Cartan Subalgebras in Lie Algebras of Associative Algebras

A Cartan subalgebra of a finite-dimensional Lie algebra L is a nilpotent subalgebra H of L that coincides with its normalizer NL H . Such subalgebras occupy an important place in the structure theory of finite-dimensional Lie algebras and their properties have been explored in many articles (see, e.g., Barnes, 1967; Benkart, 1986; Wilson, 1977; Winter, 1969). In general (more precisely, when the ground field has few elements with respect to the dimension: cf. Siciliano, 2003), it is not clear whether a Lie algebra has Cartan subalgebras. Nevertheless, the combination of the results in Barnes (1967) and Winter (1969) ensures that the Lie algebras arising from associative algebras always have Cartan subalgebras. For an associative algebra A, we denote by ALie the Lie algebra associated with A by means of the Lie product x y = xy − yx, for every x y ∈ A. The aim of this article is to describe the Cartan subalgebras of ALie and their relationship with the associative structure of A. In Section 1, we introduce a definition of torus for associative algebras defined over arbitrary fields. By means of this, a complete description of the Cartan subalgebras of the Lie algebras arising from associative algebras is obtained. Indeed, we prove the following result.