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Gelfand-Kirillov dimension for non-associative algebras. (English) Zbl 1013.17500

An analogue of the Gelfand-Kirillov dimension for nonassociative algebras is defined as follows: \[ \text{GK-}\dim A=\limsup_{n\to \infty} \log_n d_V(n), \] where \(A\) denotes a not necessarily associative \(k\)-algebra, \(\text{char }k=0\), \(V_n\) the \(k\)-subspace of \(A\) spanned by all monomials in the finite generality subset \(V\) (of \(A\)) of length \(\leq n\) and \(d_V(n)=\dim_k V_n\).
The author shows that for a Lie algebra \(L\) GK-\(\dim L\) can take any natural number, that no Lie algebra can have GK-\(\dim L\) strictly between 0 and 1, and GK-\(\dim L =0\) in case \(L\) is a finitely generated nilpotent Lie algebra.

MSC:

17A99 General nonassociative rings
16P90 Growth rate, Gelfand-Kirillov dimension
17B30 Solvable, nilpotent (super)algebras
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