AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Jordan Structures in Lie Algebras
About this Title
Antonio Fernández López, Universidad de Málaga, Málaga, Spain
Publication: Mathematical Surveys and Monographs
Publication Year:
2019; Volume 240
ISBNs: 978-1-4704-5086-1 (print); 978-1-4704-5362-6 (online)
DOI: https://doi.org/10.1090/surv/240
MathSciNet review: 3967698
MSC: Primary 17B05; Secondary 17C10
Table of Contents
Download chapters as PDF
Front/Back Matter
Chapters
- Introduction
- Nonassociative algebras
- General facts on Lie algebras
- Absolute zero divisors
- Jordan elements
- von Neumann regular elements
- Extremal elements
- A characterization of strong primeness
- From Lie algebras to Jordan algebras
- The Kostrikin radical
- Algebraic Lie algebras and local finiteness
- From Lie algebras to Jordan pairs
- An Artinian theory for Lie algebras
- Inner ideal structure of Lie algebras
- Classical infinite-dimensional Lie algebras
- Classical Banach–Lie algebras
- Adel Alahmadi, Hamed Alsulami, S. K. Jain, and Efim Zelmanov, Finite generation of Lie algebras associated with associative algebras, J. Algebra 426 (2015), 69–78. MR 3301901, DOI 10.1016/j.jalgebra.2014.10.056
- José A. Anquela and Teresa Cortés, Primitive Jordan pairs and triple systems, J. Algebra 184 (1996), no. 2, 632–678. MR 1409234, DOI 10.1006/jabr.1996.0280
- José A. Anquela and Teresa Cortés, Local-to-global inheritance of primitivity in Jordan algebras, Arch. Math. (Basel) 70 (1998), no. 3, 219–227. MR 1604064, DOI 10.1007/s000130050187
- José A. Anquela, Teresa Cortés, Esther García, and Miguel Gómez-Lozano, Polynomial identities and speciality of Martindale-like covers of Jordan algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 1–10. MR 2163397, DOI 10.1016/j.jpaa.2005.01.004
- José A. Anquela, Teresa Cortés, Ottmar Loos, and Kevin McCrimmon, An elemental characterization of strong primeness in Jordan systems, J. Pure Appl. Algebra 109 (1996), no. 1, 23–36. MR 1386811, DOI 10.1016/0022-4049(95)00086-0
- José A. Anquela, Teresa Cortés, and Fernando Montaner, Local inheritance in Jordan algebras, Arch. Math. (Basel) 64 (1995), no. 5, 393–401. MR 1323915, DOI 10.1007/BF01197216
- José A. Anquela, Teresa Cortés, and Efim Zelmanov, The Kurosh problem for Jordan nil systems over arbitrary rings of scalars, J. Algebra 444 (2015), 313–327. MR 3406180, DOI 10.1016/j.jalgebra.2015.08.004
- José A. Anquela, Teresa Cortés, and Efim Zelmanov, Local nilpotency of the McCrimmon radical of a Jordan system, Tr. Mat. Inst. Steklova 292 (2016), no. Algebra, Geometriya i Teoriya Chisel, 7–15; English transl., Proc. Steklov Inst. Math. 292 (2016), no. 1, 1–9. MR 3665922, DOI 10.1134/S0371968516010015
- Bruce Allison, John Faulkner, and Oleg Smirnov, Weyl images of Kantor pairs, Canad. J. Math. 69 (2017), no. 4, 721–766. MR 3679693, DOI 10.4153/CJM-2016-047-1
- J. C. Alexander, Compact Banach algebras, Proc. London Math. Soc. (3) 18 (1968), 1–18. MR 229040, DOI 10.1112/plms/s3-18.1.1
- José A. Anquela, Fernando Montaner, and Teresa Cortés, On primitive Jordan algebras, J. Algebra 163 (1994), no. 3, 663–674. MR 1265856, DOI 10.1006/jabr.1994.1036
- Bernard Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Functional Analysis 47 (1982), no. 1, 1–6. MR 663830, DOI 10.1016/0022-1236(82)90097-0
- Juri A. Bahturin, Simple Lie algebras satisfying a nontrivial identity, Serdica 2 (1976), no. 3, 241–246. MR 447358
- A. A. Baranov, Finitary simple Lie algebras, J. Algebra 219 (1999), no. 1, 299–329. MR 1707673, DOI 10.1006/jabr.1999.7856
- Alexander Baranov, Simple locally finite Lie algebras of diagonal type, Lie algebras and related topics, Contemp. Math., vol. 652, Amer. Math. Soc., Providence, RI, 2015, pp. 47–60. MR 3453048, DOI 10.1090/conm/652/12970
- Y. A. Bahturin, A. A. Baranov, and A. E. Zalesski, Simple Lie subalgebras of locally finite associative algebras, J. Algebra 281 (2004), no. 1, 225–246. MR 2091969, DOI 10.1016/j.jalgebra.2004.06.006
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
- Georgia Benkart, The Lie inner ideal structure of associative rings, J. Algebra 43 (1976), no. 2, 561–584. MR 435149, DOI 10.1016/0021-8693(76)90127-7
- Georgia Benkart, On inner ideals and ad-nilpotent elements of Lie algebras, Trans. Amer. Math. Soc. 232 (1977), 61–81. MR 466242, DOI 10.1090/S0002-9947-1977-0466242-6
- Khalid Bouhya and Antonio Fernandez Lopez, Jordan-$*$-triples with minimal inner ideals and compact $\textrm {JB}*$-triples, Proc. London Math. Soc. (3) 68 (1994), no. 2, 380–398. MR 1253508, DOI 10.1112/plms/s3-68.2.380
- Georgia Benkart and Antonio Fernández López, The Lie inner ideal structure of associative rings revisited, Comm. Algebra 37 (2009), no. 11, 3833–3850. MR 2573222, DOI 10.1080/00927870802545729
- Matej Brešar and Antonio Fernández López, On a class of finitary Lie algebras characterized through derivations, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4161–4166. MR 2680042, DOI 10.1090/S0002-9939-2010-10644-5
- A. A. Baranov and Antonio Fernández López, Maximal zero product subrings and inner ideals of simple rings, Preprint(ArXiv:1609.04609) (2016).
- Jose Brox, Antonio Fernández López, and Miguel Gómez Lozano, Inner ideals of Lie algebras of skew elements of prime rings with involution, Proc. Amer. Math. Soc. 144 (2016), no. 7, 2741–2751. MR 3487211, DOI 10.1090/proc/12903
- Jose Brox, Antonio Fernández López, and Miguel Gómez Lozano, Clifford elements in Lie algebras, J. Lie Theory 27 (2017), no. 1, 283–296. MR 3536548
- Jose Brox, Esther García, and Miguel Gómez Lozano, An elemental characterization of orthogonal ideals in Lie algebras, Mediterr. J. Math. 11 (2014), no. 4, 1061–1067. MR 3268807, DOI 10.1007/s00009-013-0344-3
- Jose Brox, Esther García, and Miguel Gómez Lozano, Jordan algebras at Jordan elements of semiprime rings with involution, J. Algebra 468 (2016), 155–181. MR 3550861, DOI 10.1016/j.jalgebra.2016.06.036
- G. M. Benkart and I. M. Isaacs, On the existence of ad-nilpotent elements, Proc. Amer. Math. Soc. 63 (1977), no. 1, 39–40. MR 432721, DOI 10.1090/S0002-9939-1977-0432721-6
- Mohamed Benslimane and El Amin Mokhtar Kaïdi, Structure des algèbres de Jordan-Banach non commutatives complexes régulières ou semi-simples à spectre fini, J. Algebra 113 (1988), no. 1, 201–206 (French). MR 928061, DOI 10.1016/0021-8693(88)90189-5
- W. E. Baxter and W. S. Martindale III, Central closure of semiprime nonassociative rings, Comm. Algebra 7 (1979), no. 11, 1103–1132. MR 535925, DOI 10.1080/00927877908822394
- K. I. Beidar, W. S. Martindale III, and A. V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, vol. 196, Marcel Dekker, Inc., New York, 1996. MR 1368853
- K. I. Beĭdar, A. V. Mikhalëv, and A. M. Slin′ko, A primality criterion for nondegenerate alternative and Jordan algebras, Trudy Moskov. Mat. Obshch. 50 (1987), 130–137, 261 (Russian); English transl., Trans. Moscow Math. Soc. (1988), 129–137. MR 912055
- Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
- A. A. Baranov and J. Rowley, Inner ideals of simple locally finite Lie algebras, J. Algebra 379 (2013), 11–30. MR 3019243, DOI 10.1016/j.jalgebra.2012.12.021
- Yu. A. Bakhturin and Kh. Shtrade, Locally finite-dimensional simple Lie algebras, Mat. Sb. 185 (1994), no. 2, 3–32 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 81 (1995), no. 1, 137–161. MR 1264772, DOI 10.1070/SM1995v081n01ABEH003618
- A. A. Baranov and H. Shlaka, Jordan-Lie inner ideals of finite dimensional associative algebras, Preprint(ArXiv:1609.05196) (2016).
- M. I. Berenguer and A. R. Villena, Continuity of Lie isomorphisms of Banach algebras, Bull. London Math. Soc. 31 (1999), no. 1, 6–10. MR 1651037, DOI 10.1112/S0024609398005128
- Juan Carlos Cabello, Miguel Cabrera, and Antonio Fernández López, $\pi$-complemented algebras through pseudocomplemented lattices, Order 29 (2012), no. 3, 463–479. MR 2979643, DOI 10.1007/s11083-011-9214-4
- J. C. Cabello, M. Cabrera, and R. Roura, $\pi$-complementation in the unitisation and multiplication algebras of a semiprime algebra, Comm. Algebra 40 (2012), no. 9, 3507–3531. MR 2981149, DOI 10.1080/00927872.2011.590954
- D. Ceretto, E. García, and M. Gómez Lozano, A radical for graded Lie algebras, Acta Math. Hungar. 136 (2012), no. 1-2, 16–29. MR 2925753, DOI 10.1007/s10474-012-0233-4
- Miguel Cabrera García and Ángel Rodríguez Palacios, Non-associative normed algebras. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 154, Cambridge University Press, Cambridge, 2014. The Vidav-Palmer and Gelfand-Naimark theorems. MR 3242640, DOI 10.1017/CBO9781107337763
- A. D. Chanyshev, Local finite-dimensionality of algebraic Lie algebras, Uspekhi Mat. Nauk 45 (1990), no. 6(276), 153–154 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 6, 143–144. MR 1101350, DOI 10.1070/RM1990v045n06ABEH002694
- V. I. Chernousov, The Hasse principle for groups of type $E_8$, Dokl. Akad. Nauk SSSR 306 (1989), no. 5, 1059–1063 (Russian); English transl., Soviet Math. Dokl. 39 (1989), no. 3, 592–596. MR 1014762
- Arjeh M. Cohen, Gábor Ivanyos, and Dan Roozemond, Simple Lie algebras having extremal elements, Indag. Math. (N.S.) 19 (2008), no. 2, 177–188. MR 2489325, DOI 10.1016/S0019-3577(09)00003-2
- P. M. Cohn, Further algebra and applications, Springer-Verlag London, Ltd., London, 2003. MR 1953987, DOI 10.1007/978-1-4471-0039-3
- Arjeh M. Cohen, Anja Steinbach, Rosane Ushirobira, and David Wales, Lie algebras generated by extremal elements, J. Algebra 236 (2001), no. 1, 122–154. MR 1808349, DOI 10.1006/jabr.2000.8508
- Alain D’Amour, Quadratic Jordan systems of Hermitian type, J. Algebra 149 (1992), no. 1, 197–233. MR 1165208, DOI 10.1016/0021-8693(92)90013-C
- Cristina Draper, Antonio Fernández López, Esther García, and Miguel Gómez Lozano, The socle of a nondegenerate Lie algebra, J. Algebra 319 (2008), no. 6, 2372–2394. MR 2388311, DOI 10.1016/j.jalgebra.2007.10.042
- C. Draper, A. Fernández López, E. García, and M. Gómez Lozano, The inner ideals of the simple finite dimensional Lie algebras, J. Lie Theory 22 (2012), no. 4, 907–929. MR 3052677
- Pierre de la Harpe, Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space, Lecture Notes in Mathematics, Vol. 285, Springer-Verlag, Berlin-New York, 1972. MR 0476820
- Alain D’Amour and Kevin McCrimmon, The local algebras of Jordan systems, J. Algebra 177 (1995), no. 1, 199–239. MR 1356368, DOI 10.1006/jabr.1995.1294
- Alain D’Amour and Kevin McCrimmon, The structure of quadratic Jordan systems of Clifford type, J. Algebra 234 (2000), no. 1, 31–89. MR 1799478, DOI 10.1006/jabr.2000.8452
- Alberto Elduque and Mikhail Kochetov, Gradings on simple Lie algebras, Mathematical Surveys and Monographs, vol. 189, American Mathematical Society, Providence, RI; Atlantic Association for Research in the Mathematical Sciences (AARMS), Halifax, NS, 2013. MR 3087174, DOI 10.1090/surv/189
- Theodore S. Erickson, The Lie structure in prime rings with involution, J. Algebra 21 (1972), 523–534. MR 308203, DOI 10.1016/0021-8693(72)90011-7
- John R. Faulkner, On the geometry of inner ideals, J. Algebra 26 (1973), 1–9. MR 367002, DOI 10.1016/0021-8693(73)90032-X
- Antonio Fernández López, Modular annihilator Jordan algebras, Comm. Algebra 13 (1985), no. 12, 2597–2613. MR 811525, DOI 10.1080/00927878508823292
- Antonio Fernández López, Banach-Lie algebras with extremal elements, Q. J. Math. 62 (2011), no. 1, 115–129. MR 2774357, DOI 10.1093/qmath/hap032
- Antonio Fernández López, Lie inner ideals are nearly Jordan inner ideals, Proc. Amer. Math. Soc. 142 (2014), no. 3, 795–804. MR 3148514, DOI 10.1090/S0002-9939-2013-11809-5
- Antonio Fernández López, A Jordan approach to finitary Lie algebras, Q. J. Math. 67 (2016), no. 4, 565–571. MR 3609846, DOI 10.1093/qmath/haw031
- Antonio Fernández López and Artem Yu. Golubkov, Lie algebras with an algebraic adjoint representation revisited, Manuscripta Math. 140 (2013), no. 3-4, 363–376. MR 3019131, DOI 10.1007/s00229-012-0546-1
- Antonio Fernández López, Esther García, and Miguel Gómez Lozano, The Jordan socle and finitary Lie algebras, J. Algebra 280 (2004), no. 2, 635–654. MR 2089256, DOI 10.1016/j.jalgebra.2004.06.013
- Antonio Fernández López, Esther García, and Miguel Gómez Lozano, Inner ideals of finitary simple Lie algebras, J. Lie Theory 16 (2006), no. 1, 97–114. MR 2196417
- Antonio Fernández López, Esther García, and Miguel Gómez Lozano, The Jordan algebras of a Lie algebra, J. Algebra 308 (2007), no. 1, 164–177. MR 2290916, DOI 10.1016/j.jalgebra.2006.02.035
- Antonio Fernández López, Esther García, and Miguel Gómez Lozano, An Artinian theory for Lie algebras, J. Algebra 319 (2008), no. 3, 938–951. MR 2379087, DOI 10.1016/j.jalgebra.2007.10.038
- Gang Yu, Upper bounds for finite additive 2-bases, Proc. Amer. Math. Soc. 137 (2009), no. 1, 11–18. MR 2439419, DOI 10.1090/S0002-9939-08-09430-6
- Antonio Fernández López, Esther García, Miguel Gómez Lozano, and Erhard Neher, A construction of gradings of Lie algebras, Int. Math. Res. Not. IMRN 16 (2007), Art. ID rnm051, 34. MR 2353091, DOI 10.1093/imrn/rnm051
- Antonio Fernández López and Eulalia García Rus, Algebraic lattices and nonassociative structures, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3211–3221. MR 1459121, DOI 10.1090/S0002-9939-98-04443-8
- Antonio Fernández López and Eulalia García Rus, Inner ideals in quadratic Jordan algebras of infinite capacity, Int. J. Math. Game Theory Algebra 9 (1999), no. 1, 35–54. MR 1697472
- A. Fernández López, E. García Rus, M. Gómez Lozano, and M. Siles Molina, Goldie theorems for associative pairs, Comm. Algebra 26 (1998), no. 9, 2987–3020. MR 1635898, DOI 10.1080/00927879808826324
- A. Fernandez Lopez, E. García Rus, E. Sánchez Campos, and M. Siles Molina, Strong regularity and Hermitian Hilbert triples, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 177, 43–55. MR 1269289, DOI 10.1093/qmath/45.1.43
- A. Fernández López, H. Marhnine, and C. Zarhouti, Derivations on Banach-Jordan pairs, Q. J. Math. 52 (2001), no. 3, 269–283. MR 1865901, DOI 10.1093/qjmath/52.3.269
- Antonio Fernández López and Ángel Rodríguez-Palacios, On the socle of a noncommutative Jordan algebra, Manuscripta Math. 56 (1986), no. 3, 269–278. MR 856365, DOI 10.1007/BF01180768
- Antonio Fernandez Lopez and Angel Rodriguez Palacios, Primitive noncommutative Jordan algebras with nonzero socle, Proc. Amer. Math. Soc. 96 (1986), no. 2, 199–206. MR 818443, DOI 10.1090/S0002-9939-1986-0818443-3
- Antonio Fernández López and Maribel Tocón Barroso, Strongly prime Jordan pairs with nonzero socle, Manuscripta Math. 111 (2003), no. 3, 321–340. MR 1993498, DOI 10.1007/s00229-003-0372-6
- Esther García, Inheritance of primeness by ideals in Lie algebras, Int. J. Math. Game Theory Algebra 13 (2003), no. 6, 481–484. MR 2115392
- Esther García, Tits-Kantor-Koecher algebras of strongly prime Hermitian Jordan pairs, J. Algebra 277 (2004), no. 2, 559–571. MR 2067619, DOI 10.1016/j.jalgebra.2003.10.016
- Esther García and Miguel Gómez Lozano, An elemental characterization of strong primeness in Lie algebras, J. Algebra 312 (2007), no. 1, 132–141. MR 2320450, DOI 10.1016/j.jalgebra.2006.11.003
- Esther García and Miguel Gómez Lozano, A note on a result of Kostrikin, Comm. Algebra 37 (2009), no. 7, 2405–2409. MR 2536928, DOI 10.1080/00927870802255683
- Esther García and Miguel Gómez Lozano, A characterization of the Kostrikin radical of a Lie algebra, J. Algebra 346 (2011), 266–283. MR 2842081, DOI 10.1016/j.jalgebra.2011.08.013
- Dancheng Lu and Tongsuo Wu, Regularity, $a^\ast$-invariant and the associated graded module to a filtration, Comm. Algebra 40 (2012), no. 10, 3745–3752. MR 2982894, DOI 10.1080/00927872.2011.594471
- Esther García, Miguel Gómez Lozano, and Erhard Neher, Nondegeneracy for Lie triple systems and Kantor pairs, Canad. Math. Bull. 54 (2011), no. 3, 442–455. MR 2857919, DOI 10.4153/CMB-2011-023-9
- Piotr Grzeszczuk, On nilpotent derivations of semiprime rings, J. Algebra 149 (1992), no. 2, 313–321. MR 1172431, DOI 10.1016/0021-8693(92)90018-H
- A. N. Grishkov and I. P. Shestakov, A characterization of Lie algebras of skew-symmetric elements, Acta Appl. Math. 85 (2005), no. 1-3, 157–159. MR 2128909, DOI 10.1007/s10440-004-5598-0
- Paul R. Halmos, Commutators of operators, Amer. J. Math. 74 (1952), 237–240. MR 45310, DOI 10.2307/2372081
- Johanna Hennig, Simple, locally finite dimensional Lie algebras in positive characteristic, J. Algebra 413 (2014), 270–288. MR 3216608, DOI 10.1016/j.jalgebra.2014.04.027
- I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135
- I. N. Herstein, On the Lie structure of an associative ring, J. Algebra 14 (1970), 561–571. MR 255610, DOI 10.1016/0021-8693(70)90103-1
- I. N. Herstein, Rings with involution, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. MR 0442017
- I. N. Herstein, Noncommutative rings, Carus Mathematical Monographs, vol. 15, Mathematical Association of America, Washington, DC, 1994. Reprint of the 1968 original; With an afterword by Lance W. Small. MR 1449137
- Leslie Hogben and Kevin McCrimmon, Maximal modular inner ideals and the Jacobson radical of a Jordan algebra, J. Algebra 68 (1981), no. 1, 155–169. MR 604300, DOI 10.1016/0021-8693(81)90291-X
- Harald Hanche-Olsen and Erling Størmer, Jordan operator algebras, Monographs and Studies in Mathematics, vol. 21, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 755003
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- N. Jacobson, A note on three dimensional simple Lie algebras, J. Math. Mech. 7 (1958), 823–831. MR 0097432, DOI 10.1512/iumj.1958.7.57047
- Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
- Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
- Nathan Jacobson, Lie algebras, Dover Publications Inc., New York, 1979, Republication of the 1962 original. MR 559927 (80k:17001)
- Irving Kaplansky, Regular Banach algebras, J. Indian Math. Soc. (N.S.) 12 (1948), 57–62. MR 29106
- Irving Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575–580. MR 25451, DOI 10.1090/S0002-9904-1948-09049-8
- Irving Kaplansky, Linear algebra and geometry, Reprint of the 1974 revised edition, Dover Publications, Inc., Mineola, NY, 2003. A second course. MR 2001037
- A. I. Kostrikin, Around Burnside, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 20, Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by James Wiegold. MR 1075416, DOI 10.1007/978-3-642-74324-5
- T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
- Jakob Levitzki, On the structure of algebraic algebras and related rings, Trans. Amer. Math. Soc. 74 (1953), 384–409. MR 53089, DOI 10.1090/S0002-9947-1953-0053089-1
- Ottmar Loos and Erhard Neher, Complementation of inner ideals in Jordan pairs, J. Algebra 166 (1994), no. 2, 255–295. MR 1279257, DOI 10.1006/jabr.1994.1151
- Ottmar Loos and Erhard Neher, Locally finite root systems, Mem. Amer. Math. Soc. 171 (2004), no. 811, x+214. MR 2073220, DOI 10.1090/memo/0811
- Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721
- Ottmar Loos, On the socle of a Jordan pair, Collect. Math. 40 (1989), no. 2, 109–125 (1990). MR 1094683
- Ottmar Loos, Filtered Jordan systems, Comm. Algebra 18 (1990), no. 6, 1899–1924. MR 1071080, DOI 10.1080/00927879008824000
- Ottmar Loos, Finiteness conditions in Jordan pairs, Math. Z. 206 (1991), no. 4, 577–587. MR 1100842, DOI 10.1007/BF02571365
- C. Martínez, The ring of fractions of a Jordan algebra, J. Algebra 237 (2001), no. 2, 798–812. MR 1816717, DOI 10.1006/jabr.2000.8617
- Kevin McCrimmon, The radical of a Jordan algebra, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 671–678. MR 268238, DOI 10.1073/pnas.62.3.671
- Kevin McCrimmon, Inner ideals in quadratic Jordan algebras, Trans. Amer. Math. Soc. 159 (1971), 445–468. MR 279145, DOI 10.1090/S0002-9947-1971-0279145-1
- Kevin McCrimmon, Compatible Peirce decompositions of Jordan triple systems, Pacific J. Math. 103 (1982), no. 1, 57–102. MR 687964
- Kevin McCrimmon, Reduced elements in Jordan triple systems, J. Algebra 97 (1985), no. 2, 540–564. MR 813001, DOI 10.1016/0021-8693(85)90063-8
- Kevin McCrimmon, Jordan centroids, Comm. Algebra 27 (1999), no. 2, 933–954. MR 1672019, DOI 10.1080/00927879908826470
- Kevin McCrimmon, A taste of Jordan algebras, Universitext, Springer-Verlag, New York, 2004. MR 2014924
- Kurt Meyberg, Lectures on algebras and triple systems, University of Virginia, Charlottesville, Va., 1972. Notes on a course of lectures given during the academic year 1971–1972. MR 0340353
- W. S. Martindale III and C. Robert Miers, On the iterates of derivations of prime rings, Pacific J. Math. 104 (1983), no. 1, 179–190. MR 683736
- W. S. Martindale III and C. Robert Miers, Nilpotency and generalized Lie ideals, J. Algebra 127 (1989), no. 1, 244–254. MR 1029416, DOI 10.1016/0021-8693(89)90287-1
- Fernando Montaner, Local PI theory of Jordan systems, J. Algebra 216 (1999), no. 1, 302–327. MR 1694558, DOI 10.1006/jabr.1998.7755
- Consuelo Martínez and Efim Zelmanov, Nil algebras and unipotent groups of finite width, Adv. Math. 147 (1999), no. 2, 328–344. MR 1734527, DOI 10.1006/aima.1999.1848
- Erhard Neher, Jordan triple systems by the grid approach, Lecture Notes in Mathematics, vol. 1280, Springer-Verlag, Berlin, 1987. MR 911879, DOI 10.1007/BFb0078217
- Erhard Neher, Jordan pairs with finite grids, Comm. Algebra 19 (1991), no. 2, 455–478. MR 1100357, DOI 10.1080/00927879108824149
- Erhard Neher, Lie algebras graded by $3$-graded root systems and Jordan pairs covered by grids, Amer. J. Math. 118 (1996), no. 2, 439–491. MR 1385288
- J. Marshall Osborn and M. L. Racine, Jordan rings with nonzero socle, Trans. Amer. Math. Soc. 251 (1979), 375–387. MR 531985, DOI 10.1090/S0002-9947-1979-0531985-4
- S. V. Pchelintsev, Prime algebras and absolute divisors of zero, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 1, 79–100, 207 (Russian). MR 835567
- J. Pérez González, L. Rico Romero, Á. Rodríguez-Palacios, and A. R. Villena Muñoz, Prime Jordan-Banach algebras with nonzero socle, Comm. Algebra 20 (1992), no. 1, 17–53. MR 1145324, DOI 10.1080/00927879208824330
- A. A. Premet, Inner ideals in modular Lie algebras, Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 5 (1986), 11–15, 123 (Russian, with English summary). MR 876665
- A. A. Premet, Lie algebras without strong degeneration, Mat. Sb. (N.S.) 129(171) (1986), no. 1, 140–153 (Russian); English transl., Math. USSR-Sb. 57 (1987), no. 1, 151–164. MR 830100, DOI 10.1070/SM1987v057n01ABEH003060
- Ángel Rodríguez-Palacios, The uniqueness of the complete norm topology in complete normed nonassociative algebras, J. Funct. Anal. 60 (1985), no. 1, 1–15. MR 780101, DOI 10.1016/0022-1236(85)90055-2
- Ángel Rodríguez-Palacios, An approach to Jordan-Banach algebras from the theory of nonassociative complete normed algebras, Ann. Sci. Univ. Clermont-Ferrand II Math. 27 (1991), 1–57 (1992). MR 1164710
- Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York-London, 1966. MR 0210757
- G. B. Seligman, Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR 0245627
- A. I. Širšov, On rings with identity relations, Mat. Sb. (N.S.) 43(85) (1957), 277–283 (Russian). MR 0095192
- Mercedes Siles Molina, Algebras of quotients of Lie algebras, J. Pure Appl. Algebra 188 (2004), no. 1-3, 175–188. MR 2030813, DOI 10.1016/j.jpaa.2003.10.005
- Mercedes Siles Molina, Finite $\Bbb Z$-gradings of simple associative algebras, J. Pure Appl. Algebra 207 (2006), no. 3, 619–630. MR 2265543, DOI 10.1016/j.jpaa.2005.10.005
- Oleg N. Smirnov, Simple associative algebras with finite $\mathbf Z$-grading, J. Algebra 196 (1997), no. 1, 171–184. MR 1474168, DOI 10.1006/jabr.1997.7087
- Oleg N. Smirnov, Finite $\textbf {Z}$-gradings of Lie algebras and symplectic involutions, J. Algebra 218 (1999), no. 1, 246–275. MR 1704686, DOI 10.1006/jabr.1999.7880
- Ivan Shestakov and Efim Zelmanov, A finite presentation of Jordan algebras, Internat. J. Algebra Comput. 28 (2018), no. 8, 1705–1716. MR 3899231, DOI 10.1142/S0218196718400155
- Armin Thedy, $z$-closed ideals of quadratic Jordan algebras, Comm. Algebra 13 (1985), no. 12, 2537–2565. MR 811523, DOI 10.1080/00927878508823290
- È. B. Vinberg, The classification of nilpotent elements of graded Lie algebras, Dokl. Akad. Nauk SSSR 225 (1975), no. 4, 745–748 (Russian). MR 0506488
- E. I. Zel′manov, Jordan division algebras, Algebra i Logika 18 (1979), no. 3, 286–310, 385 (Russian). MR 566787
- E. I. Zel′manov, Primary Jordan algebras, Algebra i Logika 18 (1979), no. 2, 162–175, 253 (Russian). MR 566779
- E. I. Zel′manov, Absolute zero divisors and algebraic Jordan algebras, Sibirsk. Mat. Zh. 23 (1982), no. 6, 100–116, 206 (Russian). MR 682911
- E. I. Zel′manov, Lie algebras with algebraic associated representation, Mat. Sb. (N.S.) 121(163) (1983), no. 4, 545–561 (Russian). MR 716113
- E. I. Zel′manov, Prime Jordan algebras. II, Sibirsk. Mat. Zh. 24 (1983), no. 1, 89–104, 192 (Russian). MR 688595
- E. I. Zel′manov, Characterization of the McCrimmon radical, Sibirsk. Mat. Zh. 25 (1984), no. 5, 190–192 (Russian). MR 762254
- E. I. Zel′manov, Lie algebras with finite gradation, Mat. Sb. (N.S.) 124(166) (1984), no. 3, 353–392 (Russian). MR 752226
- E. I. Zel′manov, Solution of the restricted Burnside problem for groups of odd exponent, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 42–59, 221 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 41–60. MR 1044047
- E. I. Zel′manov, Solution of the restricted Burnside problem for $2$-groups, Mat. Sb. 182 (1991), no. 4, 568–592 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 543–565. MR 1119009
- E. I. Zel′manov, On periodic compact groups, Israel J. Math. 77 (1992), no. 1-2, 83–95. MR 1194787, DOI 10.1007/BF02808012
- Efim Zelmanov, Nil rings and periodic groups, KMS Lecture Notes in Mathematics, Korean Mathematical Society, Seoul, 1992. With a preface by Jongsik Kim. MR 1199575
- Efim Zelmanov, Lie algebras and torsion groups with identity, J. Comb. Algebra 1 (2017), no. 3, 289–340. MR 3681577, DOI 10.4171/JCA/1-3-2
- E. I. Zel′manov and A. I. Kostrikin, A theorem on sandwich algebras, Trudy Mat. Inst. Steklov. 183 (1990), 106–111, 225 (Russian). Translated in Proc. Steklov Inst. Math. 1991, no. 4, 121–126; Galois theory, rings, algebraic groups and their applications (Russian). MR 1092020
- E. I. Zel′manov, Jordan algebras with finiteness conditions, Algebra i Logika 17 (1978), no. 6, 693–704, 747 (Russian). MR 555097
- K. A. Ževlakov and I. P. Šestakov, Local finiteness in the sense of Širšov, Algebra i Logika 12 (1973), 41–73, 120 (Russian). MR 0399188