Simple algebras of Weyl type, II
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- by Kaiming Zhao PDF
- Proc. Amer. Math. Soc. 130 (2002), 1323-1332 Request permission
Abstract:
Over a field $\mathbb {F}$ of any characteristic, for a commutative associative algebra $A$, and for a commutative subalgebra $D$ of $\operatorname {Der}_{\mathbb {F}}(A)$, the vector space $A[D]$ which consists of polynomials of elements in $D$ with coefficients in $A$ and which is regarded as operators on $A$ forms naturally an associative algebra. It is proved that, as an associative algebra, $A[D]$ is simple if and only if $A$ is $D$-simple. Suppose $A$ is $D$-simple. Then, (a) $A[D]$ is a free left $A$-module; (b) as a Lie algebra, the subquotient $[A[D],{A[D]}]/(Z(A[D])\cap [A[D], {A[D]}])$ is simple (except for one case), where $Z(A[D])$ is the center of $A[D]$. The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.References
- Dragomir Ž. Đoković and Kaming Zhao, Derivations, isomorphisms, and second cohomology of generalized Witt algebras, Trans. Amer. Math. Soc. 350 (1998), no. 2, 643–664. MR 1390977, DOI 10.1090/S0002-9947-98-01786-3
- Dragomir Ž. Đoković and Kaiming Zhao, Generalized Cartan type $W$ Lie algebras in characteristic zero, J. Algebra 195 (1997), no. 1, 170–210. MR 1468889, DOI 10.1006/jabr.1997.7067
- Dragomir Ž. Đoković and Kaiming Zhao, Derivations, isomorphisms and second cohomology of generalized Block algebras, Algebra Colloq. 3 (1996), no. 3, 245–272. MR 1412656
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- David A. Jordan, On the simplicity of Lie algebras of derivations of commutative algebras, J. Algebra 228 (2000), no. 2, 580–585. MR 1764580, DOI 10.1006/jabr.2000.8286
- David A. Jordan, Iterated skew polynomial rings and quantum groups, J. Algebra 156 (1993), no. 1, 194–218. MR 1213792, DOI 10.1006/jabr.1993.1070
- Naoki Kawamoto, Generalizations of Witt algebras over a field of characteristic zero, Hiroshima Math. J. 16 (1986), no. 2, 417–426. MR 855169
- J. Marshall Osborn, New simple infinite-dimensional Lie algebras of characteristic $0$, J. Algebra 185 (1996), no. 3, 820–835. MR 1419725, DOI 10.1006/jabr.1996.0352
- J. Marshall Osborn and Kaiming Zhao, Generalized Poisson brackets and Lie algebras for type $H$ in characteristic $0$, Math. Z. 230 (1999), no. 1, 107–143. MR 1671866, DOI 10.1007/PL00004684
- J. Marshall Osborn and Kaiming Zhao, Generalized Cartan type $K$ Lie algebras in characteristic $0$, Comm. Algebra 25 (1997), no. 10, 3325–3360. MR 1465118, DOI 10.1080/00927879708826056
- D. S. Passman, Simple Lie algebras of Witt type, J. Algebra 206 (1998), no. 2, 682–692. MR 1637104, DOI 10.1006/jabr.1998.7444
- Y. Su, Derivations of generalized Weyl algebras, Science in China, to appear.
- Y. Su, $2$-Cocycles on the Lie algebras of generalized differential operators, Comm. Alg., to appear.
- Y. Su and K. Zhao, Simple algebras of Weyl type, Science in China (Series A) 44 (2001), 419–426.
- Y. Su, K. Zhao, Second cohomology group of generalized Witt type Lie algebras and certain representations, Comm. Alg., to appear.
- Y. Su, K. Zhao, Isomorphism classes and automorphism groups of algebras of Weyl type, Science in China (Series A), to appear.
- Xiaoping Xu, New generalized simple Lie algebras of Cartan type over a field with characteristic $0$, J. Algebra 224 (2000), no. 1, 23–58. MR 1736692, DOI 10.1006/jabr.1998.8083
- K. Zhao, Automorphisms of algebras of differential operators, J. of Capital Normal University 1 (1994), 1–8.
- K. Zhao, Lie algebras of derivations of algebras of differential operators, Chinese Science Bulletin 38 (10) (1993), 793–798.
- Kaiming Zhao, Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$, Canad. J. Math. 50 (1998), no. 1, 210–224. MR 1618823, DOI 10.4153/CJM-1998-011-6
Additional Information
- Kaiming Zhao
- Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
- Email: kzhao@math08.math.ac.cn
- Received by editor(s): August 28, 2000
- Received by editor(s) in revised form: November 20, 2000
- Published electronically: October 25, 2001
- Additional Notes: This work was supported by the Hundred Talents Program of Chinese Academy of Sciences and by NSF of China
- Communicated by: Lance W. Small
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1323-1332
- MSC (2000): Primary 16W10, 16W25, 17B20, 17B65, 17B05, 17B68
- DOI: https://doi.org/10.1090/S0002-9939-01-06218-9
- MathSciNet review: 1879953