Asymptotic properties of free finitely generated algebras of certain varieties
The author discusses the rate of growth of finitely generated algebras of certain varieties of not necessarily associative algebras. An important characteristic of such a growth is the number \(\tau\) (d) such that \(c_ 1 n^{\tau (d)}<\dim F(n)<c_ 2 n^{\tau (d)}\) (for sufficiently large n) where F(n) is the homogeneous component of degree n in a d-generated varietally free algebra. The existence of \(\tau\) (d) is shown here for the varieties generated by the algebra of \(r\times r\) matrices over a field k as well as by the Cayley-Dixon algebra over k and some related algebras. In fact this number is equal to the transcendence degree of the center of the varietally free generating algebra of the corresponding variety. The additivity of the above characteristic is shown for certain special products of algebras. This property proves useful in computing the rate of growth of free algebras of some not necessarily associative varieties.
scientific article; zbMATH DE number 3871602
scientific article