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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.

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