This article is cited in 3 papers

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

M. I. Golenishcheva-Kutuzova^a, D. R. Lebedev^b, M. A. Olshanetsky<sup>b

a</sup> University of Cambridge
^b Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: 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 $\hat {gl}(\infty)$.

 English version: Theoretical and Mathematical Physics, 1994, 100:1, 863–873

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