Homology groups of nilpotent subalgebras of the Lie superalgebra \(K(1,1)\). II (Q2759263)

Some aspects of the cohomology of Lie algebras of vector fields are by now well understood, such as the cohomology of all vector fields on a compact manifold, but others remain a mystery. Gontcharova's theorem [\textit{L. V. Goncharova}, Funct. Anal. Appl. 7, 91-97 (1973); translation from Funkts. Anal. Prilozh. 7, 6-14 (1973; Zbl 0284.17006)] on the cohomology of the nilpotent subalgebras of the Lie algebra of polynomial vector fields on the line \(W(1)\) is of the latter type; besides her proof, which explicitly calculates kernels and images of the coboundary operator, there are at least three other proofs known. \textit{I. M. Gelfand, B. L. Feigin} and \textit{D. B. Fuks} [Funkts. Anal. Prilozh. 12, 1-5 (1978; Zbl 0396.17008)] used the Laplace operator, whereas \textit{B. L. Feigin} and \textit{V. S. Retakh} [Usp. Mat. Nauk 37, 233-234 (1982; Zbl 0493.58029)] deformed the Lie algebra into one accessible by Bott-Segal's approach. Then \textit{F. V. Weinstein} [Adv. Sov. Math. 17, 155-216 (1993; Zbl 0801.17022)] put the theorem into a more transparent combinatorial framework. The author of the article under review follows this last development for generalization.NEWLINENEWLINENEWLINEThe present research announcement is the continuation \textit{Yu. Yu. Kochetkov} [Funkts. Anal. Prilozh. 25, 86-88 (1991; Zbl 0741.17007)] of computations of the homology of nilpotent subalgebras of the Lie superalgebra \(K(1,1)\), which is the Lie superalgebra of polynomial vector fields in one even and one odd variable. The result is the proof that the second homology space \(H_2\) is finite dimensional. The author also provides a conjectured combinatorial description of \(H_2\) of \(L_k\), and a conjectured estimation on the dimension of \(H_3\).