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Irreducible non-Lie modules for Malcev superalgebras. (English) Zbl 0824.17005

The authors classify the irreducible almost faithful non-Lie modules for a Malcev superalgebra \(M= M_ 0+ M_ 1\) over a ring \(\varphi\) which contains \(1/6\). Let \(V= V_ 0+ V_ 1\) be such a module. There are three cases, loosely described as follows:
(1) \(M_ 1= 0\), \(M= M_ 0\) is a simple Lie algebra of dimension 3 over its centroid and \(V\) is a certain module described in the paper.
(2) \(M_ 1= 0\), \(M= M_ 0\) is a simple non-Lie Malcev algebra, seven dimensional over its center and \(V\) is the adjoint module.
(3) \(M_ 0= 0\), there is a field \(F\) over \(\varphi\) with submodule \(T\) such that \(M_ 1\) is an isomorphic copy of \(T\), \(V_ 0\) and \(V_ 1\) are isomorphic copies of \(F\), the action of \(M\) on \(V\) is described and there is a non-zero element \(y\in F\) such that \(T^ 2 y\) generates \(F\) as a ring.

MSC:

17A70 Superalgebras
17D10 Mal’tsev rings and algebras
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