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Centralizers in free Poisson algebras
Author(s):
Leonid
Makar-Limanov;
Ualbai
Umirbaev
Journal:
Proc. Amer. Math. Soc.
135
(2007),
1969-1975.
MSC (2000):
Primary 17B63, 17A50;
Secondary 17B40, 17A36, 16S10
Posted:
February 28, 2007
MathSciNet review:
2299468
Retrieve article in:
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Additional information
Abstract:
We prove an analog of the Bergman Centralizer Theorem for free Poisson algebras over an arbitrary field of characteristic 0. Some open problems are formulated.
References:
-
- 1.
- G.M.Bergman, Centralizers in free associative algebras, Trans. Amer. Math. Soc., 137 (1969), 327-344. MR 0236208 (38:4506)
- 2.
- L.A. Bokut, Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras, Izv. Akad. Nauk SSSR, Ser. Mat., 36 (1972), 1173-1219. MR 0330250 (48:8588)
- 3.
- P.M.Cohn, Subalgebras of free associative algebras, Proc. London Math. Soc., 56 (1964), 618-632. MR 0167504 (29:4777)
- 4.
- P.M.Cohn, Free rings and their relations, 2nd ed., Academic Press, London, 1985. MR 0800091 (87e:16006)
- 5.
- A.G.Czerniakiewicz, Automorphisms of a free associative algebra of rank 2, I, II, Trans. Amer. Math. Soc., 160 (1971), 393-401; 171 (1972), 309-315.MR 0280549 (43:6269); MR 0310021 (46:9124)
- 6.
- J. Donin and L.Makar-Limanov, Quantization of quadratic Poisson brackets on a polynomial algebra of three variables, J. Pure Appl. Algebra, 129 (1998), no. 3, 247-261.MR 1631249 (99e:16036)
- 7.
- I.M.Gelfand, A.A.Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie. (French) Inst. Hautes Études Sci. Publ. Math. No. 31, 1966, 5-19.MR 0207918 (34:7731)
- 8.
- H.W.E.Jung, Uber ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161-174. MR 0008915 (5:74f)
- 9.
- S. Kaliman, L.Makar-Limanov, On the Russel-Koras contractible threefolds, Journal of Algebraic Geometry, 6 (1997), 247-268. MR 1489115 (98m:14041)
- 10.
- W.van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde, (3)1 (1953), 33-41. MR 0054574 (14:941f)
- 11.
- L.Makar-Limanov, The automorphisms of the free algebra with two generators, Funksional. Anal. i Prilozhen. 4 (1970), no. 3, 107-108; English translation: in Functional Anal. Appl. 4 (1970), 262-263. MR 0271161 (42:6044)
- 12.
- L.Makar-Limanov, Algebraically closed skew fields, J. Algebra, 93 (1985), no. 1, 117-135. MR 0780486 (86d:16028)
- 13.
- M.Nagata, On the automorphism group of , Lect. in Math., Kyoto Univ., Kinokuniya, Tokio, 1972.MR 0337962 (49:2731)
- 14.
- I.P.Shestakov, Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras, Algebra i Logika, 32 (1993), no. 5, 571-584; English translation: in Algebra and Logic, 32 (1993), no. 5, 309-317.MR 1287006 (95c:17034)
- 15.
- I.P.Shestakov and U.U.Umirbaev, The Nagata automorphism is wild, Proc. Nat. Acad. Sci. USA, 100 (2003), no. 22, 12561-12563. MR 2017754 (2004j:13036)
- 16.
- I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two generated subalgebras of rings of polynomials, Journal of the American Mathematical Society, 17 (2004), 181-196. MR 2015333 (2004k:13036)
- 17.
- I. P. Shestakov and U. U. Umirbaev, Tame and wild automorphisms of rings of polynomials in three variables, Journal of the American Mathematical Society, 17 (2004), 197-227. MR 2015334 (2004h:13022)
- 18.
- A.I.Shirshov, Subalgebras of free Lie algebras, Mat. Sbornik, 33(75) (1953), 441-452. MR 0059892 (15:596d)
- 19.
- A.I.Shirshov, Some algorithm problems for Lie algebras, Sibirsk. Mat. Z., 3 (1962), 292-296.MR 0183753 (32:1231)
- 20.
- U. U. Umirbaev, Some algorithmic questions concerning associative algebras. (Russian) Algebra i Logika 32 (1993), no. 4, 450-470, 474; English translation: in Algebra and Logic, 32 (1993), no. 4, 244-255 (1994).MR 1286789 (95e:03128)
- 21.
- U. U. Umirbaev and I. P. Shestakov, Subalgebras and automorphisms of polynomial rings, Dokl. Akad. Nauk 386 (2002), no. 6, 745-748. MR 2004473 (2004i:14068)
- 22.
- E.Witt, Die Unterringe der freien Lieschen Ringe, Math. Z., 64 (1956), 195-216. MR 0077525 (17:1050a)
- 23.
- A.Zaks, Dedekind subrings of
![$ k[x_1,x_2, \ldots,x_n]$](/proc/2007-135-07/S0002-9939-07-08678-9/gif-references0/img2.gif) are rings of polynomials, Israel J. Math., 9 (1971), 285-289. MR 0280471 (43:6191)
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Additional Information:
Leonid
Makar-Limanov
Affiliation:
Department of Mathematics & Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel -- and -- Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email:
lml@math.wayne.edu
Ualbai
Umirbaev
Affiliation:
Department of Mathematics, Eurasian National University, Astana, 010008, Kazakhstan
Email:
umirbaev@yahoo.com
DOI:
10.1090/S0002-9939-07-08678-9
PII:
S 0002-9939(07)08678-9
Keywords:
Poisson algebras,
centralizers,
subalgebras.
Received by editor(s):
May 17, 2005
Received by editor(s) in revised form:
February 20, 2006
Posted:
February 28, 2007
Additional Notes:
The first author was supported by an NSA grant
Communicated by:
Martin Lorenz
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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