×

zbMATH — the first resource for mathematics

Quantization of the Gel’fand-Dikiĭ brackets. (English. Russian original) Zbl 0703.58022
Funct. Anal. Appl. 22, No. 4, 255-262 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 1-10 (1988).
The author starts with the following explanation: Let \(L\), \(P\) be the linear differential operators:
\[ L=\partial^n_x+\sum^n_{k=1} W_k(x,t)\partial_x^{n-k}. \]
We can consider Lax relation \(\partial L/\partial t=[L,P]\) as a system of partial differential equations with respect to unknown functions \(W_k(x,t)\).
Gel’fand and Dikiĭ constructed in the space of functionals over \(W_k(x,t)\) two symplectic structures such that this system of equations becomes Hamiltonian. The Gel’fand-Dikiĭ brackets are related to the Lie algebra \(\mathfrak{gl}(n)\) and maybe reduced to the simple Lie algebra \(\mathfrak{sl}(n)\). The author and V. A. Fateev proposed its quantization [Sov. Phys. 67, 447–454 (1980); translation from Zh. Èksp. Teor. Fiz. 94, 23–37 (1988)].
For \(n=2\) the \(W_n\) algebra turns out to be the Virasoro algebra, and in general case it is a generalization of the Poisson bracket structure.
The following theorems are proved: For a system \(\{\omega_i(z)\}_{i=2,\ldots,n}\) with simple Lie algebra \(\mathfrak{sl}(n)\) there is a closed \(w_n\) operator algebra whose structure is analogous to Gel’fand-Dikiĭ algebra.
Virasoro algebra and results of Khovanova (as \(h\to 0\)) are special cases of the Gel’fand-Dikiĭ construction.
Reviewer: Vadim Komkov

MSC:
53D50 Geometric quantization
17B80 Applications of Lie algebras and superalgebras to integrable systems
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
81S10 Geometry and quantization, symplectic methods
17B68 Virasoro and related algebras
17B63 Poisson algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] I. M. Gel’fand and L. A. Dikii, ”Fractional powers of operators, and Hamiltonian systems,” Funkts. Anal. Prilozhen.,10, No. 4, 13-29 (1976).
[2] I. M. Gel’fand and L. A. Dikii, ”The resolvent and Hamiltonian systems,” Funkts. Anal. Prilozhen.,11, No. 2, 11-27 (1977).
[3] V. G. Drinfel’d and V. V. Sokolov, ”Lie algebras and equations of Korteweg?de Vries type,” Itogi Nauki i Tekhniki, Sovrem. Probl. Mat.,24, 81-180 (1984).
[4] S. L. Luk’yanov and V. A. Fateev, ”Conformally invariant models of two-dimensional quantum field theory with Zn-symmetry,” Zh. Éksp. Teor. Fiz.,94, No. 3, 23-37 (1988).
[5] A. B. Zamolodchikov, ”Infinite extra symmetries in two-dimensional conformal quantum field theory,” Teor. Mat. Fiz.,65, No. 3, 347-359 (1985).
[6] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, ”Infinite conformal symmetry in two-dimensional quantum field theory,” Nucl. Phys.,241B, No. 2, 333-380 (1984). · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X
[7] N. N. Bogolyubov (Bogoliubov) and D. V. Shirkov, Introduction to the Theory of Quantized Fields (3rd ed.), Wiley, New York (1980).
[8] L. D. Faddeev and L. A. Takhtadzhyan (L. A. Takhtajan), Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987). · Zbl 0632.58004
[9] T. G. Khovanova, ”Gelfand-Dikii Lie algebras and the Virasoro algebra,” Funkts. Anal. Prilozhen.,20, No. 4, 89-90 (1986). · Zbl 0618.58018 · doi:10.1007/BF01077262
[10] V. A. Fateev and A. B. Zamolodchikov, ”Conformal quantum field theory models in two dimensions having Z3 symmetry,” Nuclear Phys. B,280, 644-660 (1987). · doi:10.1016/0550-3213(87)90166-0
[11] B. L. Feigin and D. B. Fuks, ”Skew-symmetric invariant differential operators on the line and Verma modules over the Virasoro algebra,” Funkts. Anal. Prilozhen.,16, No. 2, 47-63 (1982). · Zbl 0493.46061 · doi:10.1007/BF01081809
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.