Pienvelopes of Lie superalgebras
Authors:
Yuri Bahturin and Susan Montgomery
Journal:
Proc. Amer. Math. Soc. 127 (1999), 28292839
MSC (1991):
Primary 17A70, 16W50
Published electronically:
April 23, 1999
MathSciNet review:
1600092
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper we find necessary and sufficient conditions on a finitedimensional Lie superalgebra under which any associative PIenvelope of is finitedimensional. We also extend M. Scheunert's result which enables one to pass from color Lie superalgebras to the ordinary ones, to the case of gradings by an arbitrary abelian group.
 [Ba 85]
Yu. A. Bahturin, On the structure of the PIenvelope of a finitedimensional Lie algebra, Soviet Math. (Iz. VUZ) 29 (1985) no. 11, 8387.
 [BMPZ ]
Yuri
A. Bahturin, Alexander
A. Mikhalev, Viktor
M. Petrogradsky, and Mikhail
V. Zaicev, Infinitedimensional Lie superalgebras, de Gruyter
Expositions in Mathematics, vol. 7, Walter de Gruyter & Co.,
Berlin, 1992. MR
1192546 (94b:17001)
 [BeC]
Jeffrey
Bergen and Miriam
Cohen, Actions of commutative Hopf algebras, Bull. London
Math. Soc. 18 (1986), no. 2, 159–164. MR 818820
(87e:16052), http://dx.doi.org/10.1112/blms/18.2.159
 [Bi]
Yu.
V. Billig, A homomorphic image of a special Lie algebra, Mat.
Sb. (N.S.) 136(178) (1988), no. 3, 320–323, 430
(Russian); English transl., Math. USSRSb. 64 (1989),
no. 2, 319–322. MR 959484
(89k:17015)
 [Bl]
R. J. Blattner, Induced and produced representations of Lie algebras, Transactions AMS 144 (1969), 457474. MR 46i:7338a
 [BZ]
Y. Bahturin, M.Zaicev, Identities of graded Lie algebras, J. Algebra 205 (1998), 112. CMP 98:15
 [CM]
M.
Cohen and S.
Montgomery, Groupgraded rings, smash products,
and group actions, Trans. Amer. Math. Soc.
282 (1984), no. 1,
237–258. MR
728711 (85i:16002), http://dx.doi.org/10.1090/S00029947198407287114
 [CR]
Miriam
Cohen and Louis
H. Rowen, Group graded rings, Comm. Algebra
11 (1983), no. 11, 1253–1270. MR 696990
(85b:16002), http://dx.doi.org/10.1080/00927878308822904
 [Du]
Michel
Duflo, Sur la classification des idéaux primitifs dans
l’algèbre enveloppante d’une algèbre de Lie
semisimple, Ann. of Math. (2) 105 (1977),
no. 1, 107–120. MR 0430005
(55 #3013)
 [Fa]
Daniel
R. Farkas, Semisimple representations and affine
rings, Proc. Amer. Math. Soc.
101 (1987), no. 2,
237–238. MR
902534 (88h:16027), http://dx.doi.org/10.1090/S00029939198709025343
 [Kac]
V.
G. Kac, Lie superalgebras, Advances in Math.
26 (1977), no. 1, 8–96. MR 0486011
(58 #5803)
 [Mo]
S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. Algebra 195 (1997), 558  579. CMP 98:01
 [Po]
H. Pop, A generalization of Scheunert's theorem on cocycle twisting of Lie color algebras, preprint, qalg 9703002.
 [S79a]
M.
Scheunert, Generalized Lie algebras, J. Math. Phys.
20 (1979), no. 4, 712–720. MR 529734
(80f:17007), http://dx.doi.org/10.1063/1.524113
 [S79b]
Manfred
Scheunert, The theory of Lie superalgebras, Lecture Notes in
Mathematics, vol. 716, Springer, Berlin, 1979. An introduction. MR 537441
(80i:17005)
 [Ba 85]
 Yu. A. Bahturin, On the structure of the PIenvelope of a finitedimensional Lie algebra, Soviet Math. (Iz. VUZ) 29 (1985) no. 11, 8387.
 [BMPZ ]
 Yu. A. Bahturin, A. Mikhalev, V. Petrogradskii, M. Zaicev, Infinite Dimensional Lie Superalgebras, Expos. Math. vol 7, Walter de Gruyter, Berlin, 1992. MR 94b:17001
 [BeC]
 J. Bergen and M. Cohen, Actions of commutative Hopf algebras, Bull. LMS 18 (1986), 159164. MR 87e:16052
 [Bi]
 Yuly Billig, On the homomorphic image of a special Lie algebra, Mat. Sc 136 (178)(1988), 320323; English transl. in Math. USSR Sb. 64 (1989) MR 89k:17015
 [Bl]
 R. J. Blattner, Induced and produced representations of Lie algebras, Transactions AMS 144 (1969), 457474. MR 46i:7338a
 [BZ]
 Y. Bahturin, M.Zaicev, Identities of graded Lie algebras, J. Algebra 205 (1998), 112. CMP 98:15
 [CM]
 M. Cohen and S. Montgomery, Groupgraded rings, smash products, and group actions, Trans. AMS 282 (1984), 237258. MR 85i:16002
 [CR]
 M. Cohen and L. Rowen, Group graded rings, Comm. Algebra 11 (1983), 1253 1270. MR 85b:16002
 [Du]
 M. Duflo, Sur la classification des ideaux primitifs dans l'algebre enveloppante d'une algebre de Lie semisimple, Ann. Math. 105 (1977), 107120. MR 55:3013
 [Fa]
 D. Farkas, Semisimple representations and affine rings, Proc. AMS 101 (1987), 237  238. MR 88h:16027
 [Kac]
 V. Kac, Lie superalgebras, Advances in Math. 26 (1977), 896. MR 58:5803
 [Mo]
 S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. Algebra 195 (1997), 558  579. CMP 98:01
 [Po]
 H. Pop, A generalization of Scheunert's theorem on cocycle twisting of Lie color algebras, preprint, qalg 9703002.
 [S79a]
 M. Scheunert, Generalized Lie algebras, J. Math Physics 20 (1979), 712720. MR 80f:17007
 [S79b]
 M. Scheunert, The Theory of Lie Superalgebras, Lecture Notes in Math., vol. 716, SpringerVerlag, Berlin, 1979. MR 80i:17005
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
17A70,
16W50
Retrieve articles in all journals
with MSC (1991):
17A70,
16W50
Additional Information
Yuri Bahturin
Affiliation:
Department of Algebra, Moscow State University, 119899 Moscow, Russia
Email:
bahturin@mech.math.msu.su
Susan Montgomery
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, California 900891113
Email:
smontgom@math.usc.edu
DOI:
http://dx.doi.org/10.1090/S000299399904825X
PII:
S 00029939(99)04825X
Received by editor(s):
May 27, 1997
Received by editor(s) in revised form:
December 11, 1997
Published electronically:
April 23, 1999
Additional Notes:
The authors were supported by NSF grant DMS9500649
Communicated by:
Lance W. Small
Article copyright:
© Copyright 1999
American Mathematical Society
