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Isomorphisms and derivations of modular Lie algebras of Cartan type. (English. Russian original) Zbl 0654.17009

Russ. Math. Surv. 42, No. 6, 245-246 (1987); translation from Usp. Mat. Nauk 42, No. 6, 201-202 (1987).
A Lie algebra L is said to be of Cartan type if one of the following inclusion relations is satisfied: (1) \(W^{(1)}\subseteq L\subseteq W\); (2) \(S(\omega)^{(1)}\subseteq L\subseteq CS(\omega)\); (3) \(H(\omega)^{(2)}\subseteq L\subseteq H(\omega)\); (4) \(K(\omega)^{(1)}\subseteq L\subseteq K(\omega)\), where \(\omega\) is the volume, the Hamiltonian and the contact form in (1), (2) and (3) respectively (the notations W, S, etc. follow V. G. Kac [Izv. Akad. Nauk SSSR, Ser. Mat. 38, 800-834 (1974; Zbl 0317.17002)] which refer to special derivation algebras of the divided power algebra \(O(m_ 1,m_ 2,...,m_ n))\). If \(p=2\), the contact algebras and the degenerate cases \(W_ 1(1)\) and \(H_ 2(m_ 1,1)\) are excluded. Let \(L_{\max}=W\), CS(\(\omega)\), CH(\(\omega)\) and K(\(\omega)\) respectively and \(\bar L_{\max}\) be the p-envelope of \(L_{\max}\) in Der O(m\({}_ 1,...,m_ n).\)
The main results are: (1) if \(p>3\), Der L is isomorphic to the normalizer of L in \(\bar L_{\max}\); (2) let \(L,L'\) be Lie algebras of Cartan type corresponding to \(O(m_ g\) from exposition and explanation of the simplest notions - fields, rings, modules, groups, the work leads to a survey of more complicated theories: representations of groups, Lie algebras, homological algebras, K-theory.

MSC:

17B50 Modular Lie (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras

Citations:

Zbl 0317.17002
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