Centralizers in free Poisson algebras
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- by Leonid Makar-Limanov and Ualbai Umirbaev PDF
- Proc. Amer. Math. Soc. 135 (2007), 1969-1975 Request permission
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
- George M. Bergman, Centralizers in free associative algebras, Trans. Amer. Math. Soc. 137 (1969), 327–344. MR 236208, DOI 10.1090/S0002-9947-1969-0236208-5
- 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 (Russian). MR 0330250
- P. M. Cohn, Subalgebras of free associative algebras, Proc. London Math. Soc. (3) 14 (1964), 618–632. MR 167504, DOI 10.1112/plms/s3-14.4.618
- P. M. Cohn, Free rings and their relations, 2nd ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
- Anastasia J. Czerniakiewicz, Automorphisms of a free associative algebra of rank $2$. I, Trans. Amer. Math. Soc. 160 (1971), 393–401. MR 280549, DOI 10.1090/S0002-9947-1971-0280549-1
- 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, DOI 10.1016/S0022-4049(97)00079-0
- I. M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 5–19 (French). MR 207918
- Heinrich W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161–174 (German). MR 8915, DOI 10.1515/crll.1942.184.161
- Shulim Kaliman and Leonid Makar-Limanov, On the Russell-Koras contractible threefolds, J. Algebraic Geom. 6 (1997), no. 2, 247–268. MR 1489115
- W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) 1 (1953), 33–41. MR 54574
- L. G. Makar-Limanov, The automorphisms of the free algebra with two generators, Funkcional. Anal. i Priložen. 4 (1970), no. 3, 107–108 (Russian). MR 0271161
- L. Makar-Limanov, Algebraically closed skew fields, J. Algebra 93 (1985), no. 1, 117–135. MR 780486, DOI 10.1016/0021-8693(85)90177-2
- Masayoshi Nagata, On automorphism group of $k[x,\,y]$, Kinokuniya Book Store Co., Ltd., Tokyo, 1972. Department of Mathematics, Kyoto University, Lectures in Mathematics, No. 5. MR 0337962
- I. P. Shestakov, Quantization of Poisson superalgebras and the specialty of Jordan superalgebras of Poisson type, Algebra i Logika 32 (1993), no. 5, 571–584, 587 (1994) (Russian, with Russian summary); English transl., Algebra and Logic 32 (1993), no. 5, 309–317 (1994). MR 1287006, DOI 10.1007/BF02261711
- Ivan P. Shestakov and Ualbai U. Umirbaev, The Nagata automorphism is wild, Proc. Natl. Acad. Sci. USA 100 (2003), no. 22, 12561–12563. MR 2017754, DOI 10.1073/pnas.1735483100
- Ivan P. Shestakov and Ualbai U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17 (2004), no. 1, 181–196. MR 2015333, DOI 10.1090/S0894-0347-03-00438-7
- Ivan P. Shestakov and Ualbai U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc. 17 (2004), no. 1, 197–227. MR 2015334, DOI 10.1090/S0894-0347-03-00440-5
- A. I. Širšov, Subalgebras of free Lie algebras, Mat. Sbornik N.S. 33(75) (1953), 441–452 (Russian). MR 0059892
- A. I. Širšov, Some algorithm problems for Lie algebras, Sibirsk. Mat. Ž. 3 (1962), 292–296 (Russian). MR 0183753
- U. U. Umirbaev, Some algorithmic questions concerning associative algebras, Algebra i Logika 32 (1993), no. 4, 450–470, 474 (Russian, with Russian summary); English transl., Algebra and Logic 32 (1993), no. 4, 244–255 (1994). MR 1286789, DOI 10.1007/BF02261749
- U. U. Umirbaev and I. P. Shestakov, Subalgebras and automorphisms of polynomial rings, Dokl. Akad. Nauk 386 (2002), no. 6, 745–748 (Russian). MR 2004473
- Ernst Witt, Die Unterringe der freien Lieschen Ringe, Math. Z. 64 (1956), 195–216 (German). MR 77525, DOI 10.1007/BF01166568
- Abraham Zaks, Dedekind subrings of $k[x_{1},\cdots ,x_{n}]$ are rings of polynomials, Israel J. Math. 9 (1971), 285–289. MR 280471, DOI 10.1007/BF02771678
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
- Received by editor(s): May 17, 2005
- Received by editor(s) in revised form: February 20, 2006
- Published electronically: February 28, 2007
- Additional Notes: The first author was supported by an NSA grant
- Communicated by: Martin Lorenz
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1969-1975
- MSC (2000): Primary 17B63, 17A50; Secondary 17B40, 17A36, 16S10
- DOI: https://doi.org/10.1090/S0002-9939-07-08678-9
- MathSciNet review: 2299468