An example of a variety of Lie algebras with a fractional exponent
Let \(V\) be a variety of linear algebras and \(F(V)\) its countable rank relatively free algebra generated by elements \(y_1,y_2,\dots\). Let \(P_n(V)\subset F(V)\) be a subspace of multilinear elements in \(y_1,\dots,y_n\in F(V)\). Denote \(c_n(V)=\text{dim}P_n(V)\), often these numbers are referred to as the (codimension) growth sequence. One defines lower and upper exponents \[ \underline{\text{EXP}}(V)=\liminf_{n\to\infty}\root n\of{c_n(V)}, \quad \overline{\text{EXP}}(V)=\limsup_{n\to\infty}\root n\of{c_n(V)}. \] For varieties of associative algebras the codimension growth is at most exponential (A. Regev). Moreover, by a recent result of M. Zaitsev and A. Giambruno, over fields of zero characteristic these two numbers always coincide and are integers. For Lie algebras the situation is more complicated. It is known that the codimension growth might be overexponential. As a main result the authors construct a solvable variety of Lie algebras \(V\) with nonintegral exponents: \(3<\underline{\text{EXP}}(V)\leq \overline{\text{EXP}}(V)<4\).
scientific article; zbMATH DE number 1315231
scientific article