Abstract
We give a description of simple nonassociative (−1,1)-superalgebras of characteristic ≠ 2, 3. It is proved that in such a superalgebra B, the even part A is a differentially simple, associative, and commutative algebra and the odd part M is a finitely generated, associative, and commutative A-bimodule, which is a projective A-module of rank 1. Multiplication in M is uniquely defined by a fixed finite, set of derivations and by elements of A. If, in addition, the bimodule M is one-generated, that is, M=Am for a suitable m∈M, then B is isomorphic to a twisted superalgebra of vector type B(Γ,D,γ). The condition M=Am is met, for instance, if A is local or isomorphic to a polynomial algebra. In particular, if B has a positive characteristic, which is the only possibility in the finite-dimensional case, then A is local and B is isomorphic to B(Γ,D,γ). In the general case, the question of whether the A-bimodule M is one-generated remains open.
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Supported by RFFR grant No. 96-01-01511.
Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 721–739, November–December, 1998.
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Shestakov, I.P. Simple (−1,1)-superalgebras. Algebr Logic 37, 411–422 (1998). https://doi.org/10.1007/BF02671695
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DOI: https://doi.org/10.1007/BF02671695