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Cohomologies of Lie algebras of generalized Jacobi matrices. (English. Russian original) Zbl 0544.17011
Funct. Anal. Appl. 17, 153-155 (1983); translation from Funkts. Anal. Prilozh. 17, No. 2, 86-87 (1983).
Let \(k\) be a field of characteristic zero. A generalized Jacobi matrix over \(k\) is a matrix \(\{a_{ij}\}\), \(i,j\in {\mathbb Z}\) with entries from \(k\) having only a finite number of non-zero diagonals. We introduce the following notations. \(J\) is the associative algebra of the generalized Jacobi matrices (with the usual multiplication). \(J_+\) is the subalgebra consisting of the matrices \(\{a_{ij}\}\) such that \(a_{ij}=0\) if \(i\leq 0\) or \(j\leq 0\). \(F\) is the ideal in \(J\) consisting of finite matrices, and \(F_+=F\cap J_+\). \(J_ n\) is the subalgebra consisting of the fixed elements of the automorphism \(\{a_{ij}\}\mapsto \{a_{i+n,j+n}\}\) of \(J\). \(J_{\infty}\) is the minimal subalgebra containing \(J_ n\) for all \(n=1,2,\ldots\). For any associative algebra \(A\), we denote by \({\mathfrak g}A\) the corresponding Lie algebra with the bracket \([a,b]=ab-ba\). (Notice that \({\mathfrak g}J_ n=\mathfrak{gl}_ n(k[t,t^{-1}])\).)
The authors present the following results (all the cohomologies are with the trivial coefficients \(k\)):
a) \(H^*({\mathfrak g}J)\cong k[c_ 1,c_ 2,\ldots],\) where \(c_ i\in H^{2i}({\mathfrak g}J)\);
b) \(H^*({\mathfrak g}(J_+/F_+))\cong H^*({\mathfrak g}J)\);
c) \(H^*({\mathfrak g}J_{\infty})\cong k[\xi_ 1,\xi_ 2,\ldots;c_ 1,c_ 2,\ldots],\) where \(\xi_ i\in H^{2i-1}({\mathfrak g}J_{\infty}),\quad c_ i\in H^{2i}({\mathfrak g}J_{\infty}),\) and the homomorphism \(H^*({\mathfrak g}J)\to H^*({\mathfrak g}J_{\infty})\) induced by the inclusion maps “\(c_ i\) into \(c_ i\)”;
d) the composition \[ H^*({\mathfrak g}J_{\infty})\to H^*({\mathfrak g}J_ n)=H^*(\mathfrak{gl}_ n(k[t,t^{-1}]))\to H^*(\mathfrak{sl}_ n(k[t,t^{-1}])) \] is surjective and its kernel is generated by the elements \(\xi_ 1,\xi_{n+1},\xi_{n+2},\ldots\); \(c_ n,c_{n+1},\ldots\);
e) \(H^ i({\mathfrak g}J_+)=0\) for \(i>0\).
The explicit forms of the generators \(c_ i\) and \(\xi_ i\) are described in the paper.
Only the proof of a) is sketched in the paper. The other results b)–e) stay without proofs.

17B56 Cohomology of Lie (super)algebras
Full Text: DOI
[1] A. Weil, Basic Number Theory, Springer-Verlag, New York?Berlin (1967). · Zbl 0176.33601
[2] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press (1956).
[3] J.-L. Verdier, Les Représentation des Algèbras de Lie Affines, Seminaire N. Bourbaki, Vol. 1981/1982, Exp. 596.
[4] B. L. Tsygan, Usp. Mat. Nauk,38, No. 2, 217-218 (1983).
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