usual sense [1], [28]. However the mapdoes not exhaust all pairs of opposite parabolic subgroups of G. Now consider the representation ϕ:G → SL(6, K) given by

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37

Sukhanov, A. A. "DESCRIPTION OF THE OBSERVABLE SUBGROUPS OF LINEAR ALGEBRAIC GROUPS." Mathematics of the USSR-Sbornik 65, no. 1 (February 28, 1990): 97–108. http://dx.doi.org/10.1070/sm1990v065n01abeh001141.

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38

Chen, Eric, J. T. Ferrara, and Liam Mazurowski. "Generic extensions and generic polynomials for linear algebraic groups." Journal of Algebra 461 (September 2016): 1–24. http://dx.doi.org/10.1016/j.jalgebra.2016.04.027.

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39

Putcha, Mohan S. "A semigroup approach to linear algebraic groups II. Roots." Journal of Pure and Applied Algebra 39 (1986): 153–63. http://dx.doi.org/10.1016/0022-4049(86)90142-8.

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40

Voskresenskii, V. E. "Birational geometry of linear algebraic groups and galois modules." Journal of Mathematical Sciences 85, no. 4 (July 1997): 2017–114. http://dx.doi.org/10.1007/bf02359883.

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41

Bate, Michael, Benjamin Martin, and Gerhard Röhrle. "Edifices: building-like spaces associated to linear algebraic groups." Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial 20, no. 2-3 (September 13, 2023): 79–134. http://dx.doi.org/10.2140/iig.2023.20.79.

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42

Allcock, Daniel. "A new approach to rank one linear algebraic groups." Journal of Algebra 321, no. 9 (May 2009): 2540–44. http://dx.doi.org/10.1016/j.jalgebra.2009.01.034.

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43

Elduque, A., and A. V. Iltyakov. "On Polynomial Invariants of Exceptional Simple Algebraic Groups." Canadian Journal of Mathematics 51, no. 3 (June 1, 1999): 506–22. http://dx.doi.org/10.4153/cjm-1999-023-2.

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AbstractWe study polynomial invariants of systems of vectors with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system (cf. [27], [28], [13]).

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44

Vavilov, Nikolai А. "St Petersburg school of linear groups. II. Early works by Suslin." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 11, no. 1 (2024): 48–83. http://dx.doi.org/10.21638/spbu01.2024.103.

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The present survey describes the contribution of St Petersburg mathematicians to the theory of linear, classical and algebraic groups. The second part is devoted to the publications by Suslin of the 1970s and the early 1980s, in the areas of classical algebraic K-theory and the theories of linear and classical groups. Also, we describe the general context of these works, state some of the most important results by Suslin himself, and his students, and some of the most closely related follow-ups and subsequent results.

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45

Li, Ying. "On Finite Subgroups in the General Linear Groups over an Algebraic Number Field." Journal of Physics: Conference Series 2287, no. 1 (June 1, 2022): 012006. http://dx.doi.org/10.1088/1742-6596/2287/1/012006.

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Abstract As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and to classify finite cyclic subgroups.

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46

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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47

Font, Juan J., and Salvador Hernández. "Algebraic characterizations of locally compact groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 62, no. 3 (June 1997): 405–20. http://dx.doi.org/10.1017/s1446788700001099.

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AbstractLet G1, G2 be locally compact real-compact spaces. A linear map T defined from C(G1) into C(G2) is said to be separating or disjointness preserving if f = g ≡ 0 implies Tf = Tg ≡ 0 f or all f, g ∈ C(G1). In this paper we prove that both a separating map which preserves non-vanishing functions and a separating bijection which satisfies condition (M) (see Definition 4) are automatically continuous and can be written as weighted composition maps. We also study the effect of separating surjections (respectively injections) on the underlying spaces G1 and G2.Next we apply the above results to give an algebraic characterization of locally compact Abelian groups, similar to the one given in [7] for compact Abelian groups in the presence of ring isomorphisms.Finally, locally compact (not necessarily Abelian) groups are considered. We provide a sharpening of a result of Edwards and study the effect of onto (respectively injective) weighted composition maps on the groups G1 and G2.

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48

Ayala, Víctor, Heriberto Román-Flores, and María Torreblanca Todco. "Control Sets of Linear Control Systems on Matrix Groups and Applications." Mathematical Problems in Engineering 2019 (July 4, 2019): 1–12. http://dx.doi.org/10.1155/2019/2963120.

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49

Xing, C. "Linear Codes From Narrow Ray Class Groups of Algebraic Curves." IEEE Transactions on Information Theory 50, no. 3 (March 2004): 541–43. http://dx.doi.org/10.1109/tit.2004.824922.

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50

Meng, Sheng, and De-Qi Zhang. "Jordan property for non-linear algebraic groups and projective varieties." American Journal of Mathematics 140, no. 4 (2018): 1133–45. http://dx.doi.org/10.1353/ajm.2018.0026.

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Journal articles on the topic 'Linear algebraic groups'

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