Maximal Dimension Of Purely Odd Subalgebras Of Jordan Superalgebras And Representations

In1905, I. Schur posed the maximal dimension of abelian subalgebras for thegenerally linear Lie superalgebra over algebraically closed fields of characteristiczero. Consequently, it was verified that the minimal dimension of faithful repre-sentations for any finite dimensional abelian Lie algebra. The minimal dimensionof faithful representations for any finite dimensional abelian Jordan algebra andpurely odd Lie superalgebra also had the corresponding conclusions. However, theminimal dimension of faithful representations is still open for any finite dimension-al abelian Jordan superalgebra. In this paper, the main purpose is to verify theminimal dimension of faithful representations for any finite dimensional purely oddJordan superalgebra. This result is helpful in determining the minimal dimensionof faithful representations for any abelian Jordan superalgebra.In the following, the fieldFis denoted as an algebraically closed fields of char-acteristic zero. In view of the method about the minimal dimension of faithfulrepresentations for purely odd Lie superalgebras, we apply the mentioned thoughton the research of Jordan superalgebra. Through the study of the maximal di-mension of purely odd subalgebras for Special Jordan superalgebras J(m, n), weconfirm the minimal dimension of faithful representations for purely odd Jordansuperalgebras.In addition, we carry on the detailed study on maximal dimension for abeliansubalgebras of Special Jordan superalgebras J(1, n). According to the study, thespecific form of odd part matrix of abelian subalgebras of J(1, n) can be dividedinto only three conditions: row standard type matrix, column standard type matrixand general type matrix. As a result, we get the maximal dimension of three kindsabelian subalgebras for Special Jordan superalgebras J(1, n).