Varieties of linear algebras with colength one. (English. Russian original) Zbl 1304.17002
Mosc. Univ. Math. Bull. 65, No. 1, 23-27 (2010); translation from Vest. Mosk. Univ. Mat. Mekh. 65, No. 1, 25-30 (2010).
Summary: In the case of characteristic zero it is proved that there exist exactly three varieties of linear algebras with the colength equal to one for all degrees. Those are the variety of all associative-commutative algebras, the variety of all metabelian Lie algebras, and the variety of soluble Jordan algebras of the step 2 with the identity \(x^{2}x \equiv 0\).
MSC:
| 17A30 | Nonassociative algebras satisfying other identities |
| 17B01 | Identities, free Lie (super)algebras |
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16R40 | Identities other than those of matrices over commutative rings |
| 17C05 | Identities and free Jordan structures |
References:
| [1] | Yu. A. Bakhturin, Identical Relations in Lie Algebras (Nauka, Moscow, 1985; VNU, Utrecht, 1987). · Zbl 0571.17001 |
| [2] | A. Berele and A. Regev, ”Applications of Hook Diagrams to P.I. Algebras,” J. Algebra, 82, 559 (1983). · Zbl 0517.16013 · doi:10.1016/0021-8693(83)90167-9 |
| [3] | M. V. Zaitsev and S. P. Mishchenko, ”Asymptotic Behavior of the Colength Growth Functions of Varieties of Lie Algebras,” Uspekhi Matem. Nauk 54(3), 161 (1999) [Russian Math. Surveys 54 (3), 660 (1999)]. · Zbl 0967.17003 · doi:10.4213/rm173 |
| [4] | S. P. Mishchenko and M. V. Zaitsev, ”Asymptotic Behavior of Colength of Varieties of Lie Algebras,” Serdica Math. J. 26, 145 (2000). · Zbl 0967.17003 |
| [5] | A. Giambruno, S. Mishchenko, and M. Zaicev, ”On the Colength of a Variety of Lie Algebras,” Int. J. Algebra and Comput. 9(5), 483 (1999). · Zbl 1055.17501 · doi:10.1142/S0218196799000291 |
| [6] | M. V. Zaitsev and S. P. Mishchenko, ”A New Extremal Property of the Variety AN 2 of Lie Algebras,” Vestn. Mosk. Univ., Matem. Mekhan., No. 2, 15 (1999). |
| [7] | A. I. Shirshov, K. A. Zhevlakov, A. M. Slin’ko, and I. P. Shestakov, Rings that are Nearly Associative (Nauka, Moscow, 1978; Acad. Press, NY, 1982). · Zbl 0445.17001 |
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