Between $\widehat {gl}(\infty )$ and $\widehat {sl}_N$ affine algebras I. Geometrical actions

We consider the central extended $\hat {gl}(\infty)$ Lie algebra and a set of its subalgebras parametrized by $|q|=1$ which coincides with the embedding of the quantum tori Lie algebras (QTLA) in $\hat {gl}(\infty)$. For $q^N=1$ there exists an ideal and a factor over this ideal is isomorphic to $\hat {sl}_N(z)$ affine algebra. For a generic value $q$ the corresponding subalgebras are dense in $\hat {gl}(\infty)$. Thus they interpolate between $\hat {gl}(\infty)$ and $\hat {sl}_N(z)$ . All these subalgebras are fixed points of automorphisms of $\hat {gl}(\infty)$. Using the automorphisms we construct geometrical actions for the subalgebras starting from the Kirillov–Kostant form and the corresponding geometrical action for $\hat {gl}(\infty)$.