Geodesic
Identities of unitary finite-dimensional algebras
Algebra i logika, Tome 50 (2011) no. 5, pp. 563-594

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We deal with growth functions of sequences of codimensions of identities in finite-dimensional algebras with unity over a field of characteristic zero. For three-dimensional algebras, it is proved that the codimension sequence grows asymptotically as $a^n$, where $a$ is $1,2$, or $3$. For arbitrary finite-dimensional algebras, it is shown that the codimension growth either is polynomial or is not slower than $2^n$. We give an example of a finite-dimensional algebra with growth rate $a^n$ with fractional exponent $a=\frac3{\sqrt[3]4}+1$.
Keywords: finite-dimensional unitary algebra, growth function of sequences of codimensions of identities. 
@article{AL_2011_50_5_a0,
     author = {M. V. Zaitsev},
     title = {Identities of unitary finite-dimensional algebras},
     journal = {Algebra i logika},
     pages = {563--594},
     publisher = {mathdoc},
     volume = {50},
     number = {5},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2011_50_5_a0/}
}
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M. V. Zaitsev. Identities of unitary finite-dimensional algebras. Algebra i logika, Tome 50 (2011) no. 5, pp. 563-594. http://geodesic.mathdoc.fr/item/AL_2011_50_5_a0/