Simple algebras of Weyl type, II
Author:
Kaiming Zhao
Journal:
Proc. Amer. Math. Soc. 130 (2002), 13231332
MSC (2000):
Primary 16W10, 16W25, 17B20, 17B65, 17B05, 17B68
Published electronically:
October 25, 2001
MathSciNet review:
1879953
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Abstract: Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is simple. Suppose is simple. Then, (a) is a free left module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.
 [DZ1]
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
(98d:17031), http://dx.doi.org/10.1090/S0002994798017863
 [DZ2]
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
(98j:17021), http://dx.doi.org/10.1006/jabr.1997.7067
 [DZ3]
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
(97g:17020)
 [H]
I.
N. Herstein, On the Lie and Jordan rings of a simple associative
ring, Amer. J. Math. 77 (1955), 279–285. MR 0067871
(16,789e)
 [J1]
David
A. Jordan, On the simplicity of Lie algebras of derivations of
commutative algebras, J. Algebra 228 (2000),
no. 2, 580–585. MR 1764580
(2001d:16052), http://dx.doi.org/10.1006/jabr.2000.8286
 [J2]
David
A. Jordan, Iterated skew polynomial rings and quantum groups,
J. Algebra 156 (1993), no. 1, 194–218. MR 1213792
(94b:16034), http://dx.doi.org/10.1006/jabr.1993.1070
 [K]
Naoki
Kawamoto, Generalizations of Witt algebras over a field of
characteristic zero, Hiroshima Math. J. 16 (1986),
no. 2, 417–426. MR 855169
(88d:17017)
 [O]
J.
Marshall Osborn, New simple infinitedimensional Lie algebras of
characteristic 0, J. Algebra 185 (1996), no. 3,
820–835. MR 1419725
(98a:17035), http://dx.doi.org/10.1006/jabr.1996.0352
 [OZ1]
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
(2000c:17038), http://dx.doi.org/10.1007/PL00004684
 [OZ2]
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
(98e:17032), http://dx.doi.org/10.1080/00927879708826056
 [P]
D.
S. Passman, Simple Lie algebras of Witt type, J. Algebra
206 (1998), no. 2, 682–692. MR 1637104
(99j:17012), http://dx.doi.org/10.1006/jabr.1998.7444
 [S1]
Y. Su, Derivations of generalized Weyl algebras, Science in China, to appear.
 [S2]
Y. Su, Cocycles on the Lie algebras of generalized differential operators, Comm. Alg., to appear.
 [SZ1]
Y. Su and K. Zhao, Simple algebras of Weyl type, Science in China (Series A) 44 (2001), 419426. CMP 2001:12
 [SZ2]
Y. Su, K. Zhao, Second cohomology group of generalized Witt type Lie algebras and certain representations, Comm. Alg., to appear.
 [SZ3]
Y. Su, K. Zhao, Isomorphism classes and automorphism groups of algebras of Weyl type, Science in China (Series A), to appear.
 [X]
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
(2001b:17021), http://dx.doi.org/10.1006/jabr.1998.8083
 [Z1]
K. Zhao, Automorphisms of algebras of differential operators, J. of Capital Normal University 1 (1994), 18.
 [Z2]
K. Zhao, Lie algebras of derivations of algebras of differential operators, Chinese Science Bulletin 38 (10) (1993), 793798.
 [Z3]
Kaiming
Zhao, Isomorphisms between generalized Cartan type 𝑊 Lie
algebras in characteristic 0, Canad. J. Math. 50
(1998), no. 1, 210–224. MR 1618823
(99e:17035), http://dx.doi.org/10.4153/CJM19980116
 [DZ1]
 D. Z. Dokovic and K. Zhao, Derivations, isomorphisms, and second cohomology of generalized Witt algebras, Trans. Amer. Math. Soc. 350 (2) (1998), 643664. MR 98d:17031
 [DZ2]
 D. Z. Dokovic and K. Zhao, Generalized Cartan type Lie algebras in characteristic zero, J. Alg. 195 (1997), 170210. MR 98j:17021
 [DZ3]
 D. Z. Djokovic and K. Zhao, Derivations, isomorphisms, and second cohomology of generalized Block algebras, Alg. Colloq. 3 (3) (1996), 245272. MR 97g:17020
 [H]
 I. N. Herstein, On the Lie and Jordan rings of a simple associative ring, Amer. J. Math. 77 (1955), 279285. MR 16:789e
 [J1]
 D. A. Jordan, On the simplicity of Lie algebras of derivations of commutative algebras, J. Alg. 228 (2000), 580585. MR 2001d:16052
 [J2]
 D. A. Jordan, Iterated skewpolynomial rings and quantum groups, J. Alg. 156 (1993), 194218. MR 94b:16034
 [K]
 N. Kawamoto, Generalizations of Witt algebras over a field of characteristic zero, Hiroshima Math. J. 16 (1986), 417462. MR 88d:17017
 [O]
 J. M. Osborn, New simple infinitedimensional Lie algebras of characteristic 0, J. Alg. 185 (1996), 820835. MR 98a:17035
 [OZ1]
 J. M. Osborn and K. Zhao, Generalized Poisson bracket and Lie algebras of type H in characteristic 0, Math. Z. 230 (1999), 107143. MR 2000c:17038
 [OZ2]
 J. M. Osborn and K. Zhao, Generalized Cartan type K Lie algebras in characteristic 0, Comm. Alg. 25 (1997), 33253360. MR 98e:17032
 [P]
 D. P. Passman, Simple Lie algebras of Witt type, J. Alg. 206 (1998), 682692. MR 99j:17012
 [S1]
 Y. Su, Derivations of generalized Weyl algebras, Science in China, to appear.
 [S2]
 Y. Su, Cocycles on the Lie algebras of generalized differential operators, Comm. Alg., to appear.
 [SZ1]
 Y. Su and K. Zhao, Simple algebras of Weyl type, Science in China (Series A) 44 (2001), 419426. CMP 2001:12
 [SZ2]
 Y. Su, K. Zhao, Second cohomology group of generalized Witt type Lie algebras and certain representations, Comm. Alg., to appear.
 [SZ3]
 Y. Su, K. Zhao, Isomorphism classes and automorphism groups of algebras of Weyl type, Science in China (Series A), to appear.
 [X]
 X. Xu, New generalized simple Lie algebras of Cartan type over a field with characteristic 0, J. Alg. 244 (2000), 2358. MR 2001b:17021
 [Z1]
 K. Zhao, Automorphisms of algebras of differential operators, J. of Capital Normal University 1 (1994), 18.
 [Z2]
 K. Zhao, Lie algebras of derivations of algebras of differential operators, Chinese Science Bulletin 38 (10) (1993), 793798.
 [Z3]
 K. Zhao, Isomorphisms between generalized Cartan type W Lie algebras in characteristic zero, Canadian J. Math. 50 (1998), 210224. MR 99e:17035
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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
DOI:
http://dx.doi.org/10.1090/S0002993901062189
PII:
S 00029939(01)062189
Keywords:
Simple Lie algebra,
simple associative algebra,
derivation
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
Article copyright:
© Copyright 2001 American Mathematical Society
