Celebratio Mathematica — Fuchs — Complete Bibliography

Complete Bibliography

Works connected to D. I. Kalinin

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I. M. Gel’fand, D. I. Kalinin, and D. B. Fuks: “Cohomology of the Lie algebra of Hamiltonian formal vector fields,” Funkcional. Anal. i Priložen. 6 : 3 (1972), pp. 25–29. An English translation was published in Funct. Anal. Appl. 6:3 (1973). MR 312531 article

D. I. Kalinin	Related
Israïl Moiseevich Gelfand	Related

I. M. Gel’fand, D. I. Kalinin, and D. B. Fuks: “Cohomology of the Lie algebra of Hamiltonian formal vector fields,” Funct. Anal. Appl. 6 : 3 (July 1973), pp. 193–196. Zbl 0259.57023 article

As noted in [Gel’fand and Fuks 1970a], computing the cohomology of the Lie algebra of Hamiltonian formal vector fields is an essentially more difficult problem than, say, computing the cohomology of the Lie algebra of all formed vector fields, which was done in [Gel’fand and Fuks 1970b]. The methods used in [1970b] allow one to find the homology of a certain direct summand of the cochain complex of the algebra of Hamiltonian fields without great difficulty, but they yield no information about the complementary summand. In order to test the hypothesis that this complementary summand is acyclic, we have made some computations on an electronic computer. As a result, the above hypothesis has been rejected: we have discovered new and nontrivial cohomology classes of the algebra of Hamiltonian formal vector fields in \( \mathbb{R}^2 \). The important difference between these classes and the cohomology classes of the algebra of all formal vector fields found in [1970b] is that the former cannot be represented by cocycles depending only on the 2-sets of their arguments (see [Gel’fand and Fuks 1970c]).

This partial result seems interesting to us for two reasons. The first is methodological: it turns out that the difficulties encountered in computing the cohomology of the algebra of Hamiltonian fields are fundamental in origin. The second is as follows. The construction of [Godbillon and Vey 1971] and [Bernshtein, and Rozenfel’d 1972] can be carried over to the Hamiltonian case, making it possible to construct, for each cohomology class of the algebra of Hamiltonian formal vector fields, a characteristic class of Hamiltonian fibers of corresponding codimension. In particular, the classes which we will point out here furnish characteristic classes of Hamiltonian fibers of codimension 2. It would be interesting to determine whether or not these characteristic classes are nontrivial.

D. I. Kalinin	Related
Israïl Moiseevich Gelfand	Related