Geodesic
Orthogonal decomposition of some affine Lie algebras in terms of their heisenberg subalgebras
Teoretičeskaâ i matematičeskaâ fizika, Tome 102 (1995) no. 1, pp. 17-31 
L. A. Ferreira; D. I. Olive; M. V. Saveliev. Orthogonal decomposition of some affine Lie algebras in terms of their heisenberg subalgebras. Teoretičeskaâ i matematičeskaâ fizika, Tome 102 (1995) no. 1, pp. 17-31. http://geodesic.mathdoc.fr/item/TMF_1995_102_1_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the present note we suggest an affinization of a theorem by Kostrikin et. al. about the decomposition of some complex simple Lie algebras $\mathcal G$ into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out that the untwisted affine Kac–Moody algebras of types $A_{p^m-1}$ ($p$ prime, $m\geq 1$), $B_r$, $C_{2^m}$, $D_r$, $G_2$, $E_7$, $E_8$ can be decomposed into the algebraic sum of pairwise orthogonal Heisenberg subalgebras. The $A_{p^m-1}$ and $G_2$ cases are discussed in great detail. Some possible applications of such decompositions are also discussed.

[1] Soviet. Math. Dokl., 24 (1981), 292 | MR | Zbl

[2] Kostrikin A. I., Kostrikin I. A., Ufnarovskii V. A., Proc. Steklov Inst. Math., 4, 1983, 113–129 | MR

[3] Soviet. Math. Dokl., 25 (1982), 23–27 | MR | Zbl

[4] Kac V. G., Peterson D. H., “112 constructions of the basic representation of the loop group of $E_8$”, Symposium on anomalies, geometry and topology (Singapore, 1985), eds. W. A. Bardeen and A. R. White, World Scientific, 276–298 | MR

[5] Patera J., Zassenhaus H., J. Math. Phys., 29:3 (1988), 665–673 | DOI | MR | Zbl

[6] Fairlie D. B., Fletcher P., Zachos C. K., J. Math. Phys., 31 (1990), 1088–1094 | DOI | MR | Zbl

[7] Kac V. G., Kazhdan D. A., Lepowsky J., Wilson R. L., Adv. in Math., 42 (1981), 83–112 | DOI | MR | Zbl

[8] Kac V. G., Infinite Dimensional Lie Algebras, Cambridge University Press, 1990 | MR

[9] Kostant B., Amer. J. Math., 81 (1959), 973–1032 | DOI | MR | Zbl

[10] Goddard P., Nahm W., Olive D., Ruegg H., Schwimmer A., Comm. Math. Phys., 112 (1987), 385–408 ; Ferreira L. A., Gomes J. F., Zimerman A. H., Phys. Lett. B, 214 (1988), 367–370 ; Ferreira L. A., Gomes J. F., Teotônio Sobrinho P., Zimerman A. H., J. Physics A, 25 (1992), 5071–5088 ; Phys. Lett. B, 234 (1990), 315–320 ; Corrigan E., Hollowood T., Phys. Lett. B, 203 (1988), 47–51 ; Commun. Math. Phys., 122 (1989), 393–410 ; Gunaydin M., Hyun J., Phys. Lett. B, 209 (1988), 498–502 | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | Zbl | DOI | MR

[11] Olive D. I., Turok N., Underwood J. W. R., Nucl. Phys. B, 401 (1993), 663–697 | DOI | MR

[12] Olive D. I., Saveliev M. V., Underwood J. W. R., Phys. Lett. B, 311 (1993), 117–122 | DOI | MR | Zbl

[13] Leznov A. N., Saveliev M. V., Group Theoretical Methods for Integration of Nonlinear Dynamical Systems, Progress in Physics, 15, Birkhaüser-Verlag, Basel, 1992 | MR | Zbl