Alternative and Jordan algebras admitting ternary derivations with invertible values
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1505-1523.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we prove analogues of H. Komatsu and A. Nakajima theorems (see [1]) for alternative and Jordan algebras. In particular, we give a description of alternative and Jordan algebras which have ternary derivations with invertible values.
Keywords: Alternative algebras, Jordan algebras, Cayley–Dickson algebra, Albert algebra, derivation, generalized derivation, ternary derivation.
@article{SEMR_2017_14_a45,
     author = {V. N. Zhelyabin and A. I. Shestakov},
     title = {Alternative and {Jordan} algebras admitting ternary derivations with invertible values},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1505--1523},
     publisher = {mathdoc},
     volume = {14},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a45/}
}
TY  - JOUR
AU  - V. N. Zhelyabin
AU  - A. I. Shestakov
TI  - Alternative and Jordan algebras admitting ternary derivations with invertible values
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2017
SP  - 1505
EP  - 1523
VL  - 14
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2017_14_a45/
LA  - en
ID  - SEMR_2017_14_a45
ER  - 
%0 Journal Article
%A V. N. Zhelyabin
%A A. I. Shestakov
%T Alternative and Jordan algebras admitting ternary derivations with invertible values
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2017
%P 1505-1523
%V 14
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2017_14_a45/
%G en
%F SEMR_2017_14_a45
V. N. Zhelyabin; A. I. Shestakov. Alternative and Jordan algebras admitting ternary derivations with invertible values. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1505-1523. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a45/

[1] H. Komatsu, A. Nakajima, “Generalized derivations with invertible values”, Comm. Algebra, 32:5 (2004), 1937–1944 | DOI | MR | Zbl

[2] J. Bergen, I.N. Herstein, C. Lanski, “Derivations with invertible values”, Canad. J. Math., 35:2 (1983), 300–310 | DOI | MR | Zbl

[3] A. Giambruno, P. Misso, C. Polcino Milies, “Derivations with invertible values in rings with involution”, Pacific J. Math., 123:1 (1986), 47–54 | DOI | MR | Zbl

[4] Jui Chi Chang, “$\alpha$-derivations with invertible values”, Bull. Inst. Math. Acad. Sinisa, 13:4 (1985), 323–333 | MR | Zbl

[5] M. Hongan, H. Komatsu, “($\sigma,\tau$)-derivations with invertible values”, Bull. Inst. Math. Acad. Sinica, 15:4 (1987), 411–415 | MR | Zbl

[6] C. Demir, E. Albas, N. Argac, A. Fosner, “Superderivations with invertible values”, J. Algebra Appl., 14:2 (2015), 1550022, 11 pp. | DOI | MR | Zbl

[7] M. Bresar, “On the distance of the composition of two derivations to the generalized derivations”, Glasgow Math. J., 33:1 (1991), 89–93 | DOI | MR | Zbl

[8] Richard D. Schafer, An introduction to non-associative algebras, Pure and Applied Mathematics, 22, Academic Press, New York–London, 1966 | MR

[9] A. Elduque, “On triality and automorphisms and derivations of composition algebras”, Linear Algebra Appl., 314 (2000), 49–74 | DOI | MR | Zbl

[10] C. Jimenez-Gestal, J. M. Perez-Izquierdo, “Ternary derivations of generalized Cayley–Dickson algebras”, Comm. Algebra, 31:10 (2003), 5071–5094 | DOI | MR | Zbl

[11] C. Jimenez-Gestal, J.M. Perez-Izquierdo, “Ternary derivations of finite-dimensional real division algebras”, Linear Algebra Appl., 428:8–9 (2008), 2192–2219 | DOI | MR | Zbl

[12] J.M. Perez-Izquierdo, “Unital algebras, ternary derivations, and local triality”, Algebras, representations and applications, Contemp. Math., 483, Amer. Math. Soc., Providence, RI, 2009, 205–220 | DOI | MR | Zbl

[13] I. P. Shestakov, “Prime alternative superalgebras of arbitrary characteristic”, Algebra i Logika, 36:6 (1997), 675–716 | DOI | MR | Zbl

[14] K.A. Zhevalkov, A.M. Slinko, I.P. Shestakov, A.I. Shirshov, Rings, close to associative, Nauka, M., 1978 | MR

[15] A.A. Popov, “Differentially simple alternative algebras”, Algebra and logic, 49:5 (2010), 670–689 | DOI | MR | Zbl

[16] I. B. Kaigorodov, U. S. Popov, “Alternative algebras admitting derivations with invertible values and invertible derivations”, Izv. RAN. Ser. matem., 78:5 (2014), 75–90 | DOI | MR

[17] A. I. Shestakov, “Ternary derivations of separable associative and Jordan algebras”, Sib. matem. zhurn., 53:5 (2012), 1178–1195 | MR | Zbl

[18] N. Jacobson, “Kronecker factorization theorem for Cayley algebras andthe exceptional simple Jordan algebra”, Amer. J. Math., 76 (1954), 447–452 | DOI | MR | Zbl

[19] Algebra Logic, 53:4 (2014), 323–348 | DOI | MR | Zbl

[20] E. I. Zel'manov, V. G. Skosyrskii, “Special Jordan nil-algebras of bounded index”, Algebra i Logika, 22:6 (1983), 626–635 | MR | Zbl

[21] E.I. Zel'manov, “Prime Jordan algebras. II”, Sib. Math. J., 24:1 (1983), 89–104 | MR | Zbl

[22] I.N. Herstein, “Jordan derivations of prime rings”, Proc. Amer. Math. Soc., 8 (1957), 1104–1110 | DOI | MR

[23] L.A. Lagutina, “Jordan homomorphisms of associative algebras with involution”, Algebra Logic, 27:4 (1988), 402–417 | DOI | MR | Zbl

[24] Ivan Kaygorodov, Artem Lopatin, Yury Popov, Jordan algebras admitting derivations with invertible values, 2015, arXiv: 1511.00742 | MR