## Invariants of semisimple Lie algebras acting on associative algebras

HTML articles powered by AMS MathViewer

- by Piotr Grzeszczuk PDF
- Proc. Amer. Math. Soc.
**131**(2003), 709-717 Request permission

## Abstract:

If ${\mathfrak g}$ is a Lie algebra of derivations of an associative algebra $R$, then the subalgebra of invariants is the set $R^{\mathfrak g} = \{ r \in R \mid \delta (r) = 0 \ \text { for all } \delta \in {\mathfrak g}\}.$ In this paper, we study the relationship between the structure of $R^{\mathfrak g}$ and the structure of $R$, where $\mathfrak g$ is a finite dimensional semisimple Lie algebra over a field of characteristic zero acting finitely on $R$, when $R$ is semiprime.## References

- K.I. Beidar,
*Rings of quotients of semiprime rings*, Vestnik Moscow. Univers. Math. (1978), 36–43. - Konstantin I. Beidar and Piotr Grzeszczu,
*Actions of Lie algebras on rings without nilpotent elements*, Algebra Colloq.**2**(1995), no. 2, 105–116. MR**1329141** - 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** - Jeffrey Bergen and S. Montgomery,
*Smash products and outer derivations*, Israel J. Math.**53**(1986), no. 3, 321–345. MR**852484**, DOI 10.1007/BF02786565 - Jeffrey Bergen,
*Constants of Lie algebra actions*, J. Algebra**114**(1988), no. 2, 452–465. MR**936982**, DOI 10.1016/0021-8693(88)90303-1 - Jeffrey Bergen,
*Invariants of domains under the actions of restricted Lie algebras*, J. Algebra**177**(1995), no. 1, 115–131. MR**1356362**, DOI 10.1006/jabr.1995.1288 - Miriam Cohen,
*Goldie centralizers of separable subalgebras*, Michigan Math. J.**23**(1976), no. 2, 185–191. MR**412219** - 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 - Piotr Grzeszczuk,
*Constants of algebraic derivations*, Comm. Algebra**21**(1993), no. 6, 1857–1868. MR**1215550**, DOI 10.1080/00927879308824657 - Nathan Jacobson,
*Lie algebras*, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR**559927** - Irving Kaplansky,
*Lie algebras and locally compact groups*, University of Chicago Press, Chicago, Ill.-London, 1971. MR**0276398** - V. K. Kharchenko,
*Automorphisms and derivations of associative rings*, Mathematics and its Applications (Soviet Series), vol. 69, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the Russian by L. Yuzina. MR**1174740**, DOI 10.1007/978-94-011-3604-4 - V. K. Kharchenko,
*Derivations of prime rings of positive characteristic*, Algebra i Logika**35**(1996), no. 1, 88–104, 120 (Russian, with Russian summary); English transl., Algebra and Logic**35**(1996), no. 1, 49–58. MR**1400715**, DOI 10.1007/BF02367194 - V. K. Kharchenko, J. Keller, and S. Rodrigues-Romo,
*Prime rings with PI rings of constants*. part B, Israel J. Math.**96**(1996), no. part B, 357–377. MR**1433695**, DOI 10.1007/BF02937311 - 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 - Wallace S. Martindale III,
*Prime rings satisfying a generalized polynomial identity*, J. Algebra**12**(1969), 576–584. MR**238897**, DOI 10.1016/0021-8693(69)90029-5 - Susan Montgomery,
*Centralizers of separable subalgebras*, Michigan Math. J.**22**(1975), 15–24. MR**384850** - Susan Montgomery and Martha K. Smith,
*Algebras with a separable subalgebra whose centralizer satisfies a polynomial identity*, Comm. Algebra**3**(1975), 151–168. MR**374186**, DOI 10.1080/00927877508822038 - A. Z. Popov,
*Derivations of prime rings*, Algebra i Logika**22**(1983), no. 1, 79–92 (Russian). MR**751651**

## Additional Information

**Piotr Grzeszczuk**- Affiliation: Institute of Computer Science, Technical University of Białystok, Wiejska 45A, 15-351 Białystok, Poland
- Email: piotrgr@cksr.ac.bialystok.pl
- Received by editor(s): October 19, 2001
- Published electronically: July 25, 2002
- Additional Notes: The author was supported by Polish scientific grant KBN no. 2 P03A 039 14.
- Communicated by: Martin Lorenz
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 709-717 - MSC (2000): Primary 16W25; Secondary 16R20, 16U20
- DOI: https://doi.org/10.1090/S0002-9939-02-06854-5
- MathSciNet review: 1937407