Geodesic
Between $\widehat {gl}(\infty )$ and $\widehat {sl}_N$ affine algebras I. Geometrical actions
Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 1, pp. 82-96

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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 $\hat {gl}(\infty)$. 
@article{TMF_1994_100_1_a7,
     author = {M. I. Golenishcheva-Kutuzova and D. R. Lebedev and M. A. Olshanetsky},
     title = {Between $\widehat {gl}(\infty )$ and $\widehat {sl}_N$ affine algebras {I.} {Geometrical} actions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {82--96},
     publisher = {mathdoc},
     volume = {100},
     number = {1},
     year = {1994},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1994_100_1_a7/}
}
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M. I. Golenishcheva-Kutuzova; D. R. Lebedev; M. A. Olshanetsky. Between $\widehat {gl}(\infty )$ and $\widehat {sl}_N$ affine algebras I. Geometrical actions. Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 1, pp. 82-96. http://geodesic.mathdoc.fr/item/TMF_1994_100_1_a7/