The entropy of graded algebras (Q1969117)

Let \(A=\bigoplus_{n\geq 0}A_n\) be a graded algebra over a field. The authors define the entropy of \(A\) by \(H(A)=\limsup_{n\to\infty}(\dim A_n)^{1/n}\). This is related to the notion of entropy in symbolic dynamics. The authors examine the entropy of quotients and subalgebras of free associative algebras and free Lie algebras and the behaviour of the entropy function under free products. They obtain also some characterizations of free algebras in terms of entropy. Theorem 3.1 of the article states that if \(I\) is a nonzero ideal in a free associative algebra \(A\) of finite rank then \(H(A/I)<H(A)\). The analogous fact is true for free Lie algebras (Corollary 3.1). In particular this means that free of finite rank associative or Lie algebras are characterized by their entropy (Corollary 3.2). As a consequence of Corollary 3.1, the authors obtain a result on the torsion-free ranks of the lower central factors of groups (Theorem 3.3). The article contains a number of other results concerning the entropy of quotients of free associative or Lie algebras (Theorems 3.2 and 3.4, Corollaries 3.3 and 3.4). In particular, Theorem 3.4 and Corollary 3.4 deal with the entropy of associative or Lie algebras given by a presentation. According to Theorem 4.1 of the article, if \(L\) is a free Lie algebra of finite rank and \(K\) is a finitely generated proper subalgebra of \(L\) then \(H(K)<H(L)\). This is not the case for associative algebras. But in Theorem 4.2 the authors characterize free finitely generated subalgebras of free associative algebras which have maximal entropy. Furthermore, the authors prove that if \(A\) and \(B\) are associative algebras generated in degree 1 then \(H(A\otimes B)\geq H(A)+H(B)\) (where \(A\otimes B\) is the free product of \(A\) and \(B\)) and equality holds if and only if both \(A\) and \(B\) are free. This gives another characterization of free algebras in terms of entropy: if \(A\) is a graded associative algebra generated in degree 1 then \(A\) is free if and only if \(H(A\otimes A)=2H(A)\) (Corollary 5.1).