Finite-dimensional Algebra Structure Constants Of The Induction And Expansion Of The Mold On The Group Algebra

In this paper, we introduce the concepts of structural constant and the cube matrix formed by those constants of finite algebras and coalgebras: Assume αl,α2,....,αn is a basis for a certain n-dimensional algebra A(coalgebra C), andαiαj= are called the structuralconstants of algebra A (coalgebra C );and the structural constants form a wxwxw cube matrix. According to this , we give the definition of structural constants of Lie algebras, Lie superalgebras, (r -graded ) e Lie algebras. So we can, at a different angle, show and study finite algebras, coalgebras, bialgebras, Hopf algebras and Lie algebras and generalized Lie algebras. And , we obtain the necessary and sufficient conditions for a cube matrix [N] to be the matrix of a n-dimensional algebra A (coalgebra C ) under the basis α1=1A,α2,......,αn(E(α1)-l;e(αk)=0,k=2,3,...,n) (Theorem 2.1.2, Theorem 2.2.2). A certain cube matrix [N] is the matrix of a n-dimensional algebra (coalgebra) under its certain basis if and only if [N] is in equivalence with a cube matrix satisfying some conditions. We obtain a main result: the isomorphic classes of n-dimensional algebras and coalgebras are both in correspondence with the equivalent classes of cube matrix satisfying some conditions; a cube matrix is the cube matrix of an algebra, if and only if it is of a coalgebra; a cube matrix is the cube matrix of a certain algebra (coalgebra )under its certain basis, then, it is the dual algebra's (coalgebra's )cube matrix under the dual basis (Theorem2.2.6). And we also respectively obtain the necessary and sufficient conditions for a linear space to be a bialgebra, Hopf algebra, Lie algebra, Lie superalgebra, (r-graded )e Lie algebra expressed via structural constants and the cube matrix formed by them. (Theorem2.3.1, Theorem 2.3.2, Theorem1.4.2, Theorem2.4.3, Theorem2.4.4).Finally, we discuss the number of the indecomposable direct summand of the induced module from absolutely indecomposable θ[H]- module, and the sufficient condition's existence for absolutely irreducible #F[H] - module's extension(Theorem 3.1.6, Theorem 3.2.1).