Minimal Faithful Representations Of Abelian Jordan Algebras And Lie Superalgebras

In1905, I. Schur posed the maximal dimension for abelian subalgebras of thegenerally linear Lie superalgebra glm(F) over algebraically closed fields of charac-teristic zero. Consequently, it was verified that the minimal faithful representationof any finite dimensional abelian Lie algebra. However, the minimal representationis still open for any finite dimensional abelian Lie superalgebra. In this paper, ourmain purpose is to verify the minimal faithful representation for any finite dimen-sional abelian Jordan algebra and purely odd Lie superalgebra(with trivial evenparts and the definition is given in the following chapter). This result is helpful indetermining the minimal faithful representations for any abelian Lie superalgebra.In the following, the field F is denoted as an algebraically closed fields of char-acteristic zero. In view of the method about the minimal faithful representationsof any finite dimensional Lie algebra, we study the maximal dimension of the sub-algebras in the matrix algebras consisting of the mutual anticommutative matix.And we give the classification about these subalgebras in the sense of conjunction.Consequently, we obtain the minimal dimension of the faithful representation forany abelian Jordan algebra over the field F. We study the minimal faithful rep-resentation of any finite dimensional Lie superalgebra using above idea. And, wecan gain the maximal dimension of purely odd subalgebras for the generally linearLie superalgebras gl(m, n) over the field F. Furthermore, we obtain the minimaldimension of faithful representations for purely odd Lie superalgebras.However, we can not verify the minimal faithful representation for any abelianLie superalgebras with nontrivial even parts. But we give the maximal dimension ofabelian subalgebras for gl(1, n). The author expects the main results of this paperwill provide a useful reference for a further investigation.