Abstract:
We define the entropy of a graded algebra $A =
\bigoplus_n A_n$ by $\limsup_{n\rightarrow\infty}\root\uproot3 n\of {\dim
A_n}$. This is related to the notion of entropy in symbolic
dynamics, and could serve as a natural dimension concept for
algebras with exponential growth. We study the entropy of
quotients and subalgebras of free associative algebras and free
Lie algebras. We also study the behaviour of the entropy function
under free products, and obtain several characterisations of free
algebras in terms of entropy.
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