Growth in enveloping algebras
Let S be an associative or a Lie algebra over a field k such that S is generated by a finite subset X and let S(X,n) denote the subspace of S spanned by all monomials on X of length less than or equal to n. The growth function of S with respect to X is defined by \(\gamma_ s(n)=\dim S(X,n)\). It is known that \(\lambda =\lim_{n\to \infty}(\gamma_ s(n)^{1/n})\) always exists and is independent of X [see for example \textit{M. K. Smith}, Proc. Am. Math. Soc. 60(1976), 22-24 (1977; Zbl 0347.17005)]. If \(\lambda >1\) then S has exponential growth, otherwise the growth is subexponential. Also S has polynomially bounded growth if there exists a polynomial p such that \(\gamma_ s(n)\leq p(n)\) for all n. It is proved in this paper that the universal enveloping algebra U of an arbitrary finitely generated solvable-by-finite Lie algebra \({\mathfrak g}\) has subexponential growth and hence any subring of U is an Ore domain. If in addition \({\mathfrak g}\) is infinite dimensional then \({\mathfrak g}\) contains a subalgebra \({\mathfrak h}\) which can be mapped homomorphically on the Lie algebra \({\mathfrak k}\) with basis \(x,y_ 1,y_ 2,y_ 3,..\). such that \([y_ i,x]=y_{i+1}\), \([y_ i,y_ j]=0\) (i,j\(\geq 1)\), and in this case U does not have polynomially bounded growth. Wreath products are a key tool in the proof that U has subexponential growth.
scientific article; zbMATH DE number 3879108
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