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On some infinite-dimensional Lie algebras. (Russian) Zbl 0747.17024

Mathematical analysis, algebra and probability theory, Collect. Sci. Works, Tashkent, 11-18 (1987).
Let \(M\) denote the Lie algebra of infinite matrices \(A=(a_{ij})\) with only finitely many nonzero coefficients from a field of characteristic zero. If \(H\) is the subalgebra of diagonal matrices and \(e_{ij}\) denotes the standard basis of \(M\), then the linear functionals \(\alpha(h)=h_ i-h_ j\), determined by \([h,e_{ij}]=\alpha(h)\cdot e_{ij}\), can be regarded as roots \((i,j)\) of \(M\) with respect to \(H\). Simple roots have the indices \((i,i+1)\), positive and negative roots are defined in an obvious way. If \(\mathbb{R}^+\) is the set of all positive roots, then a subalgebra \(P\) of \(M\) is called parabolic, if \(P\) contains the subalgebra \(B=H+\sum_{\alpha\in\mathbb{R}^+}F_ \alpha\). A subalgebra \(A\) of \(M\) is called a standard subalgebra, if it is an ideal of a parabolic subalgebra.
The author characterizes standard subalgebras and particularly standard subalgebras of nilpotent elements (nil-algebras) of \(M\) with respect to roots and root systems. This generalizes earlier results of G. B. Gurevich [Mat. Sb., Nov. Ser. 35, 437–460 (1954; Zbl 0056.26501)] and Yu. B. Khakimdzhanov [Vestn. Mosk. Univ., Ser. I 29, No. 6, 49–55 (1974; Zbl 0298.17008)].
[For the entire collection see Zbl 0714.00011.]

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
17B20 Simple, semisimple, reductive (super)algebras
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