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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

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
Full Text: DOI
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