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The universal enveloping algebras of certain Lie algebras provide examples of integral domains with growth which is subexponential but not bounded by any polynomial. 1. Suppose the (not necessarily associative) algebra S is generated over the field k by the finite set X. Let S (X, n) denote the k-subspace of S spanned by all monomial words of length less than or equal to n in the elements of X. The growth function ys (n) of S with respect to X is defined as dim S (X, n).

Universal enveloping algebras with subexponential but not polynomially bounded growth