| The main object of study in this paper is a finite dimensional nilpotent algebra A over a field F. Every such algebra has a canonical basis and we can associate with it a sequence of non-negative integers, "n(,i)", called the genus of the algebra. The index of nilpotency of A is the least integer (alpha) for which A('(alpha)) = 0. In 5 , these concepts were used to classify all nilpotent algebras of dimension (LESSTHEQ)4. In Chapter 1, we take a closer look at the genus and find necessary and sufficient conditions for "n(,i)" to be the genus of a nilpotent algebra. As a consequence, we show that there can be no nilpotent algebra of dimension n and index 2('n-1). An upper bound for the index of nilpotency is determined in terms of the dimension of the algebra and it is shown that this upper bound is achieved.; In Chapter 2, we study the structure of the automorphism group of A. This was done in 4 for the associative algebra of all n x n strictly triangular matrices over F. We show that Aut(A) contains a normal subgroup M, the group of monic automorphisms of A, which is a nilpotent group and we associate with it a sequence of subspaces of matrices whose dimensions provide invariants of the isomorphism classes of nilpotent algebras of a given genus. In the case (alpha) = 4, we construct a normal subgroup G of M, which we call the group of inner automorphisms of A. Our construction, when applied to matrices, yields the classical inner automorphism group. M is shown to be isomorphic to a quotient of the direct product of G by a matrix group. Diagonal automorphisms were also defined in 4 . A theorem about the eigenvalues of a diagonal automorphism being the product of eigenvalues concludes this chapter.; To each finite dimensional algebra A we can associate a system of differential equations, 7 , 3 , 10 . First integrals are defined for differential equations and can be used to study the behavior of solutions. In 10 , polynomial first and higher integrals were defined for an algebra. In Chapter 3 we give the appropriate definitions for non-associative as well as associative polynomial integrals of a given degree. We show that, in case A is nilpotent, the dimensions of the linear non-associative polynomial integrals coincide with the genus. Conversely, if every linear polynomial is an integral, then the algebra is nilpotent. If one considers associative integrals, then this leads to a class of algebras satisfying a certain polynomial identity in each degree. Finally, we show that the identity in degree 3 implies that the algebra is nilpotent while the identity in degree 4 implies that the algebra is solvable. |
