Abstract
A metric is said to be Lie–Einstein if it is both a one-sided invariant metric on a Lie group and also an Einstein space. In this article right-invariant metrics are used thoughout. Several Lie–Einstein metrics on six-dimension Lie groups are constructed. The associated Lie algebras are necessarily solvable and are assumed to be indecomposable. They belong to two classes studied by Mubarakzyanov and Turkowski, respectively. Since the general problem of finding Lie–Einstein metrics in dimension six appears to be intractable, a certain ansatz for the form of the metric and class of Lie algebra is adopted. Nineteen Mubarakzyanov and five Turkowski Lie algebras are shown to be Lie–Einstein.
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