Generators of Linear Algebras
E. M. Patterson · Proceedings of the London Mathematical Society · 1957
1. A SET of elements xv x2,..., xk of a linear algebra A of finite dimension is said to generate A if every element of A is linearly dependent on products of xx, x2,..., xk; the elements xx, x2,..., xk are then called generators of A. There are surprisingly few references to generators in the literature on linear algebras. M. S. Knebelman, in his paper 'Classification of Lie algebras ' (4) defined the nullity of a linear algebra to be the minimum number of linearly independent generators, and the genus to be the difference between the dimension and the nullity. A. A. Albert used the term minimum rank in place of nullity. In a short paper (1), Albert proved that a separable associative algebra over an infinite field has minimum rank 1 if it is commutative and minimum rank 2 if it is non-commutative. A corollary to this theorem is that a semi-simple associative algebra over a non-modular field (that is, a field of characteristic zero) can be generated by one element if it is commutative and by two elements if it is noncommutative. A similar result has been obtained for Lie algebras by M. Kuranishi (5), who proved that a semi-simple Lie algebra over a nonmodular field can be generated by two elements. In the general case it is not necessarily true that simple algebras can be generated by two elements. This is shown in § 6 below. In this paper we are concerned mainly with a certain type of structure appearing in Knebelman's paper. Knebelman stated that the constants of structure of a Lie algebra of dimension n and genus r, less than n—2, are given by < & = PiA+Stuk-6iUj (i,j,k = l,...,n; A = 1,..., r), (1.1) where ui are the components of a covariant vector, p$k are the components of r skew-symmetric second-order tensors, and s \\ are the components of r fixed elements of the algebra; here the summation convention for repeated indices is used.f Further conditions on ut and p*k follow from the Jacobi identity, and Knebelman used these conditions to classify Lie algebras of genus zero and one. If the constants of structure are given by (1.1), then f In the present paper we use the summation convention only when quoting from Knebelman's paper.