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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.