Some Lie superalgebras associated to the Weyl algebras
Author:
Ian M. Musson
Journal:
Proc. Amer. Math. Soc. 127 (1999), 28212827
MSC (1991):
Primary 17B35; Secondary 16W10
Published electronically:
May 4, 1999
MathSciNet review:
1616633
Fulltext PDF Free Access
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Abstract: Let be the Lie superalgebra . We show that there is a surjective homomorphism from to the Weyl algebra , and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of on and use this to show that if is made into a Lie superalgebra using its natural grading, then . In addition, we show that if and are isomorphic as Lie superalgebras, then . This answers a question of S. Montgomery.
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 Yu. Bahturin, D. Fischman and S. Montgomery, On the generalized Lie structure of associative algebras, Israel J. Math., 96 (1996), 2748. MR 98d:16048
 [CG]
 N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston 1997. CMP 9:08
 [CM]
 D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York, 1993. MR 94j:17001
 [D]
 J. Dixmier, Enveloping Algebras, North Holland, Amsterdam 1977. MR 58:16803b
 [DH]
 D. [??]Z. Djokovi\'{c} and G. Hochschild, Semisimplicity of 2graded Lie algebras, II, Illinois J. Math., 20 (1976) 134143. MR 52:8206
 [F]
 C. Fronsdal (editor), Essays on Supersymmetry, Mathematical Physics Studies 8, Reidel Publ. Comp., Dordrecht, 1986. MR 88a:81004
 [H]
 J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, SpringerVerlag, New York, 1972. MR 48:2197
 [He]
 I. N. Herstein, On the Lie and Jordan rings of a simple associative ring, Amer. J. Math. 77 (1955), 279285. MR 16:789e
 [Jan]
 J. C. Jantzen, Lectures on Quantum Groups, American Math. Society, 1996. MR 96m:17029
 [J1]
 A. Joseph, Minimal realizations and spectrum generating algebras, Comm. Math. Phys. 36 (1974), 325338. MR 49:6795
 [J2]
 A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. scient. Éc. Norm. Sup. 9 (1976), 129. MR 53:8168
 [K]
 V. Kac, Lie Superalgebras, Adv. in Math. 16 (1977), 896. MR 58:5803
 [Mo]
 S. Montgomery, Constructing simple Lie superalgebras from associative graded algebras, J. of Algebra, 195 (1997) 558579. CMP 98:01
 [M1]
 I. M. Musson, On the center of the enveloping algebra of a classical simple Lie superalgebra, J. Algebra, 193 (1997), 75101. CMP 97:14
 [M2]
 I. M. Musson, The enveloping algebra of the Lie superalgebra, , Representation Theory, an electronic journal of the AMS, 1 (1997), 405423. CMP 98:04
 [P]
 G. Pinczon, The enveloping algebra of the Lie superalgebra , J. Alg. 132 (1990), 219242. MR 91j:17014
 [PS]
 G. Pinczon and J. Simon, Nonlinear representations of inhomogeneous groups, Lett. Math. Phys. 2 (1978), 499504. MR 80a:22018
 [Sch]
 M. Scheunert, The theory of Lie superalgebras. An introduction, Lecture Notes in Mathematics, 716, SpringerVerlag, Berlin, 1979. MR 80i:17005
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Additional Information
Ian M. Musson
Affiliation:
Department of Mathematical Sciences, University of WisconsinMilwaukee, Milwaukee, Wisconsin 53201
Email:
musson@csd.uwm.edu
DOI:
http://dx.doi.org/10.1090/S000299399904976X
PII:
S 00029939(99)04976X
Keywords:
Lie superalgebras,
Weyl algebras,
Joseph ideal
Received by editor(s):
February 7, 1997
Received by editor(s) in revised form:
December 9, 1997
Published electronically:
May 4, 1999
Additional Notes:
This research was partially supported by NSF grant DMS 9500486.
Communicated by:
Ken Goodearl
Article copyright:
© Copyright 1999
American Mathematical Society
