mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державина10.20310/2686-9667-2024-29-148-425-439mathematics-82Research ArticleСТАТЬИARTICLESРазложение модулей над обобщенными алгебрами ДиксонаDecomposition of modules over generalized Dickson algebrashttps://orcid.org/0000-0002-4733-8151ЛюдковскийСергей ВикторовичLudkovskySergey V.sludkowski@mail.ruМИРЭА – Российский технологический университетРоссияMIREA – Russian Technological UniversityRussian Federation20242512202429148425439Copyright © Людковский С.В., 20242024Людковский С.В.Ludkovsky S.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathematics.elpub.ru/jour/article/view/82

Статья посвящена модулям над обобщенными алгебрами Диксона. Эти алгебры неассоциативны и в общем случае могут быть неальтернативными. Они составляют важный класс алгебр и раздел математики. В работе изучаются левые, правые и двусторонние модули над обобщенными алгебрами Диксона. Исследуется их структура и подмодули. Особое внимание уделено бимодулям c инволюцией над обобщенными алгебрами Диксона с инволюцией. Такие бимодули имеют специфические особенности, вызванные наличием инволюции. Исследуются минимальные подмодули и разложение модулей. В частности, изучаются циклические подмодули.

The article is devoted to modules over generalized Dickson algebras. These algebras are nonassociative and generally can be nonalternative. They compose an important class of algebras and an area in mathematics. Left, right and two-sided modules over generalized Dickson algebras are studied. Their structure and submodules are investigated. Bimodules with involution are scrutinized over generalized Dickson algebras with involution. Such bimodules have specific features caused by involution. Minimal submodules and decomposition of modules are investigated. In particular, cyclic submodules are studied.

модульразложениеобобщенная алгебра Диксонаинволюциякольцоmoduledecompositiongeneralized Dickson algebrainvolutionringReferencesL. E. Dickson, The Collected Mathematical Papers, Cambridge Library Collection - Mathematics, 1–5, Chelsea Publishing Co., New York, 1975.L. E. Dickson, The Collected Mathematical Papers, Cambridge Library Collection - Mathematics, 1–5, Chelsea Publishing Co., New York, 1975.K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, A. I. Shirshov, Rings that are Nearly Associative, Academic Press, New York, 1982.K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, A. I. Shirshov, Rings that are Nearly Associative, Academic Press, New York, 1982.R. B. Brown, “On generalized Cayley-Dickson algebras”, Pacific Journal of Mathematics, 20:3 (1967), 415–422.R. B. Brown, “On generalized Cayley-Dickson algebras”, Pacific Journal of Mathematics, 20:3 (1967), 415–422.J. C. Baez, “The octonions”, Bull. Amer. Math. Soc., 39:2 (2002), 145-205.J. C. Baez, “The octonions”, Bull. Amer. Math. Soc., 39:2 (2002), 145-205.R. D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York, 1966.R. D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York, 1966.S. V. Ludkowski, “Completions and homomorphisms of infinite dimensional Cayley-Dickson algebras”, Linear Multilinear Algebra, 69:11 (2021), 2040–2049.S. V. Ludkowski, “Completions and homomorphisms of infinite dimensional Cayley-Dickson algebras”, Linear Multilinear Algebra, 69:11 (2021), 2040–2049.D. Allcock, “Reflection groups and octave hyperbolic plane”, J. Algebra, 213:2 (1998), 467-498.D. Allcock, “Reflection groups and octave hyperbolic plane”, J. Algebra, 213:2 (1998), 467-498.A. Belov, L. Bokut, L. Rowen, J.-T. Yu, “The Jacobian conjecture, together with Specht and Burnside-type problems”, Automorphisms and birational affine geometry, Springer Proceedings in Mathematics and Statistics, 79, ed. I. Cheltsov, 2014, 249–285.A. Belov, L. Bokut, L. Rowen, J.-T. Yu, “The Jacobian conjecture, together with Specht and Burnside-type problems”, Automorphisms and birational affine geometry, Springer Proceedings in Mathematics and Statistics, 79, ed. I. Cheltsov, 2014, 249–285.N. Bourbaki, Algebra, Springer, Berlin, 2022.N. Bourbaki, Algebra, Springer, Berlin, 2022.N. Jacobson, “Cayley numbers and normal simple Lie algebras of type G”, Duke Math. J., 5 (1939), 775–783.N. Jacobson, “Cayley numbers and normal simple Lie algebras of type G”, Duke Math. J., 5 (1939), 775–783.M. Goto, F.D. Grosshans, Semisimple Lie Algebras, Marcel Dekker, Inc., New York, 1978.M. Goto, F.D. Grosshans, Semisimple Lie Algebras, Marcel Dekker, Inc., New York, 1978.P. Eakin, A. Sathaye, “On automoprhisms and derivations of Cayley-Dickson algebras”, J. Algebra, 129:2 (1990), 263–278.P. Eakin, A. Sathaye, “On automoprhisms and derivations of Cayley-Dickson algebras”, J. Algebra, 129:2 (1990), 263–278.H. B. Lawson, M.-L. Michelson, Spin Geometry, Princ. Univ. Press, Princeton, 1989.H. B. Lawson, M.-L. Michelson, Spin Geometry, Princ. Univ. Press, Princeton, 1989.C. Culbert, “Cayley-Dickson algebras and loops”, J. Gener. Lie Theory Appl., 1:1 (2007), 1-17.C. Culbert, “Cayley-Dickson algebras and loops”, J. Gener. Lie Theory Appl., 1:1 (2007), 1-17.S. V. Ludkovsky, “C*-algebras of meta-invariant operators in modules over Cayley-Dickson algebras”, Southeast Asian Bull. Math., 39:5 (2015), 625–684.S. V. Ludkovsky, “C*-algebras of meta-invariant operators in modules over Cayley-Dickson algebras”, Southeast Asian Bull. Math., 39:5 (2015), 625–684.S. V. Ludkovsky, W. Sprossig, “Spectral representations of operators in Hilbert spaces over quaternions and octonions”, Complex Var. Elliptic Equ., 57:12 (2012), 1301–1324.S. V. Ludkovsky, W. Sprossig, “Spectral representations of operators in Hilbert spaces over quaternions and octonions”, Complex Var. Elliptic Equ., 57:12 (2012), 1301–1324.S. V. Ludkovsky, W. Sprossig, “Spectral theory of super-differential operators of quaternion and octonion variables”, Adv. Appl. Clifford Algebr., 21:1 (2011), 165–191.S. V. Ludkovsky, W. Sprossig, “Spectral theory of super-differential operators of quaternion and octonion variables”, Adv. Appl. Clifford Algebr., 21:1 (2011), 165–191.F. Gursey, C.-H. Tze, On the Role of Division, Jordan and Related Algebras in Particle Physics, World Scientific Publ. Co., Singapore, 1996.F. Gursey, C.-H. Tze, On the Role of Division, Jordan and Related Algebras in Particle Physics, World Scientific Publ. Co., Singapore, 1996.A. V. Shatina, A. S. Borets, “A mathematical model of the gravitational potential of the planet taking into account tidal deformations”, Russian Technol J., 12:2 (2024), 77–89.A. V. Shatina, A. S. Borets, “A mathematical model of the gravitational potential of the planet taking into account tidal deformations”, Russian Technol J., 12:2 (2024), 77–89.S. V. Ludkovsky, “Integration of vector Sobolev type PDE over octonions”, Complex Var. Elliptic Equ., 61:7 (2016), 1014–1035.S. V. Ludkovsky, “Integration of vector Sobolev type PDE over octonions”, Complex Var. Elliptic Equ., 61:7 (2016), 1014–1035.S. V. Ludkovsky, “Integration of vector hydrodynamical partial differential equations over octonions”, Complex Var. Elliptic Equ., 58:5 (2013), 579–609.S. V. Ludkovsky, “Integration of vector hydrodynamical partial differential equations over octonions”, Complex Var. Elliptic Equ., 58:5 (2013), 579–609.V. N. Markov, A. V. Mikhalev, A. A. Nechaev, “Nonassociative algebraic structures in cryptography and coding”, J. Math. Sci. (N.Y.), 245:2 (2020), 178–196.V. N. Markov, A. V. Mikhalev, A. A. Nechaev, “Nonassociative algebraic structures in cryptography and coding”, J. Math. Sci. (N.Y.), 245:2 (2020), 178–196.

The authors declare that there are no conflicts of interest present.