On Lie algebras of generalized Jacobi matrices

In these lecture notes, we consider infinite dimensional Lie algebras of generalized Jacobi matrices $\mathfrak g J(k)$ and $\mathfrak {gl}_\infty (k)$, which are important in soliton theory, and their orthogonal and symplectic subalgebras. In particular, we construct the homology ring of the Lie algebra $\mathfrak g J(k)$ and of the orthogonal and symplectic subalgebras.