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On invariants of certain modular Lie algebras. (English. Russian original) Zbl 0852.17018
Russ. Math. 37, No. 7, 22-26 (1993); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1993, No. 7 (374), 24-28 (1993).
For a finite-dimensional Lie algebra \(L\) over the prime field \(\mathbb{F}_p\) a \( \mathbb{Z}\)-form \(L_\mathbb{Z}\) is considered (it is not supposed to be a Lie ring at all). If \(L\) is a split Lie algebra with respect to torus \(T\), then the root system may be lifted to \(L_\mathbb{Z}\). In some cases the Weyl group \(W\) may be defined. The usual arguments are applied to show that the restriction of invariants gives the imbedding \(i\) of \(S (L^*_\mathbb{Q})^{L_\mathbb{Q}}\) into \(S (T^*_\mathbb{Q} )^W\), where \(L_\mathbb{Q} = \mathbb{Q} \otimes L_\mathbb{Z}\). Under some extra conditions \(i(S (L^*_\mathbb{Z} )^{L_\mathbb{Z}}) \subset S(T^*_\mathbb{Z})^W\) and \(i(\mathbb{F}_p \otimes S(L^*_\mathbb{Z})^{L_\mathbb{Z}}) \subset \mathbb{F}_p \otimes S(T^*_\mathbb{Z})^W\). For Lie algebras of Cartan type \(W_n\) and \(H_n\) the invariants \(S(T^*_\mathbb{Z})^W\) are computed.

17B50 Modular Lie (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)