Maximal Dimension And Representation For Zero Multiplier Subalgebra Of Matrix Algebra M(n,F)

In1905, I. Schur posed the maximal dimension for zero multiplication sub-algebras of the generally linear Lie superalgebra gl(n, F) over algebraically closed fields of characteristic zero. Consequently, it was verified that the minimal faithful representation of any finite dimensional zero multiplication Lie algebra. However, the minimal representation is still open for any finite dimensional zero multiplica-tion Lie superalgebra. In this paper, our main purpose is to verify the maximal dimension for the minimal faithful representation of the maximal dimension for ze-ro multiplication of matrix algebra M(n, F) and any finite dimensional associative algebra. This result is helpful in determining the minimal faithful representations of any finite dimensional zero multiplication Lie algebra.In the following, the field F is denoted as an algebraically closed fields of char-acteristic zero. In viwe of the method about the minimal faithful representations of any finite dimensional Lie algebra, we study the maximal dimension of the subalge-hras in the matrix algebras M(n,F) consisiting of the mutual zero multiplier matix. And we give the classification about these subalgebras in the sense of conjunction. Consequently, we obtain the minimal dimension of the faithful representation for any zero multiplier associative algebra over the field F. At last, we study the re-lations of zero multiplier subalgebra for Jordan algebras, the generally linear Lie algebras and the matrix algebras. The author expects the main results of this paper will provide a useful reference for a further investigation on the minimal faithful representation of zero multiplication Lie superalgebras.