Simple algebras of Weyl type, II

Author:
Kaiming Zhao

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1323-1332

MSC (2000):
Primary 16W10, 16W25, 17B20, 17B65, 17B05, 17B68

Published electronically:
October 25, 2001

MathSciNet review:
1879953

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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**, 10.1090/S0002-9947-98-01786-3**[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**, 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****[H]**I. N. Herstein,*On the Lie and Jordan rings of a simple associative ring*, Amer. J. Math.**77**(1955), 279–285. MR**0067871****[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**, 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**, 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****[O]**J. Marshall Osborn,*New simple infinite-dimensional Lie algebras of characteristic 0*, J. Algebra**185**(1996), no. 3, 820–835. MR**1419725**, 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**, 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**, 10.1080/00927879708826056**[P]**D. S. Passman,*Simple Lie algebras of Witt type*, J. Algebra**206**(1998), no. 2, 682–692. MR**1637104**, 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), 419-426. 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**, 10.1006/jabr.1998.8083**[Z1]**K. Zhao,*Automorphisms of algebras of differential operators*, J. of Capital Normal University**1**(1994), 1-8.**[Z2]**K. Zhao,*Lie algebras of derivations of algebras of differential operators*, Chinese Science Bulletin**38**(10) (1993), 793-798.**[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**, 10.4153/CJM-1998-011-6

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16W10,
16W25,
17B20,
17B65,
17B05,
17B68

Retrieve articles in all journals with MSC (2000): 16W10, 16W25, 17B20, 17B65, 17B05, 17B68

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/S0002-9939-01-06218-9

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