mathematicsВестник российских университетов. МатематикаRussian Universities Reports. Mathematics2686-96672782-3342Тамбовский государственный университет имени Г.Р. Державина10.20310/2686-9667-2022-27-140-305-317mathematics-157Research ArticleНАУЧНЫЕ СТАТЬИORIGINAL ARTICLESО дифференцированиях в групповых алгебрах и других алгебраических структурахOn derivations in group algebras and other algebraic structureshttps://orcid.org/0000-0002-6878-0993АрутюновАндроник АрамовичArutyunovAndronick A.andronick.arutyunov@gmail.comФГБУН «Институт проблем управления им. В.А. Трапезникова» Российской академии наук; ФГАОУ ВО «Московский физико-технический институт (национальный исследовательский университет)»РоссияV. A. Trapeznikov Institute of Control Sciences of RAS; Moscow Institute of Physics and TechnologyRussian Federation20222612202227140305317Copyright © Арутюнов А.А., 20222022Арутюнов А.А.Arutyunov A.A.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/157

Работа посвящена обзору известных результатов, связанных с исследованиями дифференцирований в групповых алгебрах, бимодулях и других алгебраических структурах, а также различным обобщениям и вариациям данной задачи. Дается обзор результатов, посвященных дифференцированиям в алгебрах L1(G); в алгебрах фон Ноймана и в банаховых бимодулях. Обсуждаются алгебраические задачи, в частности дифференцирования в группах, (σ,τ)-дифференцирования и исчисление Фокса. Также дается обзор некоторых результатов, связанных с приложением к псевдодифференциальным операторам и построению символьного исчисления. В заключении описываются некоторые результаты, связанные с описанием дифференцирований, как характеров на группоиде присоединенного действия. Описаны также некоторые приложения: к теории кодирования, теории концов метрических пространств и грубой геометрии.

The work is devoted to a survey of known results related to the study of derivations in group algebras, bimodules and other algebraic structures, as well as to various generalizations and variations of this problem. A review of results on derivations in L1(G) algebras, in von Neumann algebras, and in Banach bimodules is given. Algebraic problems are discussed, in particular, derivations in groups, (σ,τ)-derivations, and the Fox calculus. A review of some results related to the application to pseudodifferential operators and the construction of the symbolic calculus is also given. In conclusion, some results related to the description of derivations as characters on the groupoid of the adjoint action are described. Some applications are also described: to coding theory, the theory of ends of metric spaces, and rough geometry.

дифференцированияоператорные алгебрыгрупповые алгебры(στ)-дифференцированияалгебры фон Нойманабанаховы бимодулиderivationsoperator algebrasgroup algebras(στ)-derivationsvon Neumann algebrasBanach bimodulesИсследование выполнено при поддержке Министерства науки и высшего образования Российской Федерации (госзадание № 075-00337-20-03), проект № 0714-2020-0005.The study was supported by the Ministry of Science and Higher Education of the Russian Federation (State Assignment No. 075-00337-20-03), Project No. 0714-2020-0005.ReferencesM. Nagumo, “Einige analytische Untersuchungen in linearen metrischen Ringen”, Japan J. Math., 13 (1936), 61–80.M. Nagumo, “Einige analytische Untersuchungen in linearen metrischen Ringen”, Japan J. Math., 13 (1936), 61–80.I. Gelfand, “Normierte Ringe”, Матем. сб., 9(51):1 (1941), 3–24.I. Gelfand, “Normierte Ringe”, Матем. сб., 9(51):1 (1941), 3–24.I. Gelfand, M. Neumark, “On the imbedding of normed rings into the ring of operators in Hilbert space”, Матем. сб., 12(54):2 (1943), 197–217.I. Gelfand, M. Neumark, “On the imbedding of normed rings into the ring of operators in Hilbert space”, Матем. сб., 12(54):2 (1943), 197–217.I. Kaplansky, “Modules over operator algebras”, Amer. J. Math., 75:4 (1953), 839–858.I. Kaplansky, “Modules over operator algebras”, Amer. J. Math., 75:4 (1953), 839–858.I. Kaplansky, “Projections in banach algebras”, Annals of Mathematics, 53:2 (1950), 235–249.I. Kaplansky, “Projections in banach algebras”, Annals of Mathematics, 53:2 (1950), 235–249.S. Sakai, “On a conjecture of Kaplansky”, Tohoku Math. J., 12:2 (1960), 31–33.S. Sakai, “On a conjecture of Kaplansky”, Tohoku Math. J., 12:2 (1960), 31–33.S. Sakai, “Derivations of W^*-algebras”, Annals of Mathematics, 83:2 (1966), 273–279.S. Sakai, “Derivations of W^*-algebras”, Annals of Mathematics, 83:2 (1966), 273–279.S. Sakai, “Derivations of simple C^*-algebras”, J. Functional Analysis, 2:2 (1968), 202–206.S. Sakai, “Derivations of simple C^*-algebras”, J. Functional Analysis, 2:2 (1968), 202–206.S. Sakai, C^*-algebras and W^*-algebras, Classics in Mathematics, Springer-Verlag Berlin Heidelberg, Berlin, 1998.S. Sakai, C^*-algebras and W^*-algebras, Classics in Mathematics, Springer-Verlag Berlin Heidelberg, Berlin, 1998.C.A. Akemann, G.A. Elliott, G.K. Pedersen, J. Tomiyama, “Derivations and multipliers of C^* -algebras”, Amer. J. Math., 98:3 (1976), 679–708.C.A. Akemann, G.A. Elliott, G.K. Pedersen, J. Tomiyama, “Derivations and multipliers of C^* -algebras”, Amer. J. Math., 98:3 (1976), 679–708.C.A. Akemann, G.K. Pedersen, “Central sequences and inner derivations of separable C^*-algebras”, Amer. J. Math., 101 (1979), 1047–1061.C.A. Akemann, G.K. Pedersen, “Central sequences and inner derivations of separable C^*-algebras”, Amer. J. Math., 101 (1979), 1047–1061.B.E. Johnson, A.M. Sinclair, “Continuity of derivations and a problem of Kaplansky”, Amer. J. Math., 90:4 (1968), 1067–1073.B.E. Johnson, A.M. Sinclair, “Continuity of derivations and a problem of Kaplansky”, Amer. J. Math., 90:4 (1968), 1067–1073.G.A. Elliott, “Some C^*-algebras with outer derivations, III”, Annals of Mathematics, 106 (1977), 121–143.G.A. Elliott, “Some C^*-algebras with outer derivations, III”, Annals of Mathematics, 106 (1977), 121–143.C^*-Algebras and Their Automorphism Groups. V. 14, ed. G.K. Pedersen, Academic Press, 1979.C^*-Algebras and Their Automorphism Groups. V. 14, ed. G.K. Pedersen, Academic Press, 1979.I. Kaplansky, “Derivations of Banach Algebras”, Seminars on Analytic Functions, Princeton Univ. Press, Princeton, N.J.: Seminar V. Analytic functions as related to Banach algebras, II, 1958, 254–258.I. Kaplansky, “Derivations of Banach Algebras”, Seminars on Analytic Functions, Princeton Univ. Press, Princeton, N.J.: Seminar V. Analytic functions as related to Banach algebras, II, 1958, 254–258.S. Sakai, Operator Algebras in Dynamical Systems, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1991.S. Sakai, Operator Algebras in Dynamical Systems, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1991.R.V. Kadison, “Derivations of operator algebras”, Ann. of Math., 83 (1966), 280–293.R.V. Kadison, “Derivations of operator algebras”, Ann. of Math., 83 (1966), 280–293.R.V. Kadison, J.R. Ringrose, “Derivations of operator group algebras”, Amer. J. Math., 88:3 (1966), 562–576.R.V. Kadison, J.R. Ringrose, “Derivations of operator group algebras”, Amer. J. Math., 88:3 (1966), 562–576.R.V. Kadison, J.R. Ringrose, “Derivations and automorphisms of operator algebras”, Commun. Math. Phys., 4 (1967), 32–63.R.V. Kadison, J.R. Ringrose, “Derivations and automorphisms of operator algebras”, Commun. Math. Phys., 4 (1967), 32–63.B.E. Johnson, J.R. Ringrose, “Derivations of operator algebras and discrete group algebras”, Bull. London Math. Soc., 1 (1969), 70–74.B.E. Johnson, J.R. Ringrose, “Derivations of operator algebras and discrete group algebras”, Bull. London Math. Soc., 1 (1969), 70–74.B.E. Johnson, Cohomology in Banach Algebras, Mamoirs of the American Mathematical Society, 127, American Mathematical Society, Providence, Rhode Island, USA, 1972.B.E. Johnson, Cohomology in Banach Algebras, Mamoirs of the American Mathematical Society, 127, American Mathematical Society, Providence, Rhode Island, USA, 1972.H.G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, 24, Oxford University Press, New York, 2000, 928 pp.H.G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, 24, Oxford University Press, New York, 2000, 928 pp.B.E. Johnson, “The derivation problem for group algebras of connected locally compact groups”, Journal of the London Mathematical Society, 63:2 (2001), 441–452.B.E. Johnson, “The derivation problem for group algebras of connected locally compact groups”, Journal of the London Mathematical Society, 63:2 (2001), 441–452.V. Losert, “The derivation problem for group algebras”, Annals of Mathematics, 168:1 (2008), 221–246.V. Losert, “The derivation problem for group algebras”, Annals of Mathematics, 168:1 (2008), 221–246.B.E. Johnson, S. Parrott, “Operators commuting with a von Neumann algebras modulo the set of compact operators”, Journal of Functional Analysis, 11 (1972), 39–61.B.E. Johnson, S. Parrott, “Operators commuting with a von Neumann algebras modulo the set of compact operators”, Journal of Functional Analysis, 11 (1972), 39–61.S. Popa, “The commutant modulo the set of compact operators of a von Neumann algebra”, Journal of Functional Analysis, 71 (1987), 393–408.S. Popa, “The commutant modulo the set of compact operators of a von Neumann algebra”, Journal of Functional Analysis, 71 (1987), 393–408.A. Ber, J. Huang, G. Letvina, F. Sukochev, “Derivations with values in ideals of semifinite von Neumann algebras”, Journal of Functional Analysis, 272:12 (2017), 4984–4997.A. Ber, J. Huang, G. Letvina, F. Sukochev, “Derivations with values in ideals of semifinite von Neumann algebras”, Journal of Functional Analysis, 272:12 (2017), 4984–4997.J. Huang, Derivations with values into ideals of a semifinite von Neumann algebra, Australia’s Global University, School of Mathematics and Statistics Faculty of Science, 2019, http://hdl.handle.net/1959.4/63344.J. Huang, Derivations with values into ideals of a semifinite von Neumann algebra, Australia’s Global University, School of Mathematics and Statistics Faculty of Science, 2019, http://hdl.handle.net/1959.4/63344.A. Ber, J. Huang, K. Kudaybergenov, F. Sukochev, “Non-existence of translation-invariant derivations on algebras of measurable functions”, Quaestiones Mathematicae, 45:10 (2022).A. Ber, J. Huang, K. Kudaybergenov, F. Sukochev, “Non-existence of translation-invariant derivations on algebras of measurable functions”, Quaestiones Mathematicae, 45:10 (2022).A.Ya. Helemskii, The Homology of Banach and Topological Algebras, Mathematics and its Applications, 41, Kluwer Academic Publishers, Amsterdam, 1989.A.Ya. Helemskii, The Homology of Banach and Topological Algebras, Mathematics and its Applications, 41, Kluwer Academic Publishers, Amsterdam, 1989.A.Ya. Helemskii, Banach and Locally Convex Algebras, Oxford Mathematical Monographs, Claredon Press, 1993.A.Ya. Helemskii, Banach and Locally Convex Algebras, Oxford Mathematical Monographs, Claredon Press, 1993.G. Hochschild, “On the cohomology groups of an associative algebra”, Annals of Mathematics, 46:1 (1945), 58–67.G. Hochschild, “On the cohomology groups of an associative algebra”, Annals of Mathematics, 46:1 (1945), 58–67.G. Hochschild, B. Kostant, A. Rosenberg, “Differential forms on regular affine algebras”, Trans. Amer. Math. Soc, 102:3 (1962), 383–408.G. Hochschild, B. Kostant, A. Rosenberg, “Differential forms on regular affine algebras”, Trans. Amer. Math. Soc, 102:3 (1962), 383–408.S. Witherspoon, An Introduction to Hochschild Cohomology, Texas AM University, 2017.S. Witherspoon, An Introduction to Hochschild Cohomology, Texas AM University, 2017.D. Burghelea, “The cyclic homology of the group rings”, Commentarii Mathematici Helvetici, 60 (1985), 354–365.D. Burghelea, “The cyclic homology of the group rings”, Commentarii Mathematici Helvetici, 60 (1985), 354–365.A.S. Mischenko, “Derivations of group algebras and Hochschild cohomology”, 2020, http://arxiv.org/abs/1811.02439.A.S. Mischenko, “Derivations of group algebras and Hochschild cohomology”, 2020, http://arxiv.org/abs/1811.02439.A.S. Mischenko, “Geometric description of the Hochschild Cohomology of group algebras”, Topology, Geometry, and Dynamics: Rokhlin Memorial, Contemporary Mathematics, 772, AMS, Providence, R.I., etc., United States, 2021, 267–279.A.S. Mischenko, “Geometric description of the Hochschild Cohomology of group algebras”, Topology, Geometry, and Dynamics: Rokhlin Memorial, Contemporary Mathematics, 772, AMS, Providence, R.I., etc., United States, 2021, 267–279.A.S. Mischenko, “Description of outer derivations of the group algebras”, Topology and its Applications, 275 (2020), 107013.A.S. Mischenko, “Description of outer derivations of the group algebras”, Topology and its Applications, 275 (2020), 107013.M. Lorentz, R. Willett, “Bounded derivations on uniform Roe algebras”, Rocky Mountain J. Math., 50 (2020), 1747–1758.M. Lorentz, R. Willett, “Bounded derivations on uniform Roe algebras”, Rocky Mountain J. Math., 50 (2020), 1747–1758.V. Manuilov, “On Hochschild homology of uniform Roe algebras with coefficients in uniform Roe bimodules”, https://arxiv.org/abs/2201.06488.V. Manuilov, “On Hochschild homology of uniform Roe algebras with coefficients in uniform Roe bimodules”, https://arxiv.org/abs/2201.06488.M.K. Smith, “Derivations of group algebras of finitely-generated, torsion-free, nilpotent groups”, Houston J. Math., 4 (1978), 277–288.M.K. Smith, “Derivations of group algebras of finitely-generated, torsion-free, nilpotent groups”, Houston J. Math., 4 (1978), 277–288.E. Spiegel, “Derivations of integral group rings”, Communications in Algebra, 22:8 (1994), 2955–2959.E. Spiegel, “Derivations of integral group rings”, Communications in Algebra, 22:8 (1994), 2955–2959.D. Boucher, D. Ulmer, “Linear codes using skew polynomials with automorphisms and derivations”, Des. Codes Cryptogr., 70 (2014), 405–431.D. Boucher, D. Ulmer, “Linear codes using skew polynomials with automorphisms and derivations”, Des. Codes Cryptogr., 70 (2014), 405–431.L. Creedon, K. Hughes, “Derivations on group algebras with coding theory applications”, Finite Fields and Their Applications, 56 (2019), 247–265.L. Creedon, K. Hughes, “Derivations on group algebras with coding theory applications”, Finite Fields and Their Applications, 56 (2019), 247–265.O.D. Artemovych, V.A. Bovdi, M.A. Salim, “Derivations of group rings”, 2020, https://arxiv.org/abs/2003.01346.O.D. Artemovych, V.A. Bovdi, M.A. Salim, “Derivations of group rings”, 2020, https://arxiv.org/abs/2003.01346.Y. Rao, S. Kosari, A. Khan, N. Abbasizadeh, “A Study on special kinds of derivations in ordered hyperrings”, Symmetry, 14 (2022), 2205.Y. Rao, S. Kosari, A. Khan, N. Abbasizadeh, “A Study on special kinds of derivations in ordered hyperrings”, Symmetry, 14 (2022), 2205.D. Chaudhuri, “(σ,τ)-derivations of group rings”, Comm. Algebra, 47:9 (2019), 3800–3807.D. Chaudhuri, “(σ,τ)-derivations of group rings”, Comm. Algebra, 47:9 (2019), 3800–3807.O. Ore, “Theory of non-commutative polynomials”, Ann. Math., 34 (1933), 480–508.O. Ore, “Theory of non-commutative polynomials”, Ann. Math., 34 (1933), 480–508.V.K. Kharchenko, Automorphisms and Derivations of Associative Rings, Mathematics and its Applications, 69, Kluwer Academic Publishers, Amsterdam, 1991.V.K. Kharchenko, Automorphisms and Derivations of Associative Rings, Mathematics and its Applications, 69, Kluwer Academic Publishers, Amsterdam, 1991.F.A. Berezin, Introduction to Algebra and Analysis with Anticommuting Variables, Moscow University Press, Moscow, 1983.F.A. Berezin, Introduction to Algebra and Analysis with Anticommuting Variables, Moscow University Press, Moscow, 1983.L.A. Bokut, I.V. Lvov, V.K. Kharchenko, “Noncommutative Rings”, Algebra – 2, Itogi Nauki I Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 18, VINITI, Moscow, 1988, 5–116.L.A. Bokut, I.V. Lvov, V.K. Kharchenko, “Noncommutative Rings”, Algebra – 2, Itogi Nauki I Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 18, VINITI, Moscow, 1988, 5–116.P.M. Cohn, Difference Algebra, Interscience Publ., New York, 1965.P.M. Cohn, Difference Algebra, Interscience Publ., New York, 1965.P.M. Cohn, Skew Fields Constructions, London Mathematical Society Lecture Note Series, 27, Cambridge University Press, Cambridge, 1977.P.M. Cohn, Skew Fields Constructions, London Mathematical Society Lecture Note Series, 27, Cambridge University Press, Cambridge, 1977.K.R. Goodearl, R.B. Warfield, Jr, An Introduction to Non-Commutative Noetherian Rings, London Mathematical Society Student Texts, 16, Cambridge University Press, Cambridge, 1989.K.R. Goodearl, R.B. Warfield, Jr, An Introduction to Non-Commutative Noetherian Rings, London Mathematical Society Student Texts, 16, Cambridge University Press, Cambridge, 1989.D.A. Jordan, “Iterated skew polynomial rings and quantum groups”, Journal of Algebra, 156 (1993), 194–218.D.A. Jordan, “Iterated skew polynomial rings and quantum groups”, Journal of Algebra, 156 (1993), 194–218.V. Kac, P. Cheung, Quantum Calculus, Universitext (UTX), Springer, New York, 2002.V. Kac, P. Cheung, Quantum Calculus, Universitext (UTX), Springer, New York, 2002.C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155, 1st ed., Springer-Verlag, New York, 1995.C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155, 1st ed., Springer-Verlag, New York, 1995.T.Y. Lam, A. Leroy, “Algebraic conjugacy classes and skew polynomial rings”, Perspectives in Ring Theory, NATO ASI Series, 233, eds. F. van Oystaeyen, L. Le Bruyn, Springer, Dordrecht, 1988, 153-203.T.Y. Lam, A. Leroy, “Algebraic conjugacy classes and skew polynomial rings”, Perspectives in Ring Theory, NATO ASI Series, 233, eds. F. van Oystaeyen, L. Le Bruyn, Springer, Dordrecht, 1988, 153-203.J.T. Hartwig, D. Larsson, S.D. Silvestrov, “Deformations of Lie algebras using σ-derivations”, Journal of Algebra, 295:2 (2006), 314–361.J.T. Hartwig, D. Larsson, S.D. Silvestrov, “Deformations of Lie algebras using σ-derivations”, Journal of Algebra, 295:2 (2006), 314–361.D. Larsson, S.D. Silvestrov, “Quasi-deformations of sl_2 (F) using twisted derivations”, Comm. Algebra, 35:12 (2007), 4303–4318.D. Larsson, S.D. Silvestrov, “Quasi-deformations of sl_2 (F) using twisted derivations”, Comm. Algebra, 35:12 (2007), 4303–4318.Yu.I. Manin, Topics in Non-Commutative Geometry, Porter Lectures, Princeton University Press, 1991.Yu.I. Manin, Topics in Non-Commutative Geometry, Porter Lectures, Princeton University Press, 1991.L.R. Rosenberg, Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and Its Applications, 330, Springer, Dordrecht, 1995.L.R. Rosenberg, Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and Its Applications, 330, Springer, Dordrecht, 1995.J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, 30, Amer. Math. Society, Providence, Rhode Island, 1987.J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, 30, Amer. Math. Society, Providence, Rhode Island, 1987.Heide Gluesing-Luerssen, “Skew-Polynomial Rings and Skew-Cyclic Codes”, 2019, https://arxiv.org/abs/1902.03516.Heide Gluesing-Luerssen, “Skew-Polynomial Rings and Skew-Cyclic Codes”, 2019, https://arxiv.org/abs/1902.03516.D. Larsson, S.D. Silvestrov, “Quasi-hom-Lie algebras, central extensions and 2 -cocycle-like identities”, Journal of Algebra, 288:2 (2005), 321–344.D. Larsson, S.D. Silvestrov, “Quasi-hom-Lie algebras, central extensions and 2 -cocycle-like identities”, Journal of Algebra, 288:2 (2005), 321–344.O. Elchinger, K. Lundengard, A. Makhlouf, S. D. Silvestrov, “Brackets with (τ,σ)-derivations and (p; q)-deformations of Witt and Virasoro algebras”, Forum Math., 28:4 (2016), 657–673.O. Elchinger, K. Lundengard, A. Makhlouf, S. D. Silvestrov, “Brackets with (τ,σ)-derivations and (p; q)-deformations of Witt and Virasoro algebras”, Forum Math., 28:4 (2016), 657–673.G. Song, C. Xia, “Simple deformed Witt algebras”, Algebra Colloquium, 18:3 (2011), 533–540.G. Song, C. Xia, “Simple deformed Witt algebras”, Algebra Colloquium, 18:3 (2011), 533–540.G. Song, Y. Wu, B. Xin, “The σ-derivations of C[x^(±1),y^(±1) ]”, Algebra Colloquium, 22:2 (2015), 251–258.G. Song, Y. Wu, B. Xin, “The σ-derivations of C[x^(±1),y^(±1) ]”, Algebra Colloquium, 22:2 (2015), 251–258.E. Gordji, “A characterization of (σ,τ)-derivations on von Neumann algebras”, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 73 (2009).E. Gordji, “A characterization of (σ,τ)-derivations on von Neumann algebras”, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 73 (2009).R.H. Fox, “Free differential calculus. I: Derivation in the free group ring”, The Annals of Mathematics, 57:3 (1953), 547–560.R.H. Fox, “Free differential calculus. I: Derivation in the free group ring”, The Annals of Mathematics, 57:3 (1953), 547–560.R.H. Fox, “Free differential calculus. II: The isomorphism problem of groups”, The Annals of Mathematics, 59:2 (1954), 196–210.R.H. Fox, “Free differential calculus. II: The isomorphism problem of groups”, The Annals of Mathematics, 59:2 (1954), 196–210.R.H. Fox, “Free differential calculus III. Subgroups”, The Annals of Mathematics, 64:3 (1956), 407–419.R.H. Fox, “Free differential calculus III. Subgroups”, The Annals of Mathematics, 64:3 (1956), 407–419.K.T. Chen, R.H. Fox, R.C. Lyndon, “Free differential calculus IV. The quotient groups of the lower central series”, The Annals of Mathematics, 68:1 (1958), 81–95.K.T. Chen, R.H. Fox, R.C. Lyndon, “Free differential calculus IV. The quotient groups of the lower central series”, The Annals of Mathematics, 68:1 (1958), 81–95.R.H. Fox, “Free differential calculus V. The Alexander matrices re-examined”, The Annals of Mathematics, 71:3 (1960), 408–422.R.H. Fox, “Free differential calculus V. The Alexander matrices re-examined”, The Annals of Mathematics, 71:3 (1960), 408–422.G. Massuyeau, V. Turaev, “Quasi-Poisson structures on representation spaces of surfaces”, 2012, https://arxiv.org/abs/1205.4898.G. Massuyeau, V. Turaev, “Quasi-Poisson structures on representation spaces of surfaces”, 2012, https://arxiv.org/abs/1205.4898.А.Н. Паршин, “О кольце формальных псевдодифференциальных операторов”, Алгебра. Топология. Дифференциальные уравнения и их приложения, Сборник статей. К 90-летию со дня рождения академика Льва Семеновича Понтрягина, Труды МИАН, 224, Наука, МАИК «Наука/Интерпериодика», М., 1999, 291–305; англ. пер.:A. N. Parshin, “On a ring of formal pseudodifferential operators”, Proc. Steklov Inst. Math., 224 (1999), 266–280.А.Н. Паршин, “О кольце формальных псевдодифференциальных операторов”, Алгебра. Топология. Дифференциальные уравнения и их приложения, Сборник статей. К 90-летию со дня рождения академика Льва Семеновича Понтрягина, Труды МИАН, 224, Наука, МАИК «Наука/Интерпериодика», М., 1999, 291–305; англ. пер.:A. N. Parshin, “On a ring of formal pseudodifferential operators”, Proc. Steklov Inst. Math., 224 (1999), 266–280.A. Zheglov, “Schur-Sato theory for quasi-elliptic rings”, 2022, https://arxiv.org/abs/2205.06790.A. Zheglov, “Schur-Sato theory for quasi-elliptic rings”, 2022, https://arxiv.org/abs/2205.06790.В.П. Маслов, Операторные методы, Наука, М., 1973. [V.P. Maslov, Operator Methods, Nauka Publ., Moscow, 1973 (In Russian)].В.П. Маслов, Операторные методы, Наука, М., 1973. [V.P. Maslov, Operator Methods, Nauka Publ., Moscow, 1973 (In Russian)].М.А. Шубин, Псевдодифференциальные операторы и спектральная теория, Добросвет, Москва. [M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Dobrosvet Publ., Moscow (In Russian)].М.А. Шубин, Псевдодифференциальные операторы и спектральная теория, Добросвет, Москва. [M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Dobrosvet Publ., Moscow (In Russian)].M.F. Atiyah, I.M. Singer, “The index of elliptic operators on compact manifolds”, Bull. Amer. Math. Soc., 69:3 (1963), 422–433.M.F. Atiyah, I.M. Singer, “The index of elliptic operators on compact manifolds”, Bull. Amer. Math. Soc., 69:3 (1963), 422–433.M.F. Atiyah, I.M. Singer, “The index of elliptic operators I”, Annals of Mathematics, 87:3 (1968), 484–530.M.F. Atiyah, I.M. Singer, “The index of elliptic operators I”, Annals of Mathematics, 87:3 (1968), 484–530.I.N. Herstein, “Sui commutatori degli anelli semplici”, Seminario Mat. e. Fis. di Milano, 33 (1963), 80–86.I.N. Herstein, “Sui commutatori degli anelli semplici”, Seminario Mat. e. Fis. di Milano, 33 (1963), 80–86.V.K. Kharchenko, “Differential identities of prime rings”, Algebra and Logic, 17 (1978), 155–168.V.K. Kharchenko, “Differential identities of prime rings”, Algebra and Logic, 17 (1978), 155–168.P. Grzeszczuk, “On nilpotent derivations of semiprime rings”, Journal of Algebra, 149 (1992), 313–321.P. Grzeszczuk, “On nilpotent derivations of semiprime rings”, Journal of Algebra, 149 (1992), 313–321.C. Chen-Lian, L. Tsiu-Kwen, “Nilpotent derivations”, Journal of Algebra, 287:2 (2005), 381–401.C. Chen-Lian, L. Tsiu-Kwen, “Nilpotent derivations”, Journal of Algebra, 287:2 (2005), 381–401.D. Wright, “On the Jacobian Conjecture”, Illinois J. Math, 25 (1981), 423–440.D. Wright, “On the Jacobian Conjecture”, Illinois J. Math, 25 (1981), 423–440.M. Nagata, “On the fourteenth problem of Hilbert”, Proceedings of the International Congress of Mathematicians, Cambridge University Press., Edinburgh, 1958, 459–462.M. Nagata, “On the fourteenth problem of Hilbert”, Proceedings of the International Congress of Mathematicians, Cambridge University Press., Edinburgh, 1958, 459–462.M. Nagata, Lectures on the fourteenth problem of Hilbert, 31, Tata Institute of Fundamental Research Lectures on Mathematics, Bombai, 1965, 65 pp.M. Nagata, Lectures on the fourteenth problem of Hilbert, 31, Tata Institute of Fundamental Research Lectures on Mathematics, Bombai, 1965, 65 pp.M. Ferrero, Y. Lequain, A. Nowicki, “A note on locally nilpotent derivations”, Journal of Pure and Applied Algebra, 79 (1992), 45–50.M. Ferrero, Y. Lequain, A. Nowicki, “A note on locally nilpotent derivations”, Journal of Pure and Applied Algebra, 79 (1992), 45–50.D. Daigle, “Locally nilpotent derivations and the structure of rings”, Journal of Pure and Applied Algebra, 224:4 (2020), 106–201.D. Daigle, “Locally nilpotent derivations and the structure of rings”, Journal of Pure and Applied Algebra, 224:4 (2020), 106–201.S. Maubach, “The commuting derivations conjecture”, Journal of Pure and Applied Algebra, 179:1–2 (2003), 159–168.S. Maubach, “The commuting derivations conjecture”, Journal of Pure and Applied Algebra, 179:1–2 (2003), 159–168.Jiantao Li, Xiankun Du, “Pairwise commuting derivations of polynomial rings”, Linear Algebra and its Applications, 436:7, 2375–2379.Jiantao Li, Xiankun Du, “Pairwise commuting derivations of polynomial rings”, Linear Algebra and its Applications, 436:7, 2375–2379.А.А. Арутюнов, А.С. Мищенко, А.И. Штерн, “Деривации групповых алгебр”, Фундамент. И прикл. матем., 21:6 (2016), 65–78; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, A.I. Shtern, “Derivations of group algebras”, Journal of Mathematical Sciences, 248 (2020), 709–718.А.А. Арутюнов, А.С. Мищенко, А.И. Штерн, “Деривации групповых алгебр”, Фундамент. И прикл. матем., 21:6 (2016), 65–78; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, A.I. Shtern, “Derivations of group algebras”, Journal of Mathematical Sciences, 248 (2020), 709–718.А.А. Арутюнов, А.С. Мищенко, “Гладкая версия проблемы Джонсона о деривациях групповых алгебр”, Матем. сб., 210:6 (2019), 3–29; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, “A smooth version of Johnson’s problem on derivations of group algebras”, Sb. Math., 210:6 (2019), 756–782.А.А. Арутюнов, А.С. Мищенко, “Гладкая версия проблемы Джонсона о деривациях групповых алгебр”, Матем. сб., 210:6 (2019), 3–29; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, “A smooth version of Johnson’s problem on derivations of group algebras”, Sb. Math., 210:6 (2019), 756–782.А.А. Арутюнов, “Алгебра дифференцирований в некоммутативных групповых алгебрах”, Дифференциальные уравнения и динамические системы, Сборник статей, Труды МИАН, 308, МИАН, М., 2020, 28–41; англ. пер.: A.A. Arutyunov, “Derivation algebra in noncommutative group algebras”, Proc. Steklov Inst. Math., 308 (2020), 22–34.А.А. Арутюнов, “Алгебра дифференцирований в некоммутативных групповых алгебрах”, Дифференциальные уравнения и динамические системы, Сборник статей, Труды МИАН, 308, МИАН, М., 2020, 28–41; англ. пер.: A.A. Arutyunov, “Derivation algebra in noncommutative group algebras”, Proc. Steklov Inst. Math., 308 (2020), 22–34.A.A. Arutyunov, A.V. Alekseev, “Cohomology of n -categories and derivations in group algebras”, Topology and its Applications, 275 (2019).A.A. Arutyunov, A.V. Alekseev, “Cohomology of n -categories and derivations in group algebras”, Topology and its Applications, 275 (2019).A. Arutyunov, “A combinatorial view on derivations in bimodules”, 2022, https://arxiv.org/abs/2208.05478.A. Arutyunov, “A combinatorial view on derivations in bimodules”, 2022, https://arxiv.org/abs/2208.05478.А.А. Арутюнов, А.С. Мищенко, “Редукция исчисления псевдодифференциальных операторов на некомпактном многообразии к исчислению на компактном многообразии удвоенной размерности”, Матем. заметки, 94:4 (2013), 488–505; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, “Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension”, Math. Notes, 94:4 (2013), 455–469.А.А. Арутюнов, А.С. Мищенко, “Редукция исчисления псевдодифференциальных операторов на некомпактном многообразии к исчислению на компактном многообразии удвоенной размерности”, Матем. заметки, 94:4 (2013), 488–505; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, “Reduction of the calculus of pseudodifferential operators on a noncompact manifold to the calculus on a compact manifold of doubled dimension”, Math. Notes, 94:4 (2013), 455–469.А.А. Арутюнов, А.С. Мищенко, “Редукция ПДО исчисления на некомпактном многообразии к компактному многообразию удвоенной размерности”, Доклад. РАН, 451:4 (2013), 369–373; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, “Reduction of PDO calculus on a noncompact manifold to a double-dimensional compact manifold”, Doklady Mathematics, 88:1 (2013), 427–430.А.А. Арутюнов, А.С. Мищенко, “Редукция ПДО исчисления на некомпактном многообразии к компактному многообразию удвоенной размерности”, Доклад. РАН, 451:4 (2013), 369–373; англ. пер.: A.A. Arutyunov, A.S. Mishchenko, “Reduction of PDO calculus on a noncompact manifold to a double-dimensional compact manifold”, Doklady Mathematics, 88:1 (2013), 427–430.А.А. Арутюнов, “Редукция нелокальных псевдодифференциальных операторов на некомпактном многообразии к классическим псевдодифференциальным операторам на компактном многообразии удвоенной размерности”, Матем. заметки, 97:4 (2015), 493–502; англ. пер.: A.A. Arutyunov, “Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold”, Math. Notes, 97:4 (2015), 502–509.А.А. Арутюнов, “Редукция нелокальных псевдодифференциальных операторов на некомпактном многообразии к классическим псевдодифференциальным операторам на компактном многообразии удвоенной размерности”, Матем. заметки, 97:4 (2015), 493–502; англ. пер.: A.A. Arutyunov, “Reduction of nonlocal pseudodifferential operators on a noncompact manifold to classical pseudodifferential operators on a double-dimensional compact manifold”, Math. Notes, 97:4 (2015), 502–509.

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