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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.

##### MSC:
 17B50 Modular Lie (super)algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
##### Keywords:
invariants; Lie algebras of Cartan type