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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

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1572-9176
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Volume 26, Issue 2

Issues

Action theory of alternative algebras

José Manuel Casas / Tamar Datuashvili
  • Corresponding author
  • A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6 Tamarashvii Str., Tbilisi 0177, Georgia
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/ Manuel Ladra
  • Department of Mathematics, Institute of Mathematics, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain
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Published Online: 2019-04-06 | DOI: https://doi.org/10.1515/gmj-2019-2015

Abstract

We present the category of alternative algebras as a category of interest. This kind of approach enables us to describe derived actions in this category, study their properties and construct a universal strict general actor of any alternative algebra. We apply the results obtained in this direction to investigate the problem of the existence of an actor in the category of alternative algebras.

Keywords: Alternative algebra; action; universal strict general actor; actor; category of interest

MSC 2010: 08C05; 17D05

Dedicated to Academician Nodar Berikashvili on the occasion of his 90th birthday

References

  • [1]

    M. Atık, A. Aytekın and E. Uslu, Representability of actions in the category of (pre)crossed modules in Leibniz algebras, Comm. Algebra 45 (2017), no. 5, 1825–1841. CrossrefGoogle Scholar

  • [2]

    F. Borceux and D. Bourn, Split extension classifier and centrality, Categories in Algebra, Geometry and Mathematical Physics, Contemp. Math. 431, American Mathematical Society, Providence, (2007), 85–104. Google Scholar

  • [3]

    F. Borceux, D. Bourn and P. Johnstone, Initial normal covers in bi-Heyting toposes, Arch. Math. (Brno) 42 (2006), no. 4, 335–356. Google Scholar

  • [4]

    F. Borceux, G. Janelidze and G. M. Kelly, Internal object actions, Comment. Math. Univ. Carolin. 46 (2005), no. 2, 235–255. Google Scholar

  • [5]

    F. Borceux, G. Janelidze and G. M. Kelly, On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14 (2005), 244–286. Google Scholar

  • [6]

    D. Bourn, Action groupoid in protomodular categories, Theory Appl. Categ. 16 (2006), 46–87. Google Scholar

  • [7]

    D. Bourn and G. Janelidze, Protomodularity, descent, and semidirect products, Theory Appl. Categ. 4 (1998), 37–46. Google Scholar

  • [8]

    Y. Boyaci, J. M. Casas, T. Datuashvili and E. O. Uslu, Actions in modified categories of interest with application to crossed modules, Theory Appl. Categ. 30 (2015), 882–908. Google Scholar

  • [9]

    J. M. Casas, T. Datuashvili and M. Ladra, Actors in categories of interest, preprint (2007), https://arxiv.org/abs/math/0702574v2.

  • [10]

    J. M. Casas, T. Datuashvili and M. Ladra, Actor of an alternative algebra, preprint (2009), https://arxiv.org/abs/0910.0550v1.

  • [11]

    J. M. Casas, T. Datuashvili and M. Ladra, Actor of a precrossed module, Comm. Algebra 37 (2009), no. 12, 4516–4541. CrossrefGoogle Scholar

  • [12]

    J. M. Casas, T. Datuashvili and M. Ladra, Universal strict general actors and actors in categories of interest, Appl. Categ. Structures 18 (2010), no. 1, 85–114. Web of ScienceCrossrefGoogle Scholar

  • [13]

    J. M. Casas, T. Datuashvili and M. Ladra, Actor of a Lie–Leibniz algebra, Comm. Algebra 41 (2013), no. 4, 1570–1587. CrossrefGoogle Scholar

  • [14]

    J. M. Casas, T. Datuashvili and M. Ladra, Left-right noncommutative Poisson algebras, Cent. Eur. J. Math. 12 (2014), no. 1, 57–78. Web of ScienceGoogle Scholar

  • [15]

    J. M. Casas, T. Datuashvili, M. Ladra and E. O. Uslu, Actions in the category of precrossed modules in Lie algebras, Comm. Algebra 40 (2012), no. 8, 2962–2982. CrossrefGoogle Scholar

  • [16]

    J. M. Casas, R. Fernández-Casado, X. García-Martínez and E. Khmaladze, Actor of a crossed module of Leibniz algebras, Theory Appl. Categ. 33 (2018), 23–42. Google Scholar

  • [17]

    J. M. Casas and M. Ladra, The actor of a crossed module in Lie algebras, Comm. Algebra 26 (1998), no. 7, 2065–2089. CrossrefGoogle Scholar

  • [18]

    T. Datuashvili, Categorical, homological, and homotopical properties of algebraic objects, J. Math. Sci. (N.Y.) 225 (2017), no. 3, 383–533. CrossrefGoogle Scholar

  • [19]

    I. P. de Guzman, Annihilator alternative algebras, Pacific J. Math. 107 (1983), no. 1, 89–94. CrossrefGoogle Scholar

  • [20]

    P. J. Higgins, Groups with multiple operators, Proc. Lond. Math. Soc. (3) 6 (1956), 366–416. Google Scholar

  • [21]

    G. Hochschild, Cohomology and representations of associative algebras, Duke Math. J. 14 (1947), 921–948. CrossrefGoogle Scholar

  • [22]

    A. G. Kurosh, Lectures on General Algebra, Chelsea, New York, 1963. Google Scholar

  • [23]

    R. Lavendhomme and T. Lucas, On modules and crossed modules, J. Algebra 179 (1996), no. 3, 936–963. Web of ScienceCrossrefGoogle Scholar

  • [24]

    S. Lichtenbaum and M. Schlessinger, The cotangent complex of a morphism, Trans. Amer. Math. Soc. 128 (1967), 41–70. CrossrefGoogle Scholar

  • [25]

    J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3–4, 269–293. Google Scholar

  • [26]

    J.-L. Loday, Algèbres ayant deux opérations associatives (digèbres), C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 2, 141–146. Google Scholar

  • [27]

    J.-L. Loday, Dialgebras, Dialgebras and Related Operads, Lecture Notes in Math. 1763, Springer, Berlin (2001), 7–66. Google Scholar

  • [28]

    J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory, Contemp. Math. 346, American Mathematical Society, Providence (2004), 369–398. Google Scholar

  • [29]

    A. S.-T. Lue, Crossed homomorphisms of Lie algebras, Proc. Cambridge Philos. Soc. 62 (1966), 577–581. CrossrefGoogle Scholar

  • [30]

    A. S.-T. Lue, Non-abeliian cohomology of associative algebras, Quart. J. Math. Oxford Ser. (2) 19 (1968), 159–180. CrossrefGoogle Scholar

  • [31]

    S. Mac Lane, Extensions and obstructions for rings, Illinois J. Math. 2 (1958), 316–345. CrossrefGoogle Scholar

  • [32]

    A. Montoli, Action accessibility for categories of interest, Theory Appl. Categ. 23 (2010), 7–21. Google Scholar

  • [33]

    K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France 118 (1990), no. 2, 129–146. CrossrefGoogle Scholar

  • [34]

    G. Orzech, Obstruction theory in algebraic categories. I, J. Pure Appl. Algebra 2 (1972),287–3141. CrossrefGoogle Scholar

  • [35]

    G. Orzech, Obstruction theory in algebraic categories. II, J. Pure Appl. Algebra 2 (1972), 315–340. CrossrefGoogle Scholar

  • [36]

    T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. (2) 30 (1987), no. 3, 373–381. CrossrefGoogle Scholar

  • [37]

    C. M. Ringel, PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51–88. Google Scholar

  • [38]

    R. D. Schafer, An Introduction to Nonassociative Algebras, Pure Appl. Math. 22, Academic Press, New York, 1966. Google Scholar

  • [39]

    K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov and A. I. Shirshov, Rings that are Nearly Associative, Pure Appl. Math. 104, Academic Press, New York, 1982. Google Scholar

  • [40]

    M. Zorn, Theorie der alternativen ringe, Abh. Math. Sem. Univ. Hamburg 8 (1931), no. 1, 123–147. CrossrefGoogle Scholar

About the article

Received: 2018-04-17

Revised: 2018-10-31

Accepted: 2018-11-02

Published Online: 2019-04-06

Published in Print: 2019-06-01


Funding Source: Agencia Estatal de Investigación

Award identifier / Grant number: MTM2016-79661-P

Funding Source: Shota Rustaveli National Science Foundation

Award identifier / Grant number: GNSF/ST09 730 3-105

The authors were supported by Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER support included, UE). The second author is grateful to Santiago de Compostela and Vigo Universities and to the Rustaveli National Science Foundation for a financial support, grant GNSF/ST09 730 3-105.


Citation Information: Georgian Mathematical Journal, Volume 26, Issue 2, Pages 177–197, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2015.

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