Communicated by Alexander PremetTo Holger Petersson on his sixtieth birthday Finite dimensional simple Jordan superalgebras over algebraically closed fields of characteristic 0 have been classified by Kac [4,9] (see also Kantor [10]). Kac used his classification of Lie superalgebras to obtain his Jordan results. In [13] O. Kühn obtains a classification for unital normal Jordan superalgebras over arbitrary fields of characteristic not 2 or 3, where normal means that the even part possesses a linear form satisfying certain nondegeneracy conditions. Our purpose is to obtain a classification of simple Jordan superalgebras over fields of characteristic different from 2 whose even part is semisimple. Our proof is not as elegant as that of Kac, but since our methods are Jordan theoretic, they are less sensitive to the characteristic of the base field (modular Lie superalgebras have not yet been classified over an algebraically closed field). The main ingredients, Peirce decomposition and representation theory, can be found in [7]. Since the characteristic plays no role in most arguments we obtain the same superalgebras as Kac. However, in characteristic 3, there exists a 12-dimensional (i-exceptional) simple superalgebra having the 3 × 3 symmetric matrices as even part and a 21-dimensional (i-exceptional) simple superalgebra having the 3 × 3 symmetric matrices with entries in the split quaternions as even part; moreover the 10-dimensional Kac superalgebra is not simple in characteristic 3 but it has a 9-dimensional i-exceptional simple subsuperalgebra which we will call the degenerate Kac algebra. Our results over an algebraically closed field have been announced in [19]. The classification of the simple superalgebras whose even part is not semisimple is obtained in [14]. Applications of Jordan superalgebras can be found in [17].
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