Primitive Lie PI-algebras

Cabrera, Miguel; López, Antonio

doi:10.1090/proc/15744

Primitive Lie PI-algebras
HTML articles powered by AMS MathViewer

by Miguel Cabrera and Antonio Fernández López PDF

Proc. Amer. Math. Soc. 150 (2022), 2277-2285 Request permission

Abstract:

A Lie algebra $L$ is called primitive if it is prime, nondegenerate, and contains a nonzero Jordan element $a$ such that the attached Jordan algebra $L_a$ is primitive. In this paper we prove that every primitive Lie PI-algebra over a field of zero characteristic is simple and finite-dimensional over its centroid.

References

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
José A. Anquela and Teresa Cortés, Local and subquotient inheritance of simplicity in Jordan systems, J. Algebra 240 (2001), no. 2, 680–704. MR 1841352, DOI 10.1006/jabr.2000.8693
Juri A. Bahturin, Simple Lie algebras satisfying a nontrivial identity, Serdica 2 (1976), no. 3, 241–246. MR 447358
Yuri Bahturin, On Lie subalgebras of associative PI-algebras, J. Algebra 67 (1980), no. 2, 257–271. MR 602062, DOI 10.1016/0021-8693(80)90159-3
Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR 886063
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
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
Miguel Cabrera, Ideals which memorize the extended centroid, J. Algebra Appl. 1 (2002), no. 3, 281–288. MR 1932152, DOI 10.1142/S0219498802000185
M. Cabrera, A. Fernández López, A. Yu. Golubkov, and A. Moreno, Algebras whose multiplication algebra is PI or GPI, J. Algebra 459 (2016), 213–237. MR 3503972, DOI 10.1016/j.jalgebra.2016.04.009
M. Cabrera and Amir A. Mohammed, Extended centroid and central closure of the multiplication algebra, Comm. Algebra 27 (1999), no. 12, 5723–5736. MR 1726273, DOI 10.1080/00927879908826787
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
P. M. Cohn, Further algebra and applications, Springer-Verlag London, Ltd., London, 2003. MR 1953987, DOI 10.1007/978-1-4471-0039-3
Antonio Fernández López, Jordan structures in Lie algebras, Mathematical Surveys and Monographs, vol. 240, American Mathematical Society, Providence, RI, 2019. MR 3967698, DOI 10.1016/j.akcej.2019.02.009
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
A. A. Kucherov, O. A. Pikhtil′kova, and S. A. Pikhtil′kov, On primitive Lie algebras, Fundam. Prikl. Mat. 17 (2011/12), no. 2, 177–182 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 186 (2012), no. 4, 651–654. MR 2904291, DOI 10.1007/s10958-012-1011-0
S. A. Pikhtilkov, Structural theory of special Lie algebras, J. Math. Sci. (N.Y.) 136 (2006), no. 4, 4090–4115. Algebra. MR 2272927, DOI 10.1007/s10958-006-0222-7
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