Higherorder Sugawara operators for affine Lie algebras
Authors:
Roe Goodman and Nolan R. Wallach
Journal:
Trans. Amer. Math. Soc. 315 (1989), 155
MSC:
Primary 17B67; Secondary 15A72, 20G45
MathSciNet review:
958893
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Abstract: Let be the affine Lie algebra associated to a simple Lie algebra . Representations of are described by current fields on the circle and . In this paper a linear map from the symmetric algebra to (formal) operator fields on a suitable category of modules is constructed. The operator fields corresponding to invariant elements of are called Sugawara fields. It is proved that they satisfy commutation relations of the form with the current fields, where is a renormalization of the central element in and is the derivative of the Dirac delta function. The higherorder terms in are studied using results from invariant theory and finitedimensional representation theory of . For suitably normalized invariants of degree or less, these terms are shown to be zero. This vanishing is also proved for and running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highestweight modules for .
 [CI]
Vyjayanthi
Chari and S.
Ilangovan, On the HarishChandra homomorphism for
infinitedimensional Lie algebras, J. Algebra 90
(1984), no. 2, 476–490. MR 760024
(85m:17007), http://dx.doi.org/10.1016/00218693(84)901856
 [CGJ]
S. Coleman, D. Gross, and R. Jackiw, Fermion avatars of the Sugawara model, Phys. Rev. 180 (1969), 13591366.
 [Fre]
I.
B. Frenkel, Two constructions of affine Lie algebra representations
and bosonfermion correspondence in quantum field theory, J. Funct.
Anal. 44 (1981), no. 3, 259–327. MR 643037
(83b:17012), http://dx.doi.org/10.1016/00221236(81)900124
 [GO]
Peter
Goddard and David
Olive, KacMoody and Virasoro algebras in relation to quantum
physics, Internat. J. Modern Phys. A 1 (1986),
no. 2, 303–414. MR 864165
(87m:17027), http://dx.doi.org/10.1142/S0217751X86000149
 [Goo]
Roe
W. Goodman, Nilpotent Lie groups: structure and applications to
analysis, Lecture Notes in Mathematics, Vol. 562, SpringerVerlag,
BerlinNew York, 1976. MR 0442149
(56 #537)
 [GW1]
Roe
Goodman and Nolan
R. Wallach, Whittaker vectors and conical vectors, J. Funct.
Anal. 39 (1980), no. 2, 199–279. MR 597811
(82i:22018), http://dx.doi.org/10.1016/00221236(80)900130
 [GW2]
Roe
Goodman and Nolan
R. Wallach, Classical and quantummechanical systems of Toda
lattice type. I, Comm. Math. Phys. 83 (1982),
no. 3, 355–386. MR 649809
(84e:58031)
 [GW3]
Roe
Goodman and Nolan
R. Wallach, Structure and unitary cocycle representations of loop
groups and the group of diffeomorphisms of the circle, J. Reine Angew.
Math. 347 (1984), 69–133. MR 733047
(86g:22024a), http://dx.doi.org/10.1515/crll.1984.347.69
 [GW4]
Roe
Goodman and Nolan
R. Wallach, Classical and quantum mechanical systems of
Todalattice type. III. Joint eigenfunctions of the quantized systems,
Comm. Math. Phys. 105 (1986), no. 3, 473–509.
MR 848652
(88d:58035)
 [Har]
HarishChandra,
Differential operators on a semisimple Lie algebra, Amer. J. Math.
79 (1957), 87–120. MR 0084104
(18,809d)
 [Kac]
Victor
G. Kac, Laplace operators of infinitedimensional Lie algebras and
theta functions, Proc. Nat. Acad. Sci. U.S.A. 81
(1984), no. 2, Phys. Sci., 645–647. MR 735060
(85j:17025), http://dx.doi.org/10.1073/pnas.81.2.645
 [Kos]
Bertram
Kostant, On Whittaker vectors and representation theory,
Invent. Math. 48 (1978), no. 2, 101–184. MR 507800
(80b:22020), http://dx.doi.org/10.1007/BF01390249
 [KK]
V.
G. Kac and D.
A. Kazhdan, Structure of representations with highest weight of
infinitedimensional Lie algebras, Adv. in Math. 34
(1979), no. 1, 97–108. MR 547842
(81d:17004), http://dx.doi.org/10.1016/00018708(79)900665
 [LW]
James
Lepowsky and Robert
Lee Wilson, The structure of standard modules. I. Universal
algebras and the RogersRamanujan identities, Invent. Math.
77 (1984), no. 2, 199–290. MR 752821
(85m:17008), http://dx.doi.org/10.1007/BF01388447
 [Mur]
Francis
D. Murnaghan, The theory of group representations, Dover
Publications, Inc., New York, 1963. MR 0175982
(31 #258)
 [PS]
Andrew
Pressley and Graeme
Segal, Loop groups, Oxford Mathematical Monographs, The
Clarendon Press, Oxford University Press, New York, 1986. Oxford Science
Publications. MR
900587 (88i:22049)
 [Sug]
H. Sugawara, A field theory of currents, Phys. Rev. 170 (1968), 16591662.
 [Wal]
Nolan
R. Wallach, A class of nonstandard modules for affine Lie
algebras, Math. Z. 196 (1987), no. 3,
303–313. MR
913657 (89c:17035), http://dx.doi.org/10.1007/BF01200353
 [Wey]
Hermann
Weyl, The classical groups, Princeton Landmarks in
Mathematics, Princeton University Press, Princeton, NJ, 1997. Their
invariants and representations; Fifteenth printing; Princeton Paperbacks.
MR
1488158 (98k:01049)
 [CI]
 V. Chari and S. Ilangovan, On the HarishChandra homomorphism for infinitedimensional Lie algebras, J. Algebra 90 (1984), 476490. MR 760024 (85m:17007)
 [CGJ]
 S. Coleman, D. Gross, and R. Jackiw, Fermion avatars of the Sugawara model, Phys. Rev. 180 (1969), 13591366.
 [Fre]
 I. B. Frenkel, Two constructions of affine Lie algebra representations and bosonfermion correspondence in quantum field theory, J. Funct. Anal. 44 (1981), 259327. MR 643037 (83b:17012)
 [GO]
 P. Goddard and D. Olive, KacMoody and Virasoro algebras in relation to quantum physics, Internat. J. Mod. Phys. A 1 (1986), 303414. MR 864165 (87m:17027)
 [Goo]
 R. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Math., vol. 562, SpringerVerlag, Berlin, 1976. MR 0442149 (56:537)
 [GW1]
 R. Goodman and N. R. Wallach, Whittaker vectors and conical vectors, J. Funct. Anal. 39 (1980), 199279. MR 597811 (82i:22018)
 [GW2]
 , Classical and quantummechanical systems of Toda lattice type. I, Comm. Math. Phys. 83 (1982), 355386. MR 649809 (84e:58031)
 [GW3]
 , Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. Reine Angew. Math. 347 (1984), 69133; erratum, 352, 220. MR 733047 (86g:22024a)
 [GW4]
 , Classical and quantummechanical systems of Toda lattice type. III. Comm. Math. Phys. 105 (1986), 473509. MR 848652 (88d:58035)
 [Har]
 HarishChandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87120. MR 0084104 (18:809d)
 [Kac]
 V. Kac, Laplace operators of infinitedimensional Lie algebras and theta functions, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 645647. MR 735060 (85j:17025)
 [Kos]
 B. Kostant, Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101184. MR 507800 (80b:22020)
 [KK]
 V. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinitedimensional Lie algebra, Adv. in Math. 34 (1979), 97108. MR 547842 (81d:17004)
 [LW]
 J. Lepowsky and R. L. Wilson, The structure of standard modules, I: universal algebras and the RogersRamanujan identities, Invent. Math. 77 (1984), 199290. MR 752821 (85m:17008)
 [Mur]
 F. D. Murnaghan, The theory of group representations, Dover, New York, 1963. MR 0175982 (31:258)
 [PS]
 A. Pressley and G. Segal, Loop groups, Oxford Univ. Press, New York, 1986. MR 900587 (88i:22049)
 [Sug]
 H. Sugawara, A field theory of currents, Phys. Rev. 170 (1968), 16591662.
 [Wal]
 N. R. Wallach, A class of nonstandard modules for affine Lie algebras, Math. Z. 196 (1987), 303313. MR 913657 (89c:17035)
 [Wey]
 H. Weyl, The classical groups, Princeton Univ. Press, Princeton, N.J., 1946. MR 1488158 (98k:01049)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198909588935
PII:
S 00029947(1989)09588935
Article copyright:
© Copyright 1989
American Mathematical Society
