On $\mathbf {3}$-graded Lie algebras, Jordan pairs and the canonical kernel function
Author:
M. P. de Oliveira
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 142-151
MSC (2000):
Primary 32M15; Secondary 22E46, 46E22
DOI:
https://doi.org/10.1090/S1079-6762-03-00122-7
Published electronically:
December 17, 2003
MathSciNet review:
2029475
Full-text PDF Free Access
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Abstract: We present several embedding results for $3$-graded Lie algebras and KKT algebras that are generated by two homogeneous elements of degrees $1$ and $-1$. We also propose the canonical kernel function for a “universal Bergman kernel” which extends the usual Bergman kernel on a bounded symmetric domain to a group-valued function or, in terms of formal series, to an element in the formal completion of the universal enveloping algebra of the free $3$-graded Lie algebra in a pair of generators.
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- Marcelo P. De Oliveira, Some formulas for the canonical kernel function, Geom. Dedicata 86 (2001), no. 1-3, 227–247. MR 1856428, DOI https://doi.org/10.1023/A%3A1011915708964
- M. P. de Oliveira, On commutation relations for 3-graded Lie algebras, New York J. Math. 7 (2001), 71–86. MR 1856712
DO3 M. P. De Oliveira, On $3$-graded Lie algebras in a pair of generators: a classification, J. Pure Appl. Algebra 178 (2003), no. 1, 73–85.
- M. P. de Oliveira and O. O. Luciano, A characterization of 3-graded Lie algebras generated by a pair, J. Pure Appl. Algebra 176 (2002), no. 2-3, 175–194. MR 1933714, DOI https://doi.org/10.1016/S0022-4049%2802%2900073-7
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HC3 Harish-Chandra, Representations of semisimple Lie groups VI. Integrable and Square-Integrable Representations, Amer. J. Math. 78 (1956), 564–628.
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DF M. G. Davidson and R. C. Fabec, Geometric realizations for highest weight representations, Contemp. Math., vol. 191, 1995, pp. 13–31.
DO1 M. P. De Oliveira, Some formulas for the canonical kernel function, Geom. Dedicata 86 (2001), no. 1, 227–247.
DO2 M. P. De Oliveira, On commutation relations for $3$-graded Lie algebras, New York J. Math. 7 (2001), 71–86.
DO3 M. P. De Oliveira, On $3$-graded Lie algebras in a pair of generators: a classification, J. Pure Appl. Algebra 178 (2003), no. 1, 73–85.
DOL M. P. De Oliveira and O. O. Luciano, A characterization of $3$-graded Lie algebras generated by a pair, J. Pure Appl. Algebra 176 (2002), no. 2-3, 175–194.
FA J. Faraut et al, Analysis and Geometry on Complex Homogeneous Domains, Progress in Mathematics, vol. 185, Birkhäuser, 2000.
FK J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 64–89.
HC3 Harish-Chandra, Representations of semisimple Lie groups VI. Integrable and Square-Integrable Representations, Amer. J. Math. 78 (1956), 564–628.
LA R. P. Langlands, The dimension of spaces of automorphic forms, Amer. J. Math. 85 (1963), 99–125.
LO O. Loos, Jordan Pairs, Lect. Notes Math, vol. 460, Springer, Berlin-Heidelberg-New York, 1975.
NE E. Neher, Generators and Relations for $3$-Graded Lie Algebras, J. Algebra 155 (1993), 1–35.
SA0 I. Satake, On unitary representations of a certain group extension, Sugaku
, Math. Soc., Japan 21 (1969), 241–253.
SA1 ---, Factors of automorphy and Fock representations, Adv. in Math. 7 (1971), no. 2, 83–110.
SA ---, Algebraic structures of symmetric domains, Iwanami Shoten, Tokyo and Princeton University Press, Princeton, NJ, 1980.
W N. Wallach, The analytic continuation of the discrete series I, Trans. Amer. Math. Soc. 251 (1979), 1–17.
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Additional Information
M. P. de Oliveira
Affiliation:
Department of Mathematics, University of Toronto, Canada
Email:
mpdeoliv@math.toronto.edu
Keywords:
Bergman kernel,
symmetric domain,
$3$-graded Lie algebra
Received by editor(s):
October 11, 2001
Received by editor(s) in revised form:
October 6, 2003
Published electronically:
December 17, 2003
Additional Notes:
The author has been partially supported by FAPESP
Communicated by:
Efim Zelmanov
Article copyright:
© Copyright 2003
American Mathematical Society
