Introduction

Defining an algebra by structure constants gives a very general set-up, but many structural concepts are restricted to Lie algebras.Therefore, Magma provides a special category for structure constant algebras which are known to be Lie. See [dG00] for more details on the theory of Lie algebras and algorithms for them.

Lie algebras are viewed as free modules over a base ring R with a multiplication satisfying the usual axioms. Some functions require additional conditions on the base ring; for example, quotients can only be constructed over fields, since otherwise the quotient module is not necessarily free.

Lie algebras in Magma are structure constant algebras. That is, the Lie algebra L of dimension n over a ring R is defined in Magma by giving the n^3 structure constants a_(ij)^k in R ( 1 <= i, j, k <= n) such that, if (e_1, e_2, ..., e_n) is the basis of L, e_i * e_j = sum_(k = 1)^n a_(ij)^k * e_k.