On derivations in group algebras and other algebraic structures

A. A. Arutyunov

Vestnik rossijskih universitetov. Matematika, Tome 27 (2022) no. 140, pp. 305-317

Résumé

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 $L_1(G)$ algebras, in von Neumann algebras, and in Banach bimodules is given. Algebraic problems are discussed, in particular, derivations in groups, $(\sigma,\tau)$-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.

Détail

Keywords: derivations, operator algebras, $(\sigma,\tau)$-derivations
Mots-clés : group algebras, von Neumann algebras, Banach bimodules.

Bibliographie

[1] M. Nagumo, “Einige analytische Untersuchungen in linearen metrischen Ringen”, Japan J. Math., 13 (1936), 61–80

[2] I. Gelfand, “Normierte Ringe”, Matem. sb., 9(51):1 (1941), 3–24 | MR | Zbl

[3] I. Gelfand, M. Neumark, “On the imbedding of normed rings into the ring of operators in Hilbert space”, Matem. sb., 12(54):2 (1943), 197–217 | MR | Zbl

[4] I. Kaplansky, “Modules over operator algebras”, Amer. J. Math., 75:4 (1953), 839–858

[5] I. Kaplansky, “Projections in banach algebras”, Annals of Mathematics, 53:2 (1950), 235–249

[6] S. Sakai, “On a conjecture of Kaplansky”, Tohoku Math. J., 12:2 (1960), 31–33

[7] S. Sakai, “Derivations of $W^*$-algebras”, Annals of Mathematics, 83:2 (1966), 273–279

[8] S. Sakai, “Derivations of simple $C^*$-algebras”, J. Functional Analysis, 2:2 (1968), 202–206

[9] S. Sakai, $C^*$-algebras and $W^*$-algebras, Classics in Mathematics, Springer-Verlag Berlin Heidelberg, Berlin, 1998

[10] 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

[11] C. A. Akemann, G. K. Pedersen, “Central sequences and inner derivations of separable $C^*$-algebras”, Amer. J. Math., 101 (1979), 1047–1061

[12] B. E. Johnson, A. M. Sinclair, “Continuity of derivations and a problem of Kaplansky”, Amer. J. Math., 90:4 (1968), 1067–1073

[13] G. A. Elliott, “Some $C^*$-algebras with outer derivations, III”, Annals of Mathematics, 106 (1977), 121–143

[14] $C^*$-Algebras and Their Automorphism Groups, v. 14, ed. G. K. Pedersen, Academic Press, 1979

[15] I. Kaplansky, “Derivations of Banach Algebras”, Analytic functions as related to Banach algebras, v. II, Seminars on Analytic Functions, V, Princeton Univ. Press, Princeton, N.J., 1958, 254–258

[16] S. Sakai, Operator Algebras in Dynamical Systems, The theory of unbounded derivations in $C^*$-algebras https://doi.org/10.1017/CBO9780511662218, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1991

[17] R. V. Kadison, “Derivations of operator algebras”, Ann. of Math., 83 (1966), 280–293

[18] R. V. Kadison, J. R. Ringrose, “Derivations of operator group algebras”, Amer. J. Math., 88:3 (1966), 562–576

[19] R. V. Kadison, J. R. Ringrose, “Derivations and automorphisms of operator algebras”, Commun. Math. Phys., 4 (1967), 32–63

[20] B. E. Johnson, J. R. Ringrose, “Derivations of operator algebras and discrete group algebras”, Bull. London Math. Soc., 1 (1969), 70–74

[21] B. E. Johnson, Cohomology in Banach Algebras, Mamoirs of the American Mathematical Society, 127, American Mathematical Society, Providence, Rhode Island, USA, 1972

[22] H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, 24, Oxford University Press, New York, 2000, 928 pp.

[23] 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

[24] V. Losert, “The derivation problem for group algebras”, Annals of Mathematics, 168:1 (2008), 221–246

[25] 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

[26] S. Popa, “The commutant modulo the set of compact operators of a von Neumann algebra”, Journal of Functional Analysis, 71 (1987), 393–408

[27] 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

[28] 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

[29] A. Ber, J. Huang, K. Kudaybergenov, F. Sukochev, “Non-existence of translation-invariant derivations on algebras of measurable functions”, Quaestiones Mathematicae, 45:10 (2022) | DOI

[30] A. Ya. Helemskii, The Homology of Banach and Topological Algebras, Mathematics and its Applications, 41, Kluwer Academic Publishers, Amsterdam, 1989

[31] A. Ya. Helemskii, Banach and Locally Convex Algebras, Oxford Mathematical Monographs, Claredon Press, 1993

[32] G. Hochschild, “On the cohomology groups of an associative algebra”, Annals of Mathematics, 46:1 (1945), 58–67

[33] G. Hochschild, B. Kostant, A. Rosenberg, “Differential forms on regular affine algebras”, Trans. Amer. Math. Soc, 102:3 (1962), 383–408, Springer Science+Business Media, New York | DOI

[34] S. Witherspoon, An Introduction to Hochschild Cohomology, Texas AM University, 2017

[35] D. Burghelea, “The cyclic homology of the group rings”, Commentarii Mathematici Helvetici, 60 (1985), 354–365

[36] A. S. Mischenko, “Derivations of group algebras and Hochschild cohomology”, Differential Equations on Manifolds and Mathematical Physics – Dedicated to the Memory of Boris Sternin, 2020, arXiv: 1811.02439

[37] 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

[38] A. S. Mischenko, “Description of outer derivations of the group algebras”, Topology and its Applications, 275 (2020), 107013, arXiv: 1811.02439

[39] M. Lorentz, R. Willett, “Bounded derivations on uniform Roe algebras”, Rocky Mountain J. Math., 50 (2020), 1747–1758

[40] V. Manuilov, On Hochschild homology of uniform Roe algebras with coefficients in uniform Roe bimodules, arXiv: 2201.06488

[41] M. K. Smith, “Derivations of group algebras of finitely-generated, torsion-free, nilpotent groups”, Houston J. Math., 4 (1978), 277–288

[42] E. Spiegel, “Derivations of integral group rings”, Communications in Algebra, 22:8 (1994), 2955–2959

[43] D. Boucher, D. Ulmer, “Linear codes using skew polynomials with automorphisms and derivations”, Des. Codes Cryptogr., 70 (2014), 405–431

[44] L. Creedon, K. Hughes, “Derivations on group algebras with coding theory applications”, Finite Fields and Their Applications, 56 (2019), 247–265

[45] O. D. Artemovych, V. A. Bovdi, M. A. Salim, Derivations of group rings, 2020, arXiv: 2003.01346

[46] Y. Rao, S. Kosari, A. Khan, N. Abbasizadeh, “A Study on special kinds of derivations in ordered hyperrings”, Symmetry, 14 (2022), 2205 https://www.researchgate.net/publication/364385863_A_Study_on_Special_Kinds_of_Derivations_in_Ordered_Hyperrings

[47] D. Chaudhuri, “$(\sigma,\tau)$-derivations of group rings”, Comm. Algebra, 47:9 (2019), 3800–3807 | DOI

[48] O. Ore, “Theory of non-commutative polynomials”, Ann. Math., 34 (1933), 480–508

[49] V. K. Kharchenko, Automorphisms and Derivations of Associative Rings, Mathematics and its Applications, 69, Kluwer Academic Publishers, Amsterdam, 1991

[50] F. A. Berezin, Introduction to Algebra and Analysis with Anticommuting Variables, Moscow University Press, Moscow, 1983

[51] 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

[52] P. M. Cohn, Difference Algebra, Interscience Publ., New York, 1965

[53] P. M. Cohn, Skew Fields Constructions, London Mathematical Society Lecture Note Series, 27, Cambridge University Press, Cambridge, 1977

[54] 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

[55] D. A. Jordan, “Iterated skew polynomial rings and quantum groups”, Journal of Algebra, 156 (1993), 194–218

[56] V. Kac, P. Cheung, Quantum Calculus, Universitext (UTX), Springer, New York, 2002

[57] C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155, 1st ed., Springer-Verlag, New York, 1995

[58] 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

[59] J. T. Hartwig, D. Larsson, S. D. Silvestrov, “Deformations of Lie algebras using $\sigma$-derivations”, Journal of Algebra, 295:2 (2006), 314–361 | DOI

[60] D. Larsson, S. D. Silvestrov, “Quasi-deformations of $sl_2(\mathbb{F})$ using twisted derivations”, Comm. Algebra, 35:12 (2007), 4303–4318 | DOI

[61] Yu. I. Manin, Topics in Non-Commutative Geometry, Porter Lectures, Princeton University Press, 1991

[62] L. R. Rosenberg, Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and Its Applications, 330, Springer, Dordrecht, 1995

[63] J. C. McConnell, J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, 30, Amer. Math. Society, Providence, Rhode Island, 1987

[64] Heide Gluesing-Luerssen, Skew-Polynomial Rings and Skew-Cyclic Codes, 2019, arXiv: 1902.03516

[65] D. Larsson, S. D. Silvestrov, “Quasi-hom-Lie algebras, central extensions and $2$-cocycle-like identities”, Journal of Algebra, 288:2 (2005), 321–344 | DOI

[66] O. Elchinger, K. Lundengard, A. Makhlouf, S. D. Silvestrov, “Brackets with $(\tau,\sigma)$-derivations and $(p,q)$-deformations of Witt and Virasoro algebras”, Forum Math., 28:4 (2016), 657–673 | DOI

[67] G. Song, C. Xia, “Simple deformed Witt algebras”, Algebra Colloquium, 18:3 (2011), 533–540 | DOI

[68] G. Song, Y. Wu, B. Xin, “The $\sigma$-derivations of $C[x^{\pm 1}, y^{\pm 1}]$”, Algebra Colloquium, 22:2 (2015), 251–258, World Scientific Pub Co Pte Lt | DOI

[69] E. Gordji, “A characterization of $(\sigma,\tau)$-derivations on von Neumann algebras”, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 73 (2009), arXiv: 0903.0830

[70] R. H. Fox, “Free differential calculus. I: Derivation in the free group ring”, The Annals of Mathematics, 57:3 (1953), 547–560, JSTOR | DOI

[71] R. H. Fox, “Free differential calculus. II: The isomorphism problem of groups”, The Annals of Mathematics, 59:2 (1954), 196–210, JSTOR | DOI

[72] R. H. Fox, “Free differential calculus {III}. Subgroups”, The Annals of Mathematics, 64:3 (1956), 407–419, JSTOR | DOI

[73] 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, JSTOR | DOI

[74] R. H. Fox, “Free differential calculus V. The Alexander matrices re-examined”, The Annals of Mathematics, 71:3 (1960), 408–422, JSTOR | DOI

[75] G. Massuyeau, V. Turaev, Quasi-Poisson structures on representation spaces of surfaces, 2012, arXiv: 1205.4898

[76] A. N. Parshin, “O koltse formalnykh psevdodifferentsialnykh operatorov”, Algebra. Topologiya. Differentsialnye uravneniya i ikh prilozheniya, Sbornik statei. K 90-letiyu so dnya rozhdeniya akademika Lva Semenovicha Pontryagina, Trudy MIAN, 224, Nauka, MAIK «Nauka/Interperiodika», M., 1999, 291–305 | MR | Zbl

[77] A. Zheglov, Schur-Sato theory for quasi-elliptic rings, 2022, arXiv: 2205.06790

[78] V. P. Maslov, Operator Methods, Nauka Publ., Moscow, 1973 (In Russian)

[79] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Dobrosvet Publ., Moscow (In Russian)

[80] M. F. Atiyah, I. M. Singer, “The index of elliptic operators on compact manifolds”, Bull. Amer. Math. Soc., 69:3 (1963), 422–433 https://www.ams.org/journals/bull/1963-69-03/S0002-9904-1963-10957-X/

[81] M. F. Atiyah, I. M. Singer, “The index of elliptic operators I”, Annals of Mathematics, 87:3 (1968), 484–530

[82] I. N. Herstein, “Sui commutatori degli anelli semplici”, Seminario Mat. e. Fis. di Milano, 33 (1963), 80–86 | DOI

[83] V. K. Kharchenko, “Differential identities of prime rings”, Algebra and Logic, 17 (1978), 155–168

[84] P. Grzeszczuk, “On nilpotent derivations of semiprime rings”, Journal of Algebra, 149 (1992), 313–321

[85] C. Chen-Lian, L. Tsiu-Kwen, “Nilpotent derivations”, Journal of Algebra, 287:2 (2005), 381–401 | DOI

[86] D. Wright, “On the Jacobian Conjecture”, Illinois J. Math, 25 (1981), 423–440

[87] M. Nagata, “On the fourteenth problem of Hilbert”, Proceedings of the International Congress of Mathematicians, Cambridge University Press., Edinburgh, 1958, 459–462

[88] M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research Lectures on Mathematics, 31, Tata Institute of Fundamental Research, Bombai, 1965, 65 pp. http://www.math.tifr.res.in/p̃ubl/ln/tifr31.pdf

[89] M. Ferrero, Y. Lequain, A. Nowicki, “A note on locally nilpotent derivations”, Journal of Pure and Applied Algebra, 79 (1992), 45–50 https://www-users.mat.umk.pl/ãnow/ps-dvi/035-lok.pdf

[90] D. Daigle, “Locally nilpotent derivations and the structure of rings”, Journal of Pure and Applied Algebra, 224:4 (2020), 106–201, Elsevier BV | DOI

[91] S. Maubach, “The commuting derivations conjecture”, Journal of Pure and Applied Algebra, 179:1–2 (2003), 159–168

[92] Jiantao Li, Xiankun Du, “Pairwise commuting derivations of polynomial rings”, Linear Algebra and its Applications, 436:7, 2375–2379

[93] A. A. Arutyunov, A. S. Mischenko, A. I. Shtern, “Derivatsii gruppovykh algebr”, Fundament. i prikl. matem., 21:6 (2016), 65–78; A. A. Arutyunov, A. S. Mishchenko, A. I. Shtern, “Derivations of group algebras”, Journal of Mathematical Sciences, 248 (2020), 709–718 | DOI

[94] A. A. Arutyunov, A. S. Mischenko, “Gladkaya versiya problemy Dzhonsona o derivatsiyakh gruppovykh algebr”, Matem. sb., 210:6 (2019), 3–29 | DOI | MR

[95] A. A. Arutyunov, “Algebra differentsirovanii v nekommutativnykh gruppovykh algebrakh”, Differentsialnye uravneniya i dinamicheskie sistemy, Sbornik statei, Trudy MIAN, 308, MIAN, M., 2020, 28–41 | DOI

[96] A. A. Arutyunov, A. V. Alekseev, “Cohomology of $n$-categories and derivations in group algebras”, Topology and its Applications, 275 (2019) | DOI

[97] A. Arutyunov, A combinatorial view on derivations in bimodules, 2022, arXiv: 2208.05478

[98] A. A. Arutyunov, A. S. Mischenko, “Reduktsiya ischisleniya psevdodifferentsialnykh operatorov na nekompaktnom mnogoobrazii k ischisleniyu na kompaktnom mnogoobrazii udvoennoi razmernosti”, Matem. zametki, 94:4 (2013), 488–505 | DOI | MR | Zbl

[99] A. A. Arutyunov, A. S. Mischenko, “Reduktsiya PDO ischisleniya na nekompaktnom mnogoobrazii k kompaktnomu mnogoobraziyu udvoennoi razmernosti”, Doklad. RAN, 451:4 (2013), 369–373

[100] A. A. Arutyunov, “Reduktsiya nelokalnykh psevdodifferentsialnykh operatorov na nekompaktnom mnogoobrazii k klassicheskim psevdodifferentsialnym operatoram na kompaktnom mnogoobrazii udvoennoi razmernosti”, Matem. zametki, 97:4 (2015), 493–502 | DOI | MR | Zbl