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Growth in enveloping algebras. (English) Zbl 0552.17006

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 \(\displaystyle \lambda = \lim_{n\to \infty}(\gamma_S(n)^{1/n})\) always exists and is independent of \(X\) (see for example 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 \(x,y_1,y_2,y_3,\ldots\) 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.

MSC:

17B35 Universal enveloping (super)algebras
17B30 Solvable, nilpotent (super)algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
17B65 Infinite-dimensional Lie (super)algebras
16U10 Integral domains (associative rings and algebras)

Citations:

Zbl 0347.17005
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References:

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