Cohomologies of Lie algebras of generalized Jacobi matrices
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.
scientific article; zbMATH DE number 3865530